Introduction to the Theory of Random Processes

INTRODUCTION

TO THE THEORY OF RANDOM PROCESSES I. I. GIKHMAN A. V. SKOROKHOD Kieve State University

TRANSLATED BY SCRIPTA TECHNICA, INC.

W. B. SAUNDERS COMPANY Philadelphia

London

Toronto

W. B. Saunders Company: West Washington Square Philadelphia, Pa. 19105 12 Dyott Street London, W.C.1

1835 Yonge Street Toronto 7, Ontario

Originally published in 1965 as Vvedenie v teorige slychainich processov by the Nauka Press, Moscow

Introduction to the Theory of Random Processes © 1969 by W. B. Saunders Company. Copyright under the International Copyright Union. All rights reserved. This book is protected by copyright. No part of it may be duplicated or reproduced in any manner without written permission from the publisher. Made in the United States of America. Press of W. B. Saunders Company. Library of Congress catalog card number 68-18281.

Preface to the English Translation

The present work of Gikhman and Skorokhod is the first publication since the now classic "Stochastic Processes" by J. Doob to survey in a rigorous way the more modern results in the theory of stochastic processes and to link them to earlier developments in the subject. Not only are the standard classes of stochastic processes discussed, but much of the authors' original work on limit theorems in stochastic processes is also included. A significant feature of the book is a unique and beautiful treatment of processes with independent increments, which assumes no prior knowledge of the theory of infinitely divisible laws.

The book is appropriate for students who have a sound background in probability from a measure-theoretic viewpoint and will, undoubtedly, be welcome as a graduate text. For reference purposes the authors have included a comprehensive discussion of measure theory and of the basic ideas of probability. The authors take great care to state the topological assumptions underlying each theorem, although occasionally a result may be stated in slightly greater generality than seems warranted by the proof. The book contains a wealth of results, ideas, and techniques, the deepest appreciation of which demands a most careful reading. Certainly, this is not a book for the indolent. The English translation was reviewed by H. Dym, W. Rosenkrantz, S. Sawyer, D. Stroock, S. R. S. Varadhan, and myself. Although

it was our policy not to make major revisions of the manuscript, we corrected many small inadvertent errors. WARREN M. HIRSCH Courant Institute of Mathematical Sciences New York University

III

Introduction

The current literature on the theory of random processes quite extensive.

is

In addition to textbooks and monographs especially

devoted to this theory and its various divisions, there are many technical books, for the most part dealing with automation and radio electronics, in which considerable space is given to the theory of random processes.

From the point of view of instruction this literature can be divided into two groups, the first consisting of serious and lengthy monographs whose difficulty hinders their use as beginning textbooks,

and the second consisting of books that are either elementary or written for engineering students.

There are no books in the [Russian]

literature that are designed for rigorous exposition and are at the same time suitable for elementary instruction. The authors have therefore, decided to write the present book, based on material they have expounded in a number of courses at the University of Kiev. The first five chapters of the book are devoted to general questions in the theory of random processes (including measure theory and axiomatization of probability theory);

Chapters 6 through 9 are devoted to more specialized questions (processes with independent increments, Markov processes, and limit theorems for random processes). The book is designed for persons who have had a general course in probability theory and are ready to begin the study of the theory of random processes. The authors hope that it will prove to be useful for students in the universities and also for specialists, other than mathematicians, who wish to familiarize themselves with the fundamental methods and results of the theory in a rigorous though not the most general and exhaustive approach.

The authors have not undertaken to treat all branches of the theory. Certain questions and methods that are well covered in

current [Russian] literature are omitted. (These include the semigroup theory of Markov processes, ergodic properties of Markov processes, martingales, and generalized random processes.) On the V

INTRODUCTION

VI

other hand, questions that, up to the present, have not been included in books on the theory of random processes (such as limit theorems

for random processes) but that play an important role in contemporary theory, are considered. The theory of random processes has recently developed into a separate branch of probability theory. Because the theory of random processes is still so closely related to other divisions of probability theory, the boundaries between this theory and its divisions are often difficult to determine precisely. For example, the theory of random processes is related to the theory of summation of independent

random variables by the division of probability theory that studies processes with independent increments, and to mathematical statistics

by statistical problems in the theory of random processes.

Let us characterize the problems in the theory of random processes that, from our point of view, may be considered basic. 1. The first problem in the theory of random processes is the construction of a mathematical model that allows a rigorous (formal) definition of a random process, and the investigation of the general properties of that model. 2. A second problem is the classification of random processes. Obviously every classification is arbitrary to some extent. Therefore we need to begin with specific principles that at least indicate the direction the classification will take. The existing classification

in the theory of random processes separates from the entire set of random processes certain classes that admit a more or less constructive description. Every class is characterized by the property that only a finite number of functional characteristics need to be specified

in order to single out from the entire class an individual random process.

Sometimes, we consider classes of processes that admit a uniform solution to a specified set of problems. In considering such classes,

we are usually not interested in the difference between random processes, if the characteristics necessary for the solution of these

problems coincide for them. We might mention the following broad classes of processes: (1) processes with independent increments, (2) Markov processes, (3) Gaussian processes, (4) processes that are stationary in the narrow sense, and (5) processes that are stationary in the broad sense. (We might include in this last group processes with stationary increments.)

The third problem, closely associated with the preceding one, consists in finding, for various classes of random processes, an 3.

INTRODUCTION

V11

analytical apparatus that will enable us to calculate the probabilistic characteristics of random processes. Such an apparatus has been constructed for the simplest probabilistic characteristics, and it uses,

as a rule, either the theory of differential equations (ordinary and partial) and integrodifferential equations (in the case of Markov processes and processes with independent increments) or the theory of integral equations with symmetric kernel (in the case of Gaussian processes) or Fourier transformations and the theory of functions of a complex variable (for processes with independent increments and for stationary processes). 4. We need to single out a class of problems that has played an important role in the development of certain branches of the theory of random processes and that is of great practical significance.

In its general form, the problem consists in the best determination

of the value of some functional of a process from the values of other functionals of the same process. An example of such a problem is the problem of prediction: - From observation of a process

over a certain interval of time, determine the value of the process at some instant of time outside that interval. Under certain restrictions, prediction problems have been solved for processes that are stationary in the broad sense (see Chapter V). 5. An important class of problems in the theory of random processes is the study of various transformations of random processes.

These transformations are used to study complicated processes by reducing them to simpler ones. We might include with the study of transformations of random processes the theory of differential and integral equations involving random processes.

This class of problems also includes limit theorems for random processes since the operation of taking the limit is a

sort of transformation.

At the present time the principal fields of application of the theory of random processes are electronics (which deals primarily

with processes that are stationary in the broad sense and with Gaussian processes) and cybernetics (which deals with processes that

are stationary in the narrow sense and with Markov processes). In mathematical economics and mathematical biology we use Markov processes of a different sort. In the molecular theory of

gases we use the process of Brownian motion; in the theory of showers of cosmic particles we apply Markov processes and processes with independent increments.

In general, the methods of the theory of random processes are finding ever new fields of application, and today every one of the

INTRODUCTION

natural sciences has

felt the influence of this theory, at least to

some degree.

Let us characterize briefly the features of the contents of the The first chapter is devoted to random processes in the broad sense. This is the name we have given to the portion of the theory of random processes that deals only with distributions of finite sets of values of a random process. This portion is very close to elementary probability theory, involves no complicated mathematical concepts, and is sufficient for many applications. For a more profound study of the theory of random processes, a more highly developed theory of measure and integration is necessary. Therefore, following Chapter I we expound all the necessary information in this field (Chapter II), and on the basis of this information we construct an axiomatization of probability theory (Chapter III). We also consider the general questions in the theory of random functions and, after that, specific classes of random processes and special questions in the theory. Among random processes, extensive treatment is given to processes with independent increments (to which one chapter is devoted) and Markov processes (to which two chapters are devoted). Stationary processes are considered to some extent in Chapter I and again in Chapter V, which is devoted to linear transformations of random present book.

Chapter V also takes up the problem of linear prediction. An entire chapter is devoted to limit theorems for random processes. In this chapter basic attention is given to processes with independent increments and Markov processes. processes.

Most of the constructions are made for the case in which a random process assumes values belonging to a finite-dimensional Euclidean space. In a few cases we consider complex-valued onedimensional and multidimensional processes and also processes with values belonging to a complete metric space. We assume that the reader is familiar with the basic concepts of linear algebra, which is particularly important for the study of Gaussian processes, and the theory of Hilbert spaces, which is used in the study of linear transformations of random processes. The reader should also have some familiarity with functional analysis (complete metric spaces, compact spaces, etc.).

We have not attempted to give a complete bibliography of works on the theory of random processes. In addition to books cited in the text, the bibliography includes only the basic books on the theory of random processes and probability theory that exist in

INTRODUCTION

Ix

Russian, as well as articles in which the fundamental results in this field first appeared. The book is divided into chapters and the chapters into sections. The basic formulas and also the theorems, lemmas, and definitions are numbered afresh in each section. A reference to a theorem or

formula in the same section is indicated only by the number of the theorem or formula. If a reference is made to a theorem or formula in another section of the same chapter, the section number is added. If the reference is made to another chapter, the chapter number is added. The authors express their gratitude to colleagues and students in the Department of Probability Theory and Mathematical Statistics

of Kiev State University for the help they have given in the preparation of this book. Kiev

October 21, 1963

THE AUTHORS.

Contents

1

RANDOM PROCESSES IN THE BROAD SENSE ...................

....................................... ........ ....................... ...............

1

1. Definitions 2. Correlation Functions (Covariance Functions) 3. Gaussian Random Functions 4. Oscillations with Random Parameters 5. The Spectral Representations of the Correlation Function of a Stationary Process and of the Structural Function of a Process with Stationary Increments .......

22

MEASURE THEORY ..........................................

40

1

5 13

27

2

......................................... 2. Measurable Functions ............................. 3. Convergence in Measure ........................... 4. Integrals ......................................... 1. Measure

5. Interchanging Limits and Integrations. Lp Spaces 6. Absolute Continuity of Measures. Mappings 7. Extension of Measures 8. The Product of Two Measures

......

.......... ............................. ......................

41

50 56 61

68 76 80 93

3 AXIOMATIZATION OF PROBABILITY THEORY ................... 100 1. Probability Spaces ................................. 100

2. Construction of Probability Spaces

.................. 104

3. Independence ..................................... 111

4. Series of Independent Random Variables

.............

1 18 5. Ergodic Theorems ................................ 123

6. Conditional Probabilities and Conditional Mathematical Expectations

................................... 134 xi

Xii

CONTENTS

4

RANDOM FUNCTIONS ...................................... 144 1. Definition of a Random Function ................... 144 2. Separable Random Functions ....................... 150 3. Measurable Random Functions ..................... 156 4. Conditions for Nonexistence of Discontinuities of the

Second Kind ................................... 159

5. Continuous Random Functions ..................... 169 5

LINEAR TRANSFORMATIONS OF RANDOM PROCESSES .......... 174

1. Hilbert Spaces .................................... 174

2. Hilbert Random Functions ......................... 181

3. Stochastic Measures and Integrals ................... 190 4. Integral Representations of Random Functions 5. Linear Transformations 6. Physically Realizable Filters

........ 200

........................... 207 ........................ 216

7. Prediction and Filtering of Stationary Processes ........ 226 8. General Theorems on the Prediction of Stationary Processes ....................................... 241

6 PROCESSES WITH INDEPENDENT INCREMENTS ................. 255

1. Measures Constructed from the jumps of a Process ..... 255 2. Continuous Components of a Process with Independent Increments ..................................... 264

3. Representation of Stochastically Continuous Processes

with Independent Increments .................... 270 4. Properties of the Sample Functions of a Stochastically Continuous Process with Independent Increments ... 274 5. Processes of Brownian Motion ...................... 282 6. On the Growth of Homogeneous Processes with Independent Increments ......................... 288 7 JUMP MARKOV PROCESSES ................................. 297

1. Transition Probabilities ........................... 298 2. Homogeneous Processes with Countably Many States ... 302

xiii

CONTENTS

3. Jump Processes ................................... 311 4. Examples ........................................ 319 5. Branching Processes ............................... 327

6. The General Definition of a Markov Process .......... 343 7. The Basic Properties of Jump Processes .............. 347 8 DIFFUSION PROCESSES ..................................... 370

1. Diffusion Processes in the Broad Sense ................ 372

2. Ito's Stochastic Integral ............................ 378 3. Existence and Uniqueness of Solutions of Stochastic

Differential Equations ........................... 391

4. Differentiability of Solutions of Stochastic Equations

with Respect to Initial Conditions ................. 403 5. The Method of Differential Equations ............... 412 6. One-Dimensional Diffusion Processes with Absorption .. 420

9 LIMIT THEOREMS FOR RANDOM PROCESSES ................... 438

1. Weak Convergence of Distributions in a Metric Space .. 440

2. Limit Theorems for Continuous Processes ............ 448 3. Convergence of Sequences of Sums of Independent Random Variables to Processes of Brownian Motion .. 452 4. Convergence of a Sequence of Markov Chains to a

Diffusion Process ................................ 459

5. The Space of Functions without Discontinuities of the Second Kind ................................... 469

6. Convergence of a Sequence of Sums of Identically Distributed Independent Random Variables to a Homogeneous Process with Independent Increments . 478

7. Limit theorems for Functionals of Integral Form ....... 484 8. Application of Limit Theorems to Statistical Criteria .. 490 BIBLIOGRAPHIC NOTES ..................................... 497 BIBLIOGRAPHY ............................................. 503

INDEX OF SYMBOLS ........................................ 511

INDEX .................................................... 513

I RANDOM PROCESSES IN THE BROAD SENSE

1.

DEFINITIONS

The course of a random process, like that of a deterministic process, is described by some function (O) (which may assume real, complex,

or vector values), where 0 assumes values in a reference set O. As B varies, e(O) describes the evolution of the process. (Of course, the way in which the process evolves is random, and each of the functions describes only one of the possible ways in which the process may are called sample functions of the random develop). These functions process. For each fixed 0, the quantity E(0) is random. To be able to apply

mathematical methods to the questions that we are studying, it is natural to assume that e(O) is a random variable (possibly vectorvalued) in the probabilistic sense. Consequently, by a random process we mean a family of random variables (O) depending on a parameter B that assumes values in some set O. If the set O is arbitrary, then instead of the term "random process,"

it is more convenient to use the term "random function" and to reserve the name "random process" for those cases in which the parameter 0 is interpreted as time. When the argument of a random function is a spatial variable, this function is also called a random field. This definition of a random process, or a random function as we have just agreed to call it, needs to be made more precise. For the sake of simplicity we shall speak of a random function that assumes real values. First of all we need to make clear just what is meant by "a family of random variables depending on a parameter 0." We recall that in accordance with the principles of probability theory, a finite sequence of random variables EE

e1, S4,

,n

is completely characterized by the joint distribution function 1

RANDOM PROCESSES IN THE BROAD SENSE

2

F(xi, x2, ... , xn) = P{S 1 < xi, e2 < x2, ... , $n <

'fin}

When we turn to a probabilistic description of a random function,

we must consider the question: How should we describe mutual relationships among infinitely many random variables-the values of a random function? The simplest procedure is to say that the random function E(O) is given if all possible probabilistic relations among each finite set of values of the random function e(oi), E(02), ..., (O ), 02 e O, i 1, 2, ., n; n = 1, 2, ..., (1 ) are defined, that is, if the corresponding joint distribution functions are

given. From this point of view the random function e(o), for 0 e O, is determined by the family of distributions FO1,O 2,....

On(xi, x2, ..., xn),

Oi G O, i = 1, 2, ..., n; n - 1, 2, ..., (2 )

and each distribution function , xn) is interpreted .10n(x2, x,, as the joint distribution function of the sequence of random variables (1).

Of course for such an interpretation to be possible, the family of distributions (2) cannot be completely arbitrary. It must satisfy the following obvious conditions, which we shall call the compatibility conditions for the family of distributions (2): F01,02,. ., On,

0n+1,...,0n+P(x1' x2, .. ., xn, +00 , ..., + 00)

= FO1, 021 . FO1 02,

x2, ..., xn) ,

(3)

10n(x1, x2, ... , xn) = F0a.l,0,.2,..., 0a.n(x21 , x22 , ... , x2n)

,

(4)

where i1, i2, . . ., in is an arbitrary permutation of the indices 1, 2,. , n. The necessity of these conditions arises from the following relations: . , +00 ) P{e(o1) < x1, x(02) < x2, . . ., E(Bn) < xn,

. F0,,02, ,On, 01+11...,0n+P(xl, x2, . . . , xn, + 00, .

, e(on+P) < ca' } e(on+1) < -, = P{e(o1) < x1, e(02) < X2, ..., e(Bn) < xn}

F01, 02,...,On(xl, x2, ..., xn) ,

F(x1, x2, . . . , X,,)

W(B1) < x1, W(02) < x2, ..., e(on) < xn} = P{e(otil) < x21, $(0") < X22, ..., $(B2n) < xtin} F0q,1,022,..., 0.(X, x22, .. ., xtin)

.

Just as we can identify a single random variable with its distribution functions, so what has been said above leads us to:

1.

3

DEFINITIONS

Definition 1. A family of distributions (2) satisfying the compatibility conditions (3) and (4) is said to be a random function (O) with

real values, defined on the set O (that is, defined for 0 e O). The functions belonging to the family F01,...,0 (x1, . . . , finite-dimensional distributions of the random function.

are called

This definition of a random function is attractive because of its simplicity, and it is sufficient when we are interested in the values of the random function on a finite set of values of the argument 0. On the other hand, a serious defect in this definition is that it does not

enable us to look at the random function in its entirety, that is, to look at all its values simultaneously. In addition, in many experiments the sample function being investigated is defined by means of a graph of some curve. Our definition of a random function not only fails to enable us to construct the graph of this function but also fails to enable us to ask questions regarding such functional properties-of the functions (0) as their continuity, differentiability, and so forth. Similarly, we cannot immediately ask the question of the probability of

the event that the inequality a < (0) < b, where a < b, will be satisfied for all 0 e O. We can obtain more satisfactory definitions of a random function if we use the axiomatic approach to probability theory. Every probabilistic scheme describes the results of some experiment having random outcomes. If the result of an experiment is described by a number or a finite sequence of numbers, we say that we are observing

a random variable or a random vector. On the other hand, if the result of the experiment is described by some function, we say that we are observing a random function. Thus a random function is defined by an arbitrary probabilistic scheme describing experiments whose results are random functions. A more precise analysis of this definition will be given in Chapter IV. Let us agree to call the definition of a random function that we shall use in the present section the definition of a random function in the broad sense. Up to now, we have been speaking of a single random function.

In solving many problems, we need to deal with several distinct random functions. If we are to be able to carry out mathematical operations on them, it is not sufficient that we define each of these functions separately. How can we define a sequence of random , e.(0), each defined on the single set O? Instead of speaking about a sequence of functions, we find that it is simpler to speak of a single vector-valued function r(0), whose comfunctions E1(B), e2(0),

ponents are random functions E1(O), e2(0),

, em(B). Then we can use

the preceding definition almost without change. The role of the

4

RANDOM PROCESSES IN THE BROAD SENSE

distribution of the sequence of random variables (1) is played by the joint distribution function of the sequence of vectors C(O), VBz), .., CO.), that is, by the function of nm variables F01.0......

x12,

x,bm)

= P{01(01) < x11, e2(01) < x12, . . ., MOM) Xmn}

In what follows, the set O will usually be a set of real numbers,

and the variable 0 will be interpreted as the time t. In this case we shall let Z denote the set O, and we shall understand it to mean a finite or infinite interval (closed, open, or semi-open). We shall also consider cases in which Z is either the set of all nonnegative integers or the set of all real integers. In this case we have a sequence of random variables (vectors) {r(k)} (for k = 0, 1, 2,

or k = 0, ± 1, ±2, ... ). We shall call such a process a random process with discrete time or a random sequence. Processes with discrete time play an important role in the general theory of random processes. There are many probabilistic problems in which time is an essentially discrete quantity. Also, the study of processes with discrete time in most cases makes use of simpler tools. In addition there are a number of cases

in which processes with continuous time can be approximated by processes with discrete time. Another important class of random processes is the class of stationary processes. These are processes whose probabilistic characteristics do not change with displacement of time. More precisely, we have:

Definition 2. A random process r(t) defined on Z is said to be , tn, such that t; + t e `. stationary if for arbitrary n, t and t1, t2, (for i = 1, 2, , n), the joint distribution function of the sequence of random vectors r(t1 + t), C(t, + t), , r(t + t) is independent of t.

As a rule, stationary processes are considered on an infinite (or semi-infinite) interval of the real axis (or the set of real integers). A, useful modification in the concept of stationarity is the concept of a process with stationary increments.

Definition 3. A process r(t), for t e Z, is called a process with stationary increments if the joint distribution of the differences, S (t1 + t\),S (t3 + t) ., CO. S+ t) ,5(tn-1 + t)t)

y (t8 + t)

..

S (t2 + t), ,

is independent of t for arbitrary n, t and for t1 t2, ... , tn such that

t;+teZ (for i= 1,2, ,n). REMARK 1.

The definition of a stationary process is equivalent

2.

COVARIANCE FUNCTIONS

5

to the following: For an arbitrary bounded continuous function f(z1, z2,

, zn) of n vector-valued variables z1, z2,

, zn,

the mathe-

matical expectation M.f['(t1 + t), Vt2 + t), ... , r(t,, + t)]

is independent of t for arbitrary n, t;, and t such that t; + t e Z. Stationary processes have the following property: For every continuous function f(zl, , zn), the random process REMARK 2.

72(t) = f[r(tl + t),'(t2 + t), ..., Vtn + t)] is also stationary. The proof follows from Remark 1.

Definition 4. A sequence of random functions e,(0), where , is said to converge weakly to o(0), B e O, if B e O and n = 1, 2, the finite dimensional distribution functions of the en(B) converge to

those of e0(0). 2.

COVARIANCE FUNCTIONS

The family of joint distributions (2) of Section 1 gives a comprehensive description of a random function in the broad sense. However in many cases we need a more concise description of distributions, one that reflects other important properties of a random function. Furthermore, the solution of many probabilistic problems depends on only a small number of parameters characterizing distributions that come up in the problem. The most important numerical characteristics of distributions are their moments. In the theory of random functions, the role of moments of distributions is played by the moment functions. Definition 1. The moment functions, mi1,;2,...,(0, 02,

, B8), of

a random function c(0), where 0 e O, are defined as the functions mil, ...,;8(O1, B2, ..., 0) = M([e(0)]'1[E(02)]12 ... [ (Bs)]'8),

jk>O

(k=1,2,...,s)

(1)

if the mathematical expectation on the right is meaningful for all B; e O where i = 1, , s. The quantity q = j1 + j2 + ... +j, is called the order of the moment function. Definition - 2. A random function e(6), where B e O, is said to belong to the class L,,(O) (and we write E(0) e LP(O)) if m 1 $(0) IP < for every B G O.

Theorem 1. If $(0) e LP(O), the moment functions of order q are

6

RANDOM PROCESSES IN THE BROAD SENSE

finite for all q < p. From the inequality between the geometric and arithme-

Proof.

tic means (cf. Hardy, Littlewood, and Polya, Section 2.5) k=1

k=1

lPk-1 8

Pk?0,

ak>O,

11 akk < L Pkak

k=1

it follows that 1

Wk) I ik (1/q

k=1

JJJ

k = (E \k=1 q

I e(ok) I) / ,

q = Lk=1.l k

Using Holder's inequality (cf. Hardy, Littlewood, and Polya, Section

2.8), we obtain s

I e(ok) I L/\ qk

g'J 1g'

lk

(Liq

where 1 /q' + 1 /q = 1.

1/q

s

I

Ie(8k) IqJ 1

Thus

77e

11 I e(Bk) I9k

k=1

< Sq-'

s

k=1

I e(Bk) Iq

from which the assertion follows. If we know the characteristic function of the joint distribution of the variables e(01), , (B,), i E ake(ok) k=1

then the moment functions with integral indices can be found by differentiating: aq'11J'

= igM([e(B)]91 ... [S (O8)]j8) 21=22=... =q

(2) = igm;1...,,(B1, ..., 0.) Here, differentiation under the mathematical-expectation sign is permissible for all q = >k=1 1k < p, if E(O) e L,,(®). A precise converse of this assertion holds for moment functions with even indices. Let Da denote the operation of taking the symmetric finite difference with respect to the variable denote( its jth iterate: A

T

(81,

2s

and let li

. .. , B X1, . . . , X,)

_ *(O1, ...,O,X1, h;, '(B1, ... , B A1, ... , X; - h;, ... , ,,) ,

-

(B1, . . . , B A,1, AS (` T r=0

. . . , X,)

(-1)rC;'/ (B1, ..., B a1,

, a; + (j - 2r)h;, ..., a,) .

2.

COVARIANCE FUNCTIONS

7

(C; = j!l{r!(j - r)!} is the binomial coefficient.) Then 01'i10272 .. . ps.9a,iY (01, ... , Ba, %1 2 Jk

7s

M 11 E (- 1)rCr ei2(9k-r)hkF(Bk) 29 k k=1 r=0

M JJ (eihkf(Bk) - 2 tihkE(Bk))29k k=1 s

- II hk k(2i)2k= k=1

kM I (sin hke(Bk) )2ik [e(ok)]2jk /

1

,

hke(Bk)

k=1

or

02jk

_ \-

[7 (sln hk(Bk)

k=1 2hkjk

k=1

(Bk)]2jk

From this, by using Fatou's lemma (see Chapter II, Section 5), we get

E .k

(-

1)k=1

lim

hk-'0

k=1.2, , s

r

J1

a

k=1

M II [e(ok)]2jk k=1

11 (2hk)2jk 2=0

k=1

The expression on the left-hand side of this inequality coincides with the derivative E 2ak

ak-1

aX2j1' .. 1

aX23's 8

at the point X = 0 if 'r is 2 E81=1 jk times differentiable. Thus we have:

Theorem 2. I f the characteristic function #(0,,

, oa X,

, X.)

is p times differentiable, where p is an even integer, there exist moment functions of order q 5 p and they can be calculated from formula (2).

In addition to moment functions, we often consider central moment functions ...,;a (/Or, .. ., 0a)

M([e(01) - m1(01)l'1[V02) - m1(02)l'2 ... [ (Bs) - ml(0 )l's)

(3)

which are moment functions of a central random variable e1(0) = E(o) - m1(0) with mathematical expectation equal to 0 for arbitrary

oe0. Among the moment functions of special significance are the functions of the first two orders: m(0) = ml(0) = Me(o)

,

(4)

RANDOM PROCESSES IN THE BROAD SENSE

8 R(01, B2) = u11(01, 82)

= M([e(01) - m(01)][e(B2) - m(82)])

(5)

The function m(9) is called the mean value and R(01, 02) is called the

For 0, = 02 = 0, the covariance function gives the variance 62(9) of the random variable c(9); R(9, 0) = 0'2(9). For a stationary process (O = Z), it is obvious that m(t) = m = const , (6) covariance function.

R(t1, t2) = R(t1 - t2, 0) = R(t1 - t2)

,

(7)

that is, the covariance function depends only on the difference in the arguments. The function R(t) = R(t1 + t, t1) is also called a covariance function of a stationary process. Of course if equations (6)

and (7) are satisfied for some process, it does not follow that the process is stationary. Still, we often encounter problems whose solution depends only

on the values of the first two moments of a random function e(t). For such problems, the condition that the process be stationary reduces to conditions (6) and (7). Therefore it is natural to consider the following important class of processes (introduced by A. Ya. Khinchin).

Definition 3. A random process e(t) is said to be stationary in the broad sense if Mee(t) < oo and Me(t) = m = const; M([e(t1) - m][c(t2) - m]) = R(t1 - t2)

.

We note that for a stationary process in the broad sense the variance 62 of a random variable e(t) is independent of t: U2 = R(O) = M[e(t) - m]2

The quantity r(01, 92) =

R(01, 92) 6(01)0'(02)

=

R(01, 92)

1/R(01,

01)R(02,02)

is called the coefficient of correlation of the random variables e(91) and e(02). If e(91) and g(02) are independent, the coefficient of correlation is

0. The converse is not generally true. However in the important particular case in which the random variables e(01) and e(02) have a joint normal distribution, the variables e(01) and e(02) are independent if

the coefficient of correlation is 0 or, what amounts to the same thing, if the covariance function R(01, 02) is identically zero. In the general case, if two random variables e and 17 with finite secondorder moments satisfy the condition RE,, = M[(e - Me)(9 - My))] = 0,

2.

9

COVARIANCE FUNCTIONS

they are said to be uncorrelated.

Analogously, we may say that in those branches of the theory that deal only with first- and second-order moments, the concept of uncorrelatedness of random variables replaces the concept of independence of random variables. The coefficient of correlation of a pair of random variables is a measure of linear dependence between them; that is, the coefficient of correlation shows with what accuracy one of the random variables can be linearly expressed in terms of the other. Let us clarify this. For the measure of error aY) + b where a and b are real 8 of the approximate equation numbers, we take the quantity 8 = 1/M[e - (ay] + b)]2. Then 82

= M[(e - Me) - a(y) - My)) + (MS - aM)l -

b)]2

,

so that 82

= De + a2DlJ + (Me - aMY) - b)2 - 2aRE,n

=Ca6n- )2+01(1 -r2n)+(Me-aM)7-b)2 where Q2 = DC and 62) = D)7 are the variances of the variables respectively. This expression attains its minimum when

a=

Re* 01

=

aF 6n

r

and

QFrEmm,, 6=me-amn=me-a*

and this minimum is equal to min 32 = 62(1 - r2n). Thus the greater in absolute value the coefficient of correlation between two random

variables, the greater the accuracy with which one of them can be represented as a linear function of the other. We often consider complex-valued random functions C(0), which

can be represented in the form C(O) = C(B) + h'(B); we can also regard them as two-dimensional vector-valued random functions. For a complex-valued function, the relation C(O) e L4(0) means that M I C(0) 12 < 00 , where 0 e O, that is, e(0) e L2(®) and 72(0) e L2(®).

The covariance function of a complex random function is defined by the equation R(01, B2) = M([V01) - MV01)][C(02) - MV02)], where

the vinculum denotes the complex conjugate. Let us note certain properties of covariance functions: 1. R(0, 0) > 0 with equality holding if and only if C(B) is constant with probability 1; 2.

R(01, 02) = R(02, 01) ;

3.

1 R(01, 02) 12 < R(01, 0)R(02, 02) ;

4.

For every n, 0l, 02,

, 0n and complex numbers X1, X2, .

(8) (9) , X.1

RANDOM PROCESSES IN THE BROAD SENSE

10

j,k=1

R(O j, Bk)X ja,k > 0 .

(10)

The first two assertions are obvious. The third is obtained as a consequence of the Cauchy-Schwarz inequality (MIe77 I)2

MIeI2M177 12.

To prove 4, it suffices to note that

+

LJ R(6j,

j,k=1

y

y

n

,k = M E b(BI)b(0k)XIXk = M j,k=1

j=1

y b(B

0

We note that properties 1, 2, and 3 follow from property 4. A function R(01, 02) that satisfies property 4 is called the nonnegative-definite kernel on O. A complex-valued process C(t) is said to be stationary in the broad sense if m(t) = MC(t) = const, R(tl, t2) = R(t1 - t2). For processes that are stationary in the broad sense, pro-

perties 1 to 4 of a correlation function take the forms It. R(0) > 0 , 2'.

R(t) = R(-t)

(11)

Y.

I R(t)

R(O)

(12)

4'.

j,k=1

R(tj - tk)Xj3,k > 0 .

(13)

Let C1(0) and C2(O) denote two random functions belonging to To characterize the degree of linear dependence between two such functions, we introduce the joint covariance function. L2(O).

Definition 4. The joint covariance function of two random functions C1(0) and C2(O) in L2(O) is defined as the quantity Rs1;2(B1, 02) = M(I 1(01) - Mb1(01)][U02) - Mb2(B2)])

Suppose that O is an interval of the real axis Z. Then two processes C1(t) and C2(t) are said to be stationarily connected if each is stationary in the broad sense and 02) = R(B1 - 02). Suppose that we are given a sequence of complex-valued random functions , C,(0), C;(0) e L2(O), i = 1, 2, , r. Let us agree to C1(0), C2(0), treat this sequence as a single r-dimensional complex-valued random function r(O) = {C1(0), C2(0), , C,(0)}, 0 e O. If E and 7) are two vectors E = (E1, e2, , er), 77 = (rll, X72 , / r), we shall let er2* de-

note the matrix

fi

e,171 e1'/ 2

el' r

ef ... J7') = (ifi) -

.er iT 1 & r'/ 2

a

r'/ r

2.

COVARIANCE FUNCTIONS

11

We set m(0) = MC(0) = {MC1(0), MUO), ... , MUB)} , R(01, 02) = (Rdj(01, 02)) = MU(0) - m(01)][S(02) - m(02)]*)

i, j = 1, ..., r

= (M{[S¢(01) - m{(B1)1[Sj(02) - mj(02)l}) ,

The function m(0) is an r-dimensional complex vector-valued function. It is called the mean value of the vector-valued random function r(0). The matrix R(O1, 02) is called the covariance matrix of r(O). If O = % and m(t) = in = const, R(t1, t2) = R(t, - t2), the process r(t) is said to be stationary in the broad sense.

Corresponding to properties 1 to 4 of covariance functions are the following properties of the covariance matrix of a random function: 1.

R(O, 0) is a nonnegative-definite matrix n

2

M

L Rjk(B, j,k=1 2.

xjbj(0) j=1

R(01, 02)* = R(02, 0l)

>0;

(14)

(15)

,

where the asterisk denotes the complex conjugate matrix; 3.

I Rjk(01, 02) I2 C Rjj(01, 01)Rkk(02, 02)

,

j, k = 1, ..., r ;

(16)

, B and a sequence of complex vectors

4. For arbitrary n, 01, A1, A2, ... , An,

(R(01, Ok)Ak, A1) > 0

(17)

j,k=1

This last condition is equivalent to: 4'. For an arbitrary sequence of matrices A1, , An, the matrix j,k=1 A;R(0,, Ok)Ak is nonnegative-definite. Properties 1 and 2 are obvious. To prove property 3, let us use the Cauchy-Schwarz inequality for the mathematical expectation: Rjk(01, 02)

12

= I M[(S1(B1) - mj(B1))(bk(02) - mk(B2))l I' :_!5 Rij(011 00-'4021

To prove property 4, let us set Ak = (aki, n

n

I (R(01, Ok)Ak, Aj) =

j,k=1

, akr) .

02)

Then

r

E R21 (0j, 0k)akgajp

i

j,k=1 p,q=1

=M

n

r

Ij=1Ep=1 (bp(O,) - mp(Bj))ajp

2

> 0

In conclusion let us look at some modifications in the preceding definitions.

Let us agree to call a process C(t), for t e Z a process belonging

RANDOM PROCESSES IN THE BROAD SENSE

12

to the class L2 or Lz(1) if for arbitrary t1, t2 a Z, M I r(t2) - r(t1) I2 < -, where r(t) I denotes the norm of the vector. For processes belong-

ing to /the class L2, we introduce the vector-valued function (t1)} = m(t1, t2) = {m1(t1, t2), m2(t1, t2), ... , m,.(t1, t2)} , (t2) - V01 which we call the mean value of the increment of the process, and we introduce the matrix D(t1, t2, t3,t4)

= (D;k(t1, t2, t3,

t4))

/

yy

17( 1 = (M{[CA(t2) - C5(t1) - m;(t1, t2)][yyk(t4) - b k(t3) - mk(t3, t4)]})

(j,k=1,...,r)

which we call the structural matrix of the process r(t). If the functions m(t1, t2) and D(t1, t2, t3, t4) are independent of the displacement of the arguments, that is, if m(t1 + h, t2 + h) = m(t1, t2)

D(tl + h, t2 + h, t3 + h, t4 + h) = D(t1, t2, t3, t4)

where t, e Z, t; + h e Z, i = 1 ,

, 4, then the r(t) is called a process with stationary increments in the broad sense.

For a process with stationary increments, m(h) = m(t, t + h) is an additive function of h. If we make the additional requirement that the function m(h) be continuous or bounded on some interval, it then follows that m(h) is linear, that is, m(h) = (m1h, m2h, , mrh). For real processes the structural function D(t1, t2, t3, t4) can be expressed in terms of the simpler functions D(t1, t2) = D(t1, t2, tl, t2), which is also called the structural function of the process. Indeed, D(t1, Q = M([[y(t4) - 03) - M(49 t4) y+ '(t3) y

C(t) - m(t1, 01[04) l- S (t3) - M(49 t4) + S (t3)

- yS (t1) - m(t1, t3)1 *)

= D(t3, t4) + 2D(t1, t3, ts, t4) + D(t1, t3)

,

so that D(t2, t3, t3, t4) = ^ [D(t1, t4) - D(t1, t3) - D(t3, t4)]

2

Furthermore, D(t1, t2, t3, t4) = D(t1, t2, t2, t4) - D(t1, t2, t2, t3)

The last two formulas together express the function D(t1, t2, t3, t4) terms of the D(t1, t2).

i

3.

GAUSSIAN RANDOM FUNCTIONS

13

GAUSSIAN RANDOM FUNCTIONS

3.

In many practical problems an important role is played by random functions for which the family of joint distributions defining a random function consists of Gaussian (normal) distributions. First we shall give the definition and basic properties of a multi-dimensional Gaussian distribution. is said to _ ( 2, Definition 1. A random vector

,)

have a Gaussian (normal) distribution if the characteristic function can

be written in the form 0 1' t2, ... , t.) = M exp {i(t,

)}

= exp (i(m, t) - 2 (At, t))

,

(1 )

are vectors and and t = (t1, t2, , where m - (ml, m2, , is a nonnegative-definite real symmetric A = (Xik) for i, k = 1 , , n, matrix. Here (a, ,@) denotes the scalar product of the vectors a and ,B, so that (m, t) = L mktk ,

(At, t) = L X7kt]tk j,k=1

k=1

The following theorem serves as a formal justification of the definition that we have just given. Theorem 1. For a function -,Jr(t) exp [i(m, t) - 1/2(At, t)] to be the characteristic function of an n-dimensional random vector , it is necessary and sufficient that the real matrix A be nonnegative-definite and symmetric. The rank of the matrix A is equal to the dimension of the subspace in which the distribution of the vector a can be concentrated.

Proof of the Necessity.

Suppose that the characteristic function

of a random vector is given by formula (1). If we differentiate it first with respect to t, and then with respect to tk and then set t = 0, we see that the distribution has finite moments (Theorem 2, Section 2) and that '/r(t)

a* atj 1=0 a21r

atjatk

=iMEj=imj

-Mejek = -mjmk - X$j .

(2)

(3)

1=0

It follows from these formulas that the matrix A is real, symmetric, and nonnegative-definite:

(At, t) = M(? (ej - mj)tj) = D(e, t) > 0 . \j_1 2

(4)

RANDOM PROCESSES IN THE BROAD SENSE

14

If the rank of the matrix A is equal to r(< n), then by making a suitable change of variables tj = Ek=1 ajkc we can reduce the quadratic form to principal axes: (At, t) _

k=1

XkTk =

k=1 j=1

e (S j - mj)ajkZvk]2

mj)ajk = 0 for k = r + 1, , n with probability These relations show that there exist with probability 1, n - r

Thus 1.

linearly independent relationships among the components of the vector E and hence that its distribution is concentrated in the r-dimensional , n. hyperplane defined by E %j (xj - mj)ajk = 0, k = r + 1, Proof of the Sufficiency. Let us suppose that A is a positivedefinite symmetric matrix. The function

*(t) = exp {i(m, t) - 2 (At, t)} is absolutely integrable and differentiable. Consequently we can apply Fourier's integral formula to it:

0) =

flx) ei(z, t)dx ,

f(x) = (2 )n

L'/r'(t)e ict, x'dt

(5)

These integrals are n-dimensional, and dx and dt denote n-dimensional elements of volume. Let C denote an orthogonal matrix that re-

duces A to diagonal form, so that C*AC = D, where D = (xiaik) for i, k = 1, , n and Xi > 0, and where C* is the adjoint of C (Here, we note that C is real and orthogonal and thus C* coincides both with the transpose and with the inverse of C (that is, C* = C' = C-).) Let us make a change of variables of integration by setting

t = C u or u = C*t, where u = (u1, u2,

, un) .

Since the element of volume is not changed under an orthogonal transformation, it follows that

f(x) _ (21 L exp { - i(x - m, Cu) - I (ACu,

Cu)}du

We have (ACu, Cu) = (C*ACu, u)

a,kuk, k=1

(x - m, CU) =

*(x - m), U) _

k=1

XkUk

where xk is the kth component of the vector x* = C*(x - m). fore,

There-

3.

GAUSSIAN RANDOM FUNCTIONS

AX)

(

(2270

1

111

X

)"`

rexp{

27r J

rte,-

-l

x

- -2 n=11 ,s

(1'k=2

m exp { - 2Xk Uk - 2 X kuk}duk e Xk*2/21k

1

lI

15

k=1 1/27rXk

=

(27r)-nI2

IJ Xk

1/2e-1/2(D-lx*,x*)

Ck=1

Furthermore, Ilk=1 Xk = 0, where A is the determinant of the matrix A, and (D-lx*, x*) _ (D-1C*(x - m), C*(x - m)) _ (CD-1C*(x - m), (x - m)) ((CDC*)-1(x - m), (x - m)) = A-1(x -m), (x - m)) , where A-1 is the inverse of the matrix A. Finally we obtain (x)f

1/ (27r) n 1

1/(27r)"`.

.

exp {

exp

2

-1 `

(A-1(x -

m),

(x -

m)}

0k9(X9 - m.i)(xk - mk)

0

2 A.k=1

'

(L )

1

where the Ak; are the cofactors of the elements of the matrix A. It follows from (6) that f(x) > 0 and it follows from (5) that

L f(x)dx=*(0)= 1

.

Thus the function f(x) can be regarded as an n-dimensional distribution density, and ar(t) is its characteristic function.

Turning to the general case, let us suppose that the matrix A is a matrix of rank r (where r < n) and that C is an orthogonal transformation that reduces A to diagonal form: C*AC = D where D,. is a diagonal matrix whose diagonal elements Xk are zero for k = r + 1 , , n and positive for k = 1 , 2, , r. Suppose that X; _ X; for j = 1, , r but that X; =s for j = r+ 1, ,n. Then, A, = CD,C* is a positive-definite matrix and

're(t) = exp {i(m, t) - Z (A,t, t)} is the characteristic function of some distribution. Ass 0, the function *,(t) converges uniformly to ar(t). Hence ar(t) is the characteristic function of some distribution. As shown above, this distribution is concentrated in an r-dimensional hyperplane, so that it has no density. Such a distribution is called an improper Gaussian

RANDOM PROCESSES IN THE BROAD SENSE

16

distribution.

Corollary 1. In the expression (1) for the characteristic function is the vector of the of a Gaussian distribution, m = (m1, m2, mathematical expectation and A is the covariance matrix:

,

m= /Ljk = M[(ej - mj)(ek - mk)] . This corollary follows immediately from formulas (2) and (3). Corollary 2. If the covariance matrix A of a Gaussian random ME,

vector x is nondegenerate, there exists an n-dimensional distribution den-

sity f(x) defined by formula (6). Corollary 3. The joint distribution of an arbitrary group of components of a Gaussian random vector is Gaussian. , ) has a Gaussian Theorem 2. If a random vector C _ ( 1, e2, _ (1, , fir) and distribution and if the random vectors m.) , E" - (cr+1, (for r < n) are uncorrelated, then the vectors C' and e" are independent.

Proof. The fact that f' and E" are uncorrelated implies that , n. Therefore , r, j = r + 1, MetiE j - Me;Me j = 0, i = 1,

(r(t) = exp {i(m', t') + i(m", t") -

1

E, Xjktjtk -

2 j,k=1

1

"+

LJ Xjktjtk

2 j,k=r+1

,

where (m1, m2, ... , my), m" = (mr+i, ... , m,z) ,

m'

t' _ (t1, ... ' tr)' t" - (tr+1' ..., tn) The preceding formula can be rewritten in the form (t) = M exp {i(t', C') + i(t", C")} = M exp i(t', C')M exp i(t", E")

where *'(t') and *"(t") are the characteristic functions of the vectors E' and i". This relation proves the independence of El and Ell. , n) denote an , h and k= 1, Let W = J a j k JJ (for j= 1, arbitrary rectangular matrix, and set rJ = WC; that is f

99 =

n+

l"/1, ..., /h}, )7j = Lk=1ajkek ,

j = 1, .. ., h .

The vector 7) is a linear transform of the vector C. Theorem 3. Linear transformations of random vectors map Gaussian distributions into Gaussian distributions. Proof. Let *,,(t,, the vector '2. Then

, t,) denote the characteristic function of

3.

GAUSSIAN RANDOM FUNCTIONS

17 h

9(t1, ... , th) = M exp )i E

2=1

tR»»1J

n (/

M exp i' (

h+

k=1 \j=1

\j jllll tjajk)ek1

= exp {i(t, Win) - 2 (1XAW't, t)}

,

(7)

that is, 72 has a Gaussian distribution with mathematical expectation Win and with variance-covariance matrix A, = %A%'. , r, ) denote a seTheorem 4. Let t(l) (where a = 1 , 2, quence of n-dimensional vectors having Gaussian distributions with parameters (m(a), A(a)). The sequence of distributions of the vectors era' converges weakly (converges in distribution) to some limiting distribution

if and only if MW ._._, m

A(a) ._., A .

,

(8)

Then the limiting distribution is also a Gaussian distribution with parameters m and A.

For a sequence of distributions of random vectors f(a) to con-

verge weakly to a limit, it is necessary and sufficient that the sequence of their characteristic functions *(a)(t) converge to a conwhere tinuous function. Let us consider the sequence {ln In *(a)(t) = i(m(a), t) - 1/2(A(a)t, t) in some neighborhood of the point t = 0. For this sequence to converge it is necessary and sufficient that conditions (8) be satisfied. If conditions (8) are satisfied, then ar(t) = exp {i(m, t) - 1/2(At, t)} for all t; that is, a limiting *(a'(t) distribution exists and is Gaussian.

Let us turn now to random functions. A real r-dimensional , E,(0)} is said to be Gaussian if for every n the joint distribution of all components of the random vectors random function f(O) = {e1(B),

(9)

E(el), x(02), ... , (Bn)

are Gaussian. The covariance matrix R of the joint distribution of a sequence of random vectors (9) is rn x rn and it can be partitioned into square

r x r cells as follows: 1R(01, B1) R(01, 02) ... R(01, 0n)

R = R(02, 61) R(02, 02) ... R(02, 0) J R(e n, 0 ) R(0 1

,

0 2)

... R(6 ,,,

0n )

where R(01, 02) is the covariance matrix of the function e(0). The matrix R is real and nonnegative-definite.

RANDOM PROCESSES IN THE BROAD SENSE

18

The converse is obvious. Specifically, for any real-vector-valued function m(0) and any real-vector-valued nonnegative-definite matrix function R(01, 02), where 0; e O (for i = 1, 2), there exists an rdimensional Gaussian random function (in the broad sense) for which m(0) is the mathematical-expectation vector and R(01, 02) is the covariance matrix.

The moments of a Gaussian real valued random function can be obtained from the decomposition of the characteristic function. Confining ourselves to the case of central moments, we set m(0) = 0. Then

W" ..., 0.1 t1, ..., t8) e(-112)(fh,t) = 1

- 1 (At, t) 2

i +

2

22

(At, t)2 - ...

+ (_ 1),.

(At, t)R + .. .

2 n!

where A = (R(0;, Bk)) for j, k = 1, , s. From this we obtain for an arbitrary moment function of odd order, p;1... j (01, , 0,) = 0 if Ek-1 jk = 2n + 1. For central moments of functions of even order, fc;1...;,(B1, ..., O) _

a2%

1

aril ... ats8 21ni

(At, t),

,

t-o

jk = 2n . k-1

(10)

For example, for fourth-order moment functions we have the following formulas: ,u4(B) = 3R2(0, 0), "31(01, 02) = 3R(01, 01)R(01, 02) , 1/21(01,

02,

03) = R(01, 01)R(02, 03) + 2R(01, 02)R(01, 03)

,

1/ "1111(81, 02, 03, 04) = R(01, 02)R(03, 04)

+ R(01, 03)R(02, 04) + R(01, 04)R(02, 03)

In the general case, p;1... ;8(01, ..., 0,) = ZUR(0, 0q) , (11) the structure of which can be described as follows: We write the points 0, ... , 0, in order, where Ok is repeated until it appears jk times. We partition this sequence into arbitrary pairs. Then we take the product on the right side of formula (11) over all pairs of

this partition, and we take the sum over all partitions (pairs that differ only by a permutation of the elements are considered as a The assertion follows immediately from formula (10). The fact that Gaussian random functions play an important role in practical problems can be explained in part as follows: Under single pair).

S.

GAUSSIAN RANDOM FUNCTIONS

19

quite broad conditions, the sum of a large number of independent small (in absolute value) random functions is apporoximately a Gaussian random function, regardless of the probabilistic nature of the individual terms. This so-called theorem on the normal correlation is a multi-dimensional generalization of the central limit theorem.

Here is one of its simpler formulations: Theorem S. Let {Yjn} denote a sequence of sums of random func. Suppose that the foltions 17,(0) = Ek=61 a, (0), 0 e O, n = 1, 2, lowing three conditions are satisfied: a. For fixed n, the random variables an,(01), an2(02), , a l l y i n d e p e n d e n t f o r arbitrary 01, 02, , 0mn, possess second order

maxk blk(0), 0 asn, -; moments, and Mank(O) = 0, Ma;,k(B) = b. The sequence of covariance functions Rn(O1, 02) = M[)7n(0j)1In(02)] converges as n --- 00 to some limit lim"_ Rn(01, 02) = R(01, 02); c. For every 0, the sums "(B) = Ek °'1 ank(0) satisfy Lindeberg's condition: For arbitrary positive z, m"

-B, L k=1 1

Izl>rB,

X dH%k(6, x) -4 0 ,

where nnk(B, x) is the distribution function of the random variable ank(O), and B; = Ek1 b;, k(0) = R,(0, 0). Then the sequence {i),(0)} converges weakly as n ---p 00 to a Gaussian random function with mathematical expectation zero and covariance function R(01, 02).

Proof of this theorem reduces to one of the variants of the one-dimensional central limit theorem. We recall the necessary formulation: , where Theorem 6. Suppose that n = E'_"1 Enk, for n = 1, 2, the random variables enk satisfy the following conditions: a. For fixed n, the random variables en1+ en29 1 $nmn are nondegenerate and mutually independent, Meenk = 0 and Me;,k = bnk; b. For arbitrary v > 0, Lindeberg's condition is satisfied: mn

Bn2 k=1

Izl>rBn

xtdHnk(x) -a 0

as

n -,

,

where Bn = Ek=1 b;,k, and the H,k(x) are distribution functions of the variables nk Then the sequence {C,,/B"} of the distributions of the variable Cn/B" converges as n ---p 00 to a Gaussian distribution with parameters 0 and 1 (cf. Gnedenko [1963], p. 306).

Consider the characteristic function it E t;pn(B;)

'n(B1, .. ., 0., tt1, ..., tte) = Me 9=1

RANDOM PROCESSES IN THE BROAD SENSE

20

which for t = 1 is the characteristic function of the joint distribution of the variables )7n(0,), , 7),(0J, and for fixed t1, , t8 is the characteristic function of the random variable Zj=1 Sn The quantity C. can be written in the form 'mn

s

bn = Zk=1Rnk yy

Rnk =

Here,

j=1

tjank(ej) ,

/

k = 1, ..., Ynn .

8

Mlank = 0 , (Ink = MNnk = t tjtPMLank(ej)ank(01)J 7

j,P=1

Bn = Y, bnk = k=1

ll

j,P=1

tjtPRn(ej, OP)

From this we see that max

j.P=1

k

tjtp max bnk(ej) max k

bnk(OP)

0

n --> 0 ,

as

k

and Bn BZ = Z;,P=1 t;tPR(ej, BP). If k = 0, then {C.} converges weakly to zero and 'rn(91, , t,) --. 1, which is a special , 08, t,

case of the assertion of the theorem. For B2 > 0 we verify the satisfaction of Lindeberg's conditions for the variables Rnk this case it is sufficient to show that k=1

MLgr

j=1

tjank(Bi)JI

j=1

tjank(Bj))a] --> 0

as

In

n .--* 00

for arbitrary z > 0, where gr(x) = 0 for I x I < z and gr(x) = 1 for I x I > z.

If tPcenk(BP) is the greatest in absolute value of the tjank(ej)

, s), then

(for j = 1, 2,

prtL tjank(0j)/11L tjank(0j) I 2 7= j=1

Drla(tpank(BP))s2(tPank(BP))2

1

Therefore we always have grl\j l tjank(0j) 1( L tjank(ej)) < SQ

/

j=1

j=1

grls(tjank(ej))(tjank(B,))Z

and In,

L M[ k=1

rl

j=1

tjank(0j)J(L tjank(0j)) ] j=1 m 1

j=1

lank.

k=1

Ixlgrlsltjl

x2dHnk(ej,x)-->0

as

n --, c

Thus the central limit theorem is applicable to the quantities By virtue of this theorem, -t2/2 j,P1j E t P t xce j' eP ) e-(a2t2)/2 = e

Here, if we set t = 1, we see that the sequence of characteristic

3. GAUSSIAN RANDOM FUNCTIONS

21

functions of the joint distributions of the quantities for , s, converges as n 00 to the characteristic function of j = 1, a Gaussian distribution. The continuity of the correspondence between the distributions and the characteristic functions implies the conclusion of the theorem. A vector process E(t), for t e [0, T], is called a process with independent increments if, for arbitrary t1, t2, , where 0
the random variables e(0), E(t) - e(0), e(t2) - E(t), , E(tn) are mutually independent. Chapter VI is devoted to the theory of these processes.

A Gaussian process with independent increments is called a Wiener process or a process of Brownian motion. If we observe through

a high-powered microscope a small particle suspended in a liquid, we see that the particle is in constant motion and that its path is an extremely complicated broken line with chaotically directed links. This phenomenon is explained by the collisions of molecules of the liquid with the particle. The particle is considerably larger than the molecules of the liquid, and in a single second it experiences an enormous number of such collisions. It is impossible to follow the result of each collision of the particle with the molecules. The visible motion of the particle is called Brownian motion. As a first approximation we may assume that the displacements of the particle under the influence of collisions with the molecules of the medium are independent of each other and thus we may consider Brownian motion as a process with independent increments. Since each individual displacement is small, we may assume that the central limit theorem in probability theory is applicable to their sum, and we may treat Brownian motion as a Gaussian process. Let us make some remarks regarding the structural function of a process of Brownian motion. If t, < t2 < t3 < t4, we have, from the independence of the increments f(t), the obvious equation for the structural function D(tl, t2, t3, t4\\): D(tl, t2, t3, t4) = 0 .

Let us define D(s, t) = D(s, t, s, t) and let us suppose that s < u < t. Then

D(s, t) = D(s, u) + D(u, t)

,

(12)

that is, the matrix D(s, t) is an additive function of the interval. If f(t) is a process with stationary increments, then D(s, t) = D(t - s) and the preceding formula can be written

RANDOM PROCESSES IN THE BROAD SENSE

22

D(sl)+D(s2)=D(sl+s2),

s;>0,

i = 1, 2.

(13)

If D(s) is continuous, it is easy to see that solutions of equation (13)

are of the form D(s) = Ds , where D is a nonnegative-definite matrix.

(14)

Formula (14) completely

describes the structural function of a process of Brownian motion with stationary increments. If the mathematical-expectation vector m(t) = f(0)) is continuous with respect to t, then m(t) = mt. In the one-dimensional case, Brownian motion with stationary increments is completely determined by the distribution of the quantity (0) and the parameters m and or, where a > 0. Here, MR(t) - M(0)] = mt

and D[e(t) - e(0)] = c2t. It was just this process that N. Wiener first studied in detail. 4.

OSCILLATIONS WITH RANDOM PARAMETERS

We now introduce some terminology that will be useful for a physical interpretation of random processes.

If e(t) denotes an elec-

tric current at an instant t and if we are interested in the energy dissipated by this current per unit of resistance, we have the followt2 ing natural definitions. The quantity e2(t)dt is called the energy cl of the random process e(t) in the course of an interval [t1, t2]. The intergral 2(t)dt, if it converges, is called the total energy of the process e(t) (for - oo < t < oo ). The limit

lim 1 f T

T--2T -T

e2(t)dt

is called the mean power of the random process. If the process e(t) is complex we should write I e(t) 12 in place of e2(t) in the two preceding expressions.

In the case in which the process has a different physical interpretation, this terminology may prove contrary to the usual representations. Nonetheless, we shall use it in what follows. In many questions random processes are simulated by the sum of harmonics with given frequencies and random amplitudes and phases. In other words, one considers processes of the form

e(t) _

k=1

ak cos (ukt + q.k)

(1 )

4.

OSCILLATIONS WITH RANDOM PARAMETERS

23

where the uk are given numbers and the quantities ak and p, are random. The probabilistic structure of this process is completely determined by the joint distribution of the random variables ak and , n). (for k = 1, 2, If a random process E(t) is the sum of n harmonic oscillations with amplitudes I ak and frequencies uk the set of all frequencies {uk}, for k = 1, 2, , n, considered as a set of points on the line < u < oo ), is called the spectrum of the random function e(t). In many cases it is convenient to consider complex-valued random processes of oscillatory nature,

k

y

(2)

S(t) = Z 7ke%ukt k=1

where the complex amplitudes 7k are random variables:

7k=ak+iRk Here the ak and the Rk, for k = 1, 2, , n, are real. The process C(t) can be split into real and imaginary parts: fi(t) = e(t) + i72(t)

,

where ak COS Ukt - Qk Sin ukt k=1

)I(t) _

(3) ak sin ukt + Qk COS ukt .

k=1

If we set ak = I yk I cos cpk and Rk = 7k I sin qk, we obtain an ex pression for the functions e(t) and ra(t) in the form fi(t) =

k=1

17k I COS (ukt +Cpk)

(4)

11(t) = k=1 L I yk I sin (ukt + q k)

It is easy to compute the mean power of the process C(t). CT 1

2T --T

We have

n+

()t Edt =2T1 J-T k,r=l7kyreit(Uk-nr)dt T

k=1

17k IE +k,r=1 E 7kyr k#r

sin T(uk

- ur)

T(uk - ur)

We obtain, with probability 1, * In defining a random process by means of formula (1), we have in mind a random process whose finite-dimensional distributions can be calculated by beginning with formula (1) and the joint distribution of the quantities a1, , an, (pi, -, 1pn

RANDOM PROCESSES IN THE BROAD SENSE

24

lim 2T

TT

y., 12

I at) I2dt = kE l

Thus the mean power of the oscillatory random process C(t) is equal to the sum of the mean powers of the harmonic components of the process.

We compute in analogous fashion the mean value of the random

function C(t) in an infinite interval of time. We have C(t)dt

2T

JT T

= E yk

sin Tuk Tuk

The value of (sin u)lu is taken to be 1 at u = 0. Thus if 0 is a point of the spectrum, then lim

1

T

T-.ee 2T -T

C(t)dt = yo

,

where yo is the amplitude corresponding to the frequency u = 0. Even under special assumptions regarding the distribution of the variables yk, the distribution of the variable C(t) is extremely complicated. However, the simplest characteristics of the distribution of the random variable ((t) can be obtained without difficulty. Let us suppose that the complex amplitudes yk have mathematical expectations equal to 0 and that they are uncorrelated, that is, that M'Yk=O, M'Ykyr=O,

(5)

kzzz r.

We note that if 0 is not a point of the spectrum of a random process the mathematical expectation of the function C(t), that is, its mean value in the probabilistic sense, coincides with the mean value over the infinite time interval (- oo, oo). On the other hand, if 0 is a point of the spectrum of the process, then the Then MC(t) = 0.

mean value of the sample function over the time is a random We have the following expression for the covariance function of the process C(t):

variable.

R(t1, t2) = M[C(t1)&(t2)] = ML

',

k,r=1

t1-urt2)]

ykyrei(Uk

_

Cke{uk(tl-t2)

k=1

Thus the covariance function of the process C(t) depends only on the difference t1 - t2 of the arguments:

where ck = M I yk 12 .

R(t1, t2) = R(t1 - t2) R(t) = Y, Cke;ukt k=1

,

(6)

.

(7)

Consequently, if the quantities yk in the process (2) are uncorrelated

and if they have mean values equal to 0, then the process C(t) is

4.

OSCILLATIONS WITH RANDOM PARAMETERS

25

a stationary process in the broad sense and its covariance function is given by formula (7). This formula is called the spectral distribution of the covariance function. It determines the spectrum of the random process (that is, the set of frequencies {uk}, for k = 1, 2,

, n,

of the harmonic oscillations constituting the process C(t)) and the mathematical expectations ck of the mean powers of the corresponding

harmonic components of the process in a unit of time. The quan-

tity ck can be called simply the mean value of the power of the harmonic component of the process with frequency uk. It is obtained by first averaging the power over time and then averaging it in the probabilistic sense, that is, by taking the mathematical expectation. In connection with these energy representations we introduce the following important characteristic of a random process, the spectral

function of the process. The spectral function F(u) of the process (2) is defined by

F(u)=Eck k

uk
This means that F(u) is equal to the mean power of the harmonic components of the process C(t) (whose frequencies are less than a given quantity u) in a unit of time. It completely characterizes both the mean power of each harmonic component of the process C(t) and the overall mean power of the harmonic components of the process, whose frequencies lie in a given interval. Specifically,

ck-F(uk+0)-F(uk),

,

ul
ck=F(u2)-F(ul)

By means of the spectral function, the covariance function of the process C(t) can be written in the form R(t) = r° eiutdF(u)

(8)

From a mathematical point of view, the spectral function is a nonnegative nondecreasing left-continuous function that is piecewiseconstant with finitely many jumps of magnitudes ck. We shall find

that the concept of a spectral function can be introduced for arbitrary processes that are stationary in the broad sense.

This question,

like the question of generalizing the representation (8) to arbitrary stationary processes, is considered in Section 5. Of particular interest are random processes that can be obtained from processes of the form y( S(t) _ n 7keiukt (9) k=1

by passage to the limit. The basis of this procedure is the fact that

RANDOM PROCESSES IN THE BROAD SENSE

26

the number of terms in the sum (9) increases without bound as we decrease the complex amplitudes 7k whereas the spectrum of the process

(that is, the set of all frequencies uk) fills the line (- cc, oc) ever more densely.

In passing to the limit, we can obtain a random process

that has a continuous spectrum. The more precise meaning of this term and the question of the analogue of the representation (9) for the random processes obtained in this manner will be considered in Section 5 and later in Chapter V. Here we confine ourselves to a consideration of the limiting distributions.

, n and n = Theorem 1. Let {ank} and {,l9nk} (for k = 1, ) denote two sequences of sequences of mutually independent (in each sequence) random variables 1, 2,

Mank = MRnk - 0 ,

/

cc

Dank = DNnk = - b'nk <

/ and yy b,(t)

= D=1 ynkeiunkt = en(t) + 'Y)X), Ink = ank + ll nk. Suppose that the following conditions are satisfied as n cc :

The sequence of the spectral functions F,(u) of the processes e, (t) converges on some everywhere-dense set of values on the line (- 00, 00) a.

to a bounded function F(u), such that

F(b.

0, F(+cc)=62=lim2>bnk, n-.o,

k=1

The random variables ank and S.k satisfy Lindeberg's condition

(cf. Theorem 5 of Section 3). Then the sequence of random processes C(t) converges weakly to the The characteristic

stationary process C(t), where C(t) = g(t) + ir)(t). function of the joint distribution of the variables

-

(ti), ..., e(t8), 7)(t1), ..., 7)(t8) is given by the expression *N,

..., Xe, //1, ... , p8) = exp {_4B2}

(10)

.

(11)

where

j = 1, ..., S,

B2 =j,k=1 i R(t; - tk)ZjZk, Zj = Xj -'Ili R(t) = f

eintdF(u)

.

(12)

(13)

We note that under the conditions of this theorem we can use an arbitrary bounded nondecreasing function for the function F(u). This theorem is a special case of Theorem 5 of Section 3. In this application we take 0 to be the set of points 0 = (t, q), - cc < t < cc and q = 1, 2, and we define ank(0) by

5.

SPECTRAL REPRESENTATIONS

27

. 1) = Re auk(t, 2) = Im and /9 k satisfy Lindeberg's conIt is easy to verify that if the dition, the variables also satisfy it for arbitrary 0. Using the notation of Theorem 5 of Section 3, we have 02) = R91g2(tl, t2) for Bi = (ti,g), where

ank(t,

7nkeiu,yk1,

R,1)(t1, t2) = Men(tl)e%(t2)

7uke4uukt

,

( p R12'ltl, t2) = M/ n\ti)'/n(t2)

R22'(t1, t2) =

On the basis of formulas (3), (1

Rll )(t1, t2)

R22 >'

1

_q R12 (tl, t2)

2f

21 1

t2)

cos (t2 - ti)u dFn(u)

sin (t2 - tl)u dF.(u)

Keeping Helly's theorem in mind, by letting n approach - we obtain from condition a., the existence of the limits Rif(t) = limR;;)(z, z + t), i, j = 1, 2 , R11(t) = RE2(t) = 1 _ cos to dF(u) 2

R12(t) =

J

sin to dF(u)

Thus the conditions of Theorem 5 of Section 3 are satisfied. It follows from that theorem that the characteristic function of the joint distribution of the variables (10) is #(a,1, , e, 1, ...4t.) =e -B4/2' where s+

B2 =r,k=1 L Rll(tr - tk)XkXT + 2R12(tr - tk)xkf"r + R22(tr 1

88++

2 k,r=1

/,

R(tr - tk)ZZk, Zr = Xr - iL^r,

tk)f kf^r

r = 1, ..., S

This completes the proof of the theorem. 5.

THE SPECTRAL REPRESENTATIONS OF THE COVARIANCE FUNCTION OF A STATIONARY PROCESS AND OF THE STRUCTURAL FUNCTION OF A PROCESS WITH STATIONARY INCREMENTS

In the preceding section we constructed an example of a stationary process whose covariance function was given by the formula R(t) =

eitudF(u)

(1 )

RANDOM PROCESSES IN THE BROAD SENSE

28

where F(u) was an arbitrary nonnegative bounded nondecreasing func-

tion that was everywhere continuous on the left. We find that equation (1) applies for all processes that are stationary in the broad sense. Specifically, we have: Theorem 1 (Khinchin). For a function R(t) (defined for - o < t < -) to be the covariance function of a process that is stationary in the broad sense and that satisfies the condition

(2)

MIC(t+z)-C(t)11.--*0 as r.-->0,

it is necessary and sufficient that it have a representation of the form (1).

Proof of the Necessity.

Let us set a2 = M I C(O) 11.

Then

R(t + z) - R(t) I = M([C(t + z) a{M I C(t + z) - C(t) I2}1/2

It follows from condition (2) that the function R(t) is continuous. Furthermore, it is a positive-definite function: that is, R(tk -tj)T,ka,j > 0 k, j=1

for an arbitrary choice of numbers n, t1, t2, , t, . It follows from the Bochner-Khinchin theorem (see, for example, B. V. Gnedenko,

1967, §38,) that the function has a representation of the form (1). The sufficiency of the condition of the theorem follows from the preceding section in which it was shown that for an arbitrary nonnegative nondecreasing function F(u) one can construct a normal complex-valued stationary (even in the narrow sense) process whose correlation function is given by formula (1). Actually, one can construct in a much simpler manner a process that is stationary in the broad sense with covariance function given by formula (1) for a given spectral function F(u). Suppose that

F(+ -) = o 2. Let e denote a random variable with distribution function (l/62)F(x). Let C(t) = uei('1+w), where q and e are independent and q is uniformly distributed on the interval (0, 27r). Then

= MC(t)

6

2zei(tu+(P)

w0

1 a2

dF(u)

dT = 0 27r

and

R(t + h, h) = M[C(t + h)C(h)] O

=

rzeitu

Q2 dF(u)

_=

etitdF(u) J

S.

29

SPECTRAL REPRESENTATIONS

Definition 1.

The function F(u) in the representation (1) of the

covariance function of a process that is stationary in the broad sense is called a spectral function. If F(u) is absolutely continuous,

the function f defined by

F(u) = I f(v)dv is called the spectral density of the process. Let us recall the physical meaning of the spectral function.

If

the random function e(t) represents an electric current, then the function F(u) can be interpreted as follows: The process e(t) can be represented as a "continuous sum" of simple harmonic oscillations

with random amplitudes, and the increment F(u2) - F(u), where u, < u2, is equal to the mean power of the harmonic components of the process whose frequencies lie in the half-closed half-open [ul, u2).

The spectral function can be obtained from the covariance If a and b are points of continuity of the distribution

function.

F(u), then as one can see from the theory of characteristic functions (cf. Gnedenko, 1967, §35),

F(b) - F(a) =

1

2tr

f J

m e_sta _ earn

it

R(t)dt ,

(3)

where the integral on the right should be understood in the sense of the principal value. At points of discontinuity of the function F(u), formula (3) remains valid if we replace F on the left with the function G defined by G(u) = [F(u + 0) - F(u)]/2. If the covariance function R(t) is absolutely integrable on the interval (- oo , o), that is, if I R(t) I dt < -, then the right-hand member of equation (3) is differentiable with respect to the parameter

Thus the absolute integrability of the function R(t) implies the existence of a spectral density equal to b.

f(u) = 27c -- e-'-'R(t)dt 1

that is, f(u) is the inverse Fourier transform of the correlation function R(t). This situation is analogous with multidimensional processes, in which case the role of the covariance function is played by a covariance matrix function. We introduce the following notation: The trace of a square matrix 4) = (cD,k), for j, k = 1, , n, is defined as the sum of the

RANDOM PROCESSES IN THE BROAD SENSE

30

diagonal elements of that matrix and is denoted by tr 4':

tr (D =

E!=1 41.1.7

Let 4) and IF denote two m x n matrices 4, =(cI

k= 1, ,n,

), j = 1,...,m;

k=1,...,n.

'F=(I'Jk), j=1,...,m;

The scalar product of the matrices 4' and IF is defined by (4','P) = E ;k ;k = tr &Fj=1

k=1, ,

mm

where %F* is the adjoint of the matrix %F; that is IF* = (%P* ) for

j= 1, ,m and k= 1, ,n; 'I'jk

The norm 11011 of the

matrix 4) is defined as

E I(D,kI

II4II

k=1, ,n

Theorem 2. For a matrix function R(t) = (R;k(t)), where j, k = 1, , n, to be the covariance matrix of some vector-valued process r(t) that is stationary in the broad sense and satisfies the condition

limMIIVt+v)-(t)II2=0, T-o

(4)

it is necessary and sufficient that it have a representation of the form

R(t) _

e%utdF(u)

where F(u) = (F2k(u)), for j, k = 1,

(5)

,

, n, is a matrix function sat-

isfying the following two conditions: (a) For arbitrary ul < u2, the matrix OF(u) = F(u2) - F(u1) is nonnegative-definite, and (b)

tr{F(+-)-F(--)}<-. With regard to condition (b) we note that by virtue of condition (a) the diagonal elements of the matrix F(u) are nondecreasing functions of u. Thus condition (b) is equivalent to the requirement that each diagonal element FF,(u) of the matrix F(u) be of bounded varia-

tions on the line (- oo, oo). Furthermore, the positive-definiteness of the matrix OF(u) implies that I OF;k(u) 12 AF;;(u)OFkk(u), so that s p=1

1/2

a

OF.9k(up) I <

p=1

1F.1A(up)

1/2

[ p=1 i JFkk(uP)

where OF(up) = F(up) - F(up_1), for p = 1, 2,

u0
, s, and where

From this it follows that the offdiagonal elements F,k(u) of the matrix F(u) are also functions of bounded variation. Without loss of generality we can assume that F(- oo) = 0.

S.

SPECTRAL REPRESENTATIONS

31

Let us now prove the theorem.

Proof of the Necessity. Let R(t) denote the covariance matrix function of an r-dimensional vector-valued process C(t) that is stationary in the broad sense and that satisfies condition (4). For an arbitrary r-dimensional vector c, we introduce the random variable (r(t), c).

Then

M I Ut + r) - C0(t) I2 <- I C I I {M I I Vt + v) - r(t) I I2}"2 so that limr.o M I C,(t + r) - Ca(t) I2 = 0. In addition, C,(t) is a ranI

dom process that is stationary in the broad sense: me = M C,(t) = (Mr(t), c) = const , M(}Ce(t + r) - me][be(r) - mj) = (c, R(t)c) Let Re(t) denote the covariance function of the process Ca(t).

We

have

Ra(t) _ (c, R(t)c)

(6)

.

On the basis of the preceding theorem, the covariance function Re(t) can be represented in the form Ra(t)

ea(udF,(u)

(7)

,

where FF(u) is a nondecreasing function of u and FF(u) < oo. Let e(k) denote the r-dimensional vector whose kth component is equal to one and whose other components are equal to zero, e(k)

0

p

1,

p=k.

P

k

It follows from (6) that R.(k)(t) = Rkk(t) is the covariance function of the kth component of the vector-valued process C(t). We set e(k,;) = e(k) + e(j), e(k,j) = ie(k) + e(j). From (6), we obtain

j

Re(k,j)(t) = Rkk(t) + Rkk(t) + Rkk(t) + Rjj(t) , k R;(k,j)(t) = Rkk(t) + iRka(t) - lRjk(t) + Rj9(t)

from which we get Rkk(t) =

Re(k,j)(t) - 1Re(k,j)(t) 2

-

1-Z 2

(Re(k) - Re(j))

If we set Fkj(u)

Fe(k,j)(U) - iFe(k,j)(u) 2

_ 1 - i (Fe(k)(u) - Fe(j)(U)) 2

/ = Fe(k)(u) Fkk(u)

we obtain

,

k:/ j,

RANDOM PROCESSES IN THE BROAD SENSE

32 eztudFkj(u)

Rkj(t)

k, j = 1, 2, ..., r

,

.

Furthermore, Ra(t)

= k,j=1 E CkRkj(t)c, = fJ

eitud(

E CkFk,(u)c;)

`k,j=1

By virtue of the uniqueness of the function FF(u), we obtain F,(u)

k, j=1

CkFkj(u)Cj

From this it follows that the matrix OF(u) _ (OFk,(u)), for k, j = 1,

, r, is nonnegative-definite and Fkk(+ co) < o-.

Proof of the Sufficiency. Consider the matrix function of two variables R(t1, t2) = R(t1 - t2), where R(t) is defined by formula (5).

Let us show that this function is nonnegative-definite. We have tq)Z$P Z'a =1,

.T Rk'

P,4=1, ...,n

=J

(tP,

mod( W, F(u) J3')

where

W

n

W = Li edutPZ(P) Z(P) = (ZiP), ..., ZTP)) P=1

It follows from condition (a) of the Theorem 2 that W, F(u) W is a nondecreasing function of u. This means that R(t1 - t2) is a nonnegative-definite matrix. Let us construct an r-dimensional complex Gaussian process with vector-valued mathematical expectation equal to 0 and with covariance matrix function R(t, - t2) (cf. Section 3). We obtain a process r(t) that is stationary in the strict sense. Here, M I I r(t + z) - r(t) I I 2 = 2tr [R(0) - Re R(z)] . Since the function R(t) defined by equation 5 is continuous, this last formula shows that the process r(t) satisfies condition 4 of Theorem 2. This completes the proof of the theorem. Let us look at the question of the spectral distribution of the covariance function of a stationary process with discrete time. Theorem 3. The matrix function R(t) = (Rkj(t)), for

and t=0, is the covariance function of some stationary (in the broad sense) vectorvalued process r(t) with discrete time if and only if it can be represented

in the form R(t) = 50eitudFu)

where F(u) = (Fk j (u)) , for k, j = 1,

, r, is a matrix function defined

5.

SPECTRAL REPRESENTATIONS

33

on [0, 27r) and enjoying the following properties: (a) For arbitrary u1 and u2 such that 0< u1 < u2 < 27r, the matrix OF(u) = F(u2) - F(u1) is nonnegative-definite; (b) tr [F(27r - 0] - F(0)] < o o. Proof of this theorem is analogous to that of Theorem 2. One first considers the one-dimensional case, just as in Theorem 1, except that instead of referring to the Bochner-Khinchin theorem, one needs

to refer here to a theorem of Herglotz (see Gnedenko, 1967, p. 259) on the representation of a nonnegative-definite sequence of numbers.

Then as in the proof of Theorem 2, one shifts from the onedimensional to the multidimensional case. Let us now look at processes with stationary increments (in the broad sense). For structural functions of such processes, we obtain a formula analogous to the spectral representation of the covariance function of a stationary process. Theorem 4. Let r(t) denote an r-dimensional process with stationary increments (in the broad sense). The function D(t1, t2, t t4) is the structural function of the process r(t) satisfying the relation

MIIr(t+r)-r(t)II2.-.0 as r->0,

(8)

if and only if this function possesses a representation of the form etit2u _ eitlu a-it4u -

D(t1, t2, t2, t4) = -

iu

iu

Cit3u dF(u) '

(9 )

where F(u) is a matrix function (- oo < u < 00) with the following properties: (a) For every u1 and u2 such that u2 > u1, the matrix F(u2) - F(u1) is nonnegative-definite; (b) (°°Wk(u)a[trP(u)]

< 00

JJ

(10) ,

where

k(u) =

Proof.

Let us set

rm(k)_r(k2

1)-r(

2

k = 0, ± 1, ±2, For arbitrary m, the sequence rm(k), for k = 0, ± 1, ±2, m = 1 , 2,

,

(1 1)

, is a process with discrete time that is stationary in the broad sense. On the basis of Theorem 3, the covariance matrix R.(k) of the sequence (11) has the following representation

RANDOM PROCESSES IN THE BROAD SENSE

34

=

Rm(k)

rz

eikudFm(u)

(12)

,

-rz

where the Fm(u) satisfy condition (a) and tr{Fm(7r) - Fm(-7c -0)} < 00.

It is more convenient to write formula (12) in the form

=

Rm(k)

(13)

where Fm(u) = 0 for u < -2mtc ,

Fm(u) = Fm(u) Fm(-1 l for \\2m/ \ 2m /

- 2mtc < u < 2mtc

-

-

u>2-7r.

for

Fm(u) = Fm(tc) Since

t+S+k+l, t,,+r2 11 2='E'ED(t+ k+1-1 t+k+l t+j j=11=1

Dt+

k

2-

,

2m

r

2m

I

t +

2m

2m/

a

R,(k+I- j),

j=1 t=1

it follows that for numbers t1, t2, t3, and t4 of the form k/2m (where

k = 0, +1, +2,

), we obtain (t4-t3)2m-1

E j=1

D(t1, t2, t3, t4) _

-

etiut2

(t2-t1)2m-1

E

exp iu(k + 1 - j) }dm(U) 2m

1=1

- e:ut1

ei(u12m) -1

a-tiut4

-

e-tiut3

e-:(u12m)

which can also be written in the form eiut2 - ecut1 a-iut4 D(t1, t2, t3, t4) _

-

-W

-1

dPm(u)

e_iut3

1

iu

du

k(u)

dHm(u)

,

(14)

where a2k(a)

Hm(u) = J u ae

4 sin 2

a

dFm(u)

,

(15)

2m+1

Let us now show that the sequence of matrix functions Hm(u) is weakly compact. This means that the sequence {Hm(u)} contains a subsequence {Hmk(u)} such that for an arbitrary f(u) that is bounded

and continuous on (- 00 , o), km ,f(u)d'mk(u)

- f(u)dH(u)

5.

SPECTRAL REPRESENTATIONS

35

where H(u) is some matrix whose elements are functions of bounded variation. The matrix H(u) is then the weak limit of the sequence of the matrices Hmk(u). On the basis of Helly's theorem the sequence {Hm(u)}

is weakly compact if the norms of the matrices Hm(u) are uniformly bounded (with respect to m) and L>A dHm(u) II --.0 as A-. 00 uniuniI

formly with respect to m. Let us show that these conditions can be satisfied.

We have

D(t, t + h, t, t + h) = M(['(t + h) - r(t)][ (t + h) - r(t)l *) 4 sin' uh u2k(u)

dHm(u), t = k

,

h=J

(16)

.

m

Let us set 'r(h) = M I r(t + h) - fi(t) Ill. By the hypothesis of the theorem, #(h) -. 0 as h -. 0. From this it follows that the function *(h) is continuous. This is true because I *(h")

- *(h') I < M(I I '(t + h') - r(t + h") I

I

x [II'(t+h")-r(t)II + IIr(t+h')-C'(t)Ii]) V* (j h' - h" I)!(h") + 1/-(I h' - h" I)iir(h') It follows from (16) that for A > 0, 4 sin 2

A

*(h) ? I I D(t, t, + h, t, t, + h) I I > i-A h.

1 AAdHkk)(u) ,

(17)

.

,

uh 2

>

4 h2

= 7r

U 2

for I u I < 1.

uh 2

u' If AI h I < 7r we have

where Hm(u) = (Hm-)(u)) for r, s = 1 , 4 sine

.

Consequently i/r(h) >

Furthermore, for A > I we have

*(h) > 4f

l'>A

= 2f

sin' uh dHm,k)(u) 2

(4 - cos uh)dHm,k)(u) .

J I,&I>A

Integrating this inequality with respect to h, we obtain 1

h

' #(

5.

> 2f 2(1

iul>A

-

C1 -

1 1( Ah

uh

/J

dHc(k,k)(u) Iul>A

(18)

RANDOM PROCESSES IN THE BROAD SENSE

36

Since the left-hand member of this inequality approaches 0 independently of m as h --> 0, it follows that dH.'k'(u) uniformly with respect to m as A- oo.

0

From the positive-definiteness

of the matrix OH., we have I

I OHm(u) I C tr OHm

so that dHm(u)

tr

dHm(u) -- 0

lul>A

as

lul>A

A,-

uniformly with respect to m, and by virtue of (17), iuisAdH.(u)

h I< A

4h2 *(h)

These inequalities prove the weak compactness of the sequence of matrices Hm(u).

If we now take the limit in (14) with respect to the subsequence

of indices mk such that {Hmk(u)} converges weakly to H(u), we obtain edut2

D(tl, t2, t3, t4) -

- eiut, a-tint, iu

e-tiut3 dH(u)

- iu

This equation is valid for all dyadic t1, t2, t3, t,.

k(u)

Obviously

H(u2) - H(u1)

is a nonnegative-definite matrix. Since the left and right members of the formula are continuous functions and since these two members coincide on an everywhere-dense subset of the values of t1, t2, 4, t

they are equal for all values of these variables. It only remains for us to set F(u) = %{dH(u)/k(u)} to obtain the desired result. Let us now look at a scalar random field e(x) in n-dimensional space En: x = (x1, x2, , x") where - oo < xti < o o. This field is said to be homogeneous if Me(x) = a = const, R[xl + z, x2 + z] = M([e(x, + z) - a][e(x1 + z) -a]) = R(xl, x2) If we set z = - x1 in this last condition, we obtain R(x1, x2) = R(0, x2 - x1)

.

The last equation means that the covariance between the random variables e(x) and E(x) depends only on the vector x2 - x, connecting the points x1 and x2. The function R(x) = R(z, z + x) is also called the covariance function of the homogeneous random field.

5.

SPECTRAL REPRESENTATIONS

37

It is nonnegative-definite function of n variables; that is, the quadratic form N

E R(xi - xk)XiXk

i,k=1

is nonnegative-definite for arbitrary choice of N and points x1, x2, .. . , XN .

The function R(x) is continuous if and only if the random field fi(x) satisfies the condition

z--+0.

as

MIe(x+z)-e(x)I2---->0

(19)

The Bochner-Khinchin theorem for nonnegative-definite functions

of a single variable can be carried over almost without change in the course of the proof to functions of several variables. Thus the covariance function of a homogeneous field satisfying condition (9) has a representation of the form R(x) =

ei(.,.)du(v)

where (x, v) denotes the scalar product of the n-dimensional vectors

x and v and a(A) is a finite measure in En. A random field is said to be isotropic if the covariance function R(x1, x2) depends only on x, and the distance between the points x1 and x2. If in addition it is homogeneous, then R(x1, x2) = R(p), where p is the distance between the points x1 and x2:

p=

E (xi - x2)2

We will find a representation of the covariance function of a homogeneous isotropic field. Let us look at the expression for the covariance function of a homogeneous field R(p) =

En

ei(x,v)da(v) \ //

and let us integrate this expression over the surface of a sphere SP of radius p. Reversing the order of integration, we obtain P

n

n-1 R(p) = 27cn\/2 P2I

ff

JEn{Jspei(x'U)ds}da'(v)

ff

.

(20)

Let f(x) denote an arbitrary integrable function in E. and let V denote the ball of radius p with center at a fixed point. Then dp J VP

ff(x)dx1 ... dxn =

f(x)ds

1 SP

,

38

RANDOM PROCESSES IN THE BROAD SENSE

where the integral on the right is over the surface S,o of the ball V,,. Let us use this formula to evaluate the inner integral on the right side of formula (20). Shifting to spherical coordinates in ndimensional space (cf. G. M. Fikhtengol'ts, vol. III, p. 401) and taking for q1 the angle between the vector x and v, we obtain e'(z,-)dx1 ... dx = J VP

... JJ

r, 2reirlvlcos'Pirn-1

JP f 0

0

sinn-2 P1 J

0

0

x sinn-1p2 ... sin pn_2drdp, ... ddn_2

2jr' I,(n

1)/2

(

1J0

eirIvIcosp,rn-1

sin"-2p1 drdp1

'00

2

J

Furthermore, eirlvlcosvi

T T = k=0 f

(ir I V I cos TX k sinn-2p1dp1 k!

sin%-2 m1dn11

i

0

I

YI

2k+1

1

/

2

r(2k + n) 2 f

(2k)!

k=O

)(n

2

and

I_P 0

0eirlvlcosVirn-1

sinn-2 1 dp1

0

p2ktn y lk

k=O

F( 2k

(2k)! (2k + n)

)(n

1) 2

2

F( 2K + nl 2

J

Using the formula for the gamma function for half-integral values

P(k +

_ i/-7r P(2k)

11

22k-1 I'(k)

2

we obtain

I=

1/7rI'n

(p I V I )Zk+(n/Z) 2 V

2i I,(n

1)2(n/2)-lpn/2

151. (-1)k

J./2

k=

k!r(k+ 2

+ 1)

Y/2 2 1) (I2v

I

In/2(P I v I)]

is the Bessel function of the first kind of order in. where Consequently,

5.

SPECTRAL REPRESENTATIONS

Vp

39

yJ

1n/2

ei(x'")dxl ... dx'n

J./.(p I L

From this it follows that ea(x,v)Lts

Sp

= (27tP)

n/2

/2(p I y 1)

In particular, the integral depends on I v J. We introduce the positive parameter X and we set g(?) = a(VV), a, > 0. This last formula and formula (20) yield R(p) = 2`n-2)/2 r(2)

J(.-2)/2(2p)

Jo

(

p)(n-2)/2

dg(?)

(21)

where g(%) is an increasing function, g(0) = 0, and

g(+ °°) = a(EE) = R(0) < - .

Thus we have obtained Theorem S. For R(p)(0 G p < -) to be the covariance function of a homogeneous isotropic n-dimensional random field satisfying condition (19), it is necessary and sufficient that it have a representation of the form (21), where g(X) is a bounded nondecreasing function.

For n = 2 the formula takes the following simple form R(p) = o Jo(Xp)dg(X) ,

(22)

and for n = 3,

R( p )

= 2f

o

sin P ox'

dg (X)

(23)

II

MEASURE THEORY

We assume that the reader is familiar with the elements of the set-theoretic construction of probability theory. Therefore, in the present chapter we have omitted elementary examples and

details that give the intuitive basis underlying formal definitions (see, for example, Gnedenko, 1967). The starting point in probability theory is the assumption that one can define a set U and a class C25 of subsets of U so that every event A, for which it is meaningful to speak of its probability within the framework of a particular problem, can be interpreted as some subset of the set U belonging to e. Since an arbitrary event A interpreted in this manner can be regarded as the union of elements of U that belong to A, we call the points in the set U elementary events and the set U itself the space of elementary events.

For example, if an experiment consists of drawing the graph of a continuous random function in the course of a fixed interval of time [a, b], then U can be understood as the space of continuous functions

on the interval [a, b]. In what follows the events are identified with the sets corresponding to them. Obviously a certain event then coincides with the set U and an impossible event coincides with the empty set; the union, coincidence, and difference of two or more events then coincide with the set-theoretic union, intersection, and difference of sets. The incompatibility of a class of events means that the intersection of the corresponding sets is empty. If A is an event, the complementary event A is the set-theoretic complement of A in U. Furthermore, to every event A E C25 is assigned a nonnegative number p(A) called the probability of the event A. It is natural to require that the class e of events and their probabilities (which are defined) enjoy the following properties (with which we are familiar from elementary probability theory): (a) The difference of two events or the union of an arbitrary sequence of events in the class e are events (that is, they belong to e); (b) the prob40

1.

MEASURE

41

ability of the union of an arbitrary sequence of incompatible events is equal to the sum of the probabilities of the events of the given sequence; and (c) the probability of a certain event is equal to 1. The mathematical apparatus with which we formulate the basic

assumptions and concepts of probability theory and derive the general theorems is the abstract theory of measure and integration.

The material from this theory that we shall need in this book is expounded in the present chapter.

1.

MEASURE

Let U denote an abstract set, which we shall call a space. We shall indicate subsets of U by italic letters and classes of subsets of U by German letters (capitals in both cases). We assume that the definitions and simplest properties of the algebraic operations on sets are known and we mention only the frequently used duality relationships nAk = UAk k

(1)

U Ak = n Ak

(2)

k

and

k

k

where the index k ranges over an arbitrary (finite or infinite) set of values (cf. for example, Kolmogorov and Fomin). Definition 1. A nonempty class Jt of subsets of U is called an algebra of sets of U if it enjoys the following properties: a. b.

A e Jt and B e J{ imply A U B e Jl, A e 91 implies A e R.

Let us give some of the simpler consequences of this definition. Since A U A= U, the relation A e R implies U e R. This in turn implies that the empty set belongs to the algebra of sets. Furthermore, if A e R and B G R, then on the basis of relationships ( 1 ) and (2), A n B = (J U B) e R, and A\B= A n Be R, that is, the intersection and difference of two sets belonging to the algebra 91 also belong to 91. From this it follows by induction that the union and intersection of an arbitrary finite number of sets belong-

ing to the algebra R also belong to R. With respect to the unions and intersections of a countably infinite collection of sets in R, the latter assumption generally ceases to be valid. Therefore we introduce the following definition, which plays a fundamental role.

MEASURE THEORY

42

Definition 2. An algebra of sets C is called a o-algebra if for an arbitrary sequence of sets Ak e e, where k = 1, 2, , U k=1 Ak e e. The sets A e C25 are said to be e-measurable. (Since lk=1 Ak =

Uk=1 Ak, the intersection of an arbitrary countable collection of sets belonging to e also belongs to e.) Theorem 1. For every class of sets $t there exists a smallest 6-algebra e containing K. This a-algebra is called the a-algebra generated by the class a, and is denoted by v{%}. It is easy to prove the existence of such a or-algebra. There exist 6-algebras containing W. To exhibit one, it suffices to take the class of all subsets of the set U. Noting that the intersection of an arbitrary set of a-algebra is again a o'-algebra,

we see that the intersection of all o'-algebras containing a is the minimal or-algebra containing a. Definition 3.

In a metric space, the v-algebra of sets generated

by the class 6 of open sets is called the 6-algebra of Borel sets and its elements are called Borel sets. Obviously the 6-algebra generated by the closed sets of a

metric space coincides with the o'-algebra of Borel sets. We can easily see that in a separable metric space the o-algebra

of Borel sets is the o'-algebra generated by the set of open (or closed) spheres. On the real line the 6-algebras generated by open or closed (or even half-open half-closed) intervals coincide with the

a-algebra of Borel sets. In n-dimensional Euclidean space E. we choose for a system of sets generating the 6-algebra of Borel sets the systems of closed, open, or half-open half-closed parallelepipeds (or intervals) J[a, b], J(a, b), J[a, b), J(a, b]. (If a = (a1, a2, .,a.) and b = (b1, bE, , bJ , , n}. The then J[a, b) = {(x1, xE, , x ); a; < x; < b;, i = 1, 2, other intervals are defined in analogous fashion.) Suppose that to every set A in a certain class of sets I we assign a definite number W = W(A), which may be + oo or - oo.

This defines a set function W on t into the set of real numbers,

A,W= W(A). Definition 4. A set function W is said to be additive (or finitely additive) if it assumes infinite values of only one sign and if, for an arbitrary finite sequence of sets Ak e ! (for k = 1, 2, , n) that are pairwise disjoint (that is, Ak (1 A,. = 0, for k # r where k, r = 1, 2, , n and 0 denotes the empty set) such that

1.

MEASURE

43

UAkE k-1 we find that W(U Ak) = Ek=1W(Ak) . k-1

If this equation holds for an arbitrary countable collection of sets, that is, if

W(0Ak) = > W(Ak)

for an arbitrary sequence of sets Ak E I, where A. 1 A, = 0 whenever k :j,_ r, for k, r = 1, 2,

, n,

UAke

k=1

,

such that

,

then the set function W = W(A) is said to be countably additive (or completely additive).

Definition 5. A countably additive nonnegative set function

p = p(A) defined on a u-algebra of sets e and satisfying the equation p(O) = 0 is called a measure. If a u-algebra of sets e is defined on a set U and a measure p is defined on 0, then the set U is called a space with measure {U, e, u} or a measurable space.

The latter term will be applied for

a set U with a fixed u-algebra of sets e even when the measure p is not given. We can easily see that the condition p(O) = 0 is equivalent to the condition that p(A) is not identically equal to + co for all A e An arbitrary set A E e of a space with measure {U, e, p} can .

itself be regarded as a space with measure {A, CSA, PA}, where e,

is the u-algebra of subsets A of the form A fl B for an arbitrary subset B of 0 and PA(C) = p(C) for every C E e,. We now present a few properties of measures. Theorem 2. a. If A and B D A belong to e, then jt(A) < p(B), and if p(A) # co, then j_t(B\A) = p(B) - p(A). b. If {An} is a countable or infinite sequences of sets belonging to e, then jt(Un An) < Zn Ft(An). c. If {An} is an increasing sequence of sets in 0, that is, if

An+1 D A. f o r n = 1 , 2,

, then

lim tt(An) = P(UAn) d.

If {An}, for n = 1, 2,

(3)

, is a decreasing sequence of sets

MEASURE THEORY

44

in 0 and if p(A1) < 0, then lim p(An) = n--

tt((An) n=1

(4)

.

Proof. a. Since B\A e e and B = A U (B\A) (for A c B), we have p(B) = AC(A) + ,(B\A). b. Let us set C1 = Al and C. = An\(Uk=i Ak) for n = 2, 3,

.

Then the sets C. belong to 0 and they are pairwise disjoint (that is, Cn (1 C,. = 0 for n r). Furthermore, un=1 Cn = U;,=1 A. and p(C.) < p(A,). Therefore,

(OA) = (OC) = E p(C.) < n=1 E MA.) n=1 n=1 n=1

c.

If An c An+1 for n = 1, 2,

, we obtain, in the notation

used above, C. = An\An_1 and p(C.) = U(A,) - p(An-1) if ft(An-1) 00. co for every n and Ao = 0. Then Let us suppose that rt(An)

p(C.) = i [lp(An) - p(An-1)1 = lim tt(An) p(:E An) = in=1 n=1 n-1 n-On the other hand, if p(Ano) = co for some n = no, then for n > no we have a fortiori jt(An) _ co and j_c(U;,=1 An) _ o o.

.

The sets B. d. Let us set Bn = Al\An for n = 1, 2, belong to the u-algebra e, they increase monotonically (that is, Bn c Bn+1), and from (c), 1a(Un°1 Bn) = limn-- jt(B,). hand, n z, An = Al\Ut6=1 B,,. Therefore

On the other

,t(n1An) = p(A1) - f"(V1B.) = p(A1) - lim jt(B,) = ,a(A1) - lim[ p(A1) - u(A,)} = lim p(A,) Deflnltion 6. Let {An}, for n = 1, 2, , denote an infinite sequence of sets. The limit superior lim A. of the sequence {An} is

defined as the set consisting of those points of U that belong to infinitely many of the sets An. The limit inferior lim An of the sequence {An} is defined as the set of those points of the space U that belong to all except possibly finitely many of the sets A. for

n=1,2,

Thus

lim A. = fl U Ak

,

(5)

limAn=UnAk. n=1 k=1

(6)

n=1 k=n

If {An}, for n = 1 , 2, .. , is an increasing sequence, then lim An=

1.

MEASURE

45

An. On the other hand, if {An} is a decreasing lim A. = sequence, then lim A. = lim A. = nn-1 An. It follows from (5) and (6) that the limits superior and inferior of a sequence of sets belonging to a a-algebra c also belongs to Cam. If i denotes a measure

on e, then it follows from assertions c and d of Theorem 2 that a(lim An) = lim

n-.m

p(k-n

fI(lim An) = lim lt(1 Ak) n-.m

(7)

/ Ak)

k-n1

/

,

(8)

with equation 7 holding if the measure p is finite. Definition 7. A sequence of sets {An}, for n = 1, 2, , is said to be convergent if lim A,, = lim A. In this case the common value of the limits superior and inferior of the sequence {An} is called the limit of the sequence {An}: lim An = lim An = lim A. It follows from our definition of convergence of a sequence of sets that every point u e U either belongs to only a finite number of the sets A. or belongs to all the A. from some n on. It follows from what was said above that every monotonic sequence is convergent.

Since

II(n Ak) < 1e('4,) < '2(U Ak) it follows on the basis of formulas (3) and (4) that for every convergent sequence {An} of sets A. and every finite measure j_t(lim An) = lim P(An)

.

(9)

We now introduce the following useful concept: Definition 8. A class t of sets is said to be monotonic if the convergence of an arbitrary monotonic sequence of sets An e fit, , implies tnat the limit of such a sequence belongs for n = 1, 2,

to R. Since the intersection of monotonic classes is a monotonic class, it follows that corresponding to an arbitrary class % of sets there is a minimal monotonic class m{%} containing %. Obviously, every a-algebra is a monotonic class and every algebra that is a monotonic class is a a-algebra: u,n=1 An =

In many cases we need to show that a particular class of sets contains a minimal a-algebra generated by a given algebra. For which the following theorem is useful. Theorem 3. The minimal monotonic class m{ l(} containing the limUk=1 A,

algebra % coincides with the minimal a-algebra a{%}.

MEASURE THEORY

46

On the basis of the remark made above it is sufficient to show that m{%} D c{%}, and to do this it is sufficient to show that m{a} is an algebra. Let S{A} denote the class of all sets B such that A U B e m{S2I}, A\B a m{%}; B\A e m{%}

(10)

.

The class S{A} is monotonic: if is a monotonic sequence of sets and each B. satisfies conditions (10), then A U B, A\B, are also monotonic sequences of sets and lim (A U

lim lim

A U lim B. e m{%} , A\lim B. e m{%}

lim

e m{%} ;

that is, lim B. e m{%}. If A e W, then R(A} D I. Consequently, m{%} c St (A}; that is, for every F e m{?X} we have F e S{A}. It then follows from the definition of R{F} that A e R{F}. Then just as above, it follows from the monotonicity of S{F} that m{%} c St{F}. This means that relations (10) hold for arbitrary A and B in m{%}; that is, m{%} is an algebra of sets. Let us pause to look at arbitrary countably additive functions

defined on a v-algebra of e. We shall call them charges. Since every charge is the difference between two measures the study of charges reduces to the study of measures. from Theorem 4 below.

This follows immediately

Definition 9. For arbitrary A e e, the quantities W+(A) =

sup

A'cA,A'c

W(A'); W-(A) _ -inf

A'cA,A'e

W(A')

(11)

are called respectively the positive and negative variations of the charge W on the set A, and the quantity

W I (A) = W+(A) + W (A) is called the absolute variation.

(12)

We note that for arbitrary A e e, (13) IN'I(A)? IW(A) It follows immediately from the definition that W+ and W- are nonnegative and nondecreasing set functions: If A c B, then

0 < W±(A) < W±(B) .

(14)

Furthermore, W-+(A

U A2) < W-(AJ + W±(AE) .

(15)

Throughout the remaining portion of this section we shall assume that the space U and a 6-algebra C on it are fixed. All the sets that we shall consider are assumed to be s-measurable.

1.

47

MEASURE

Lemma 1. If W(A) < °° for every A, then W+(U) < ..

To prove this, let us assume the opposite, namely that W+ (u) _ °° . Let us show that in this case there exists for arbitrary c > 0, a set A such that W(A) > c and W+(A) = °°. We also prove this assertion by contradiction. If it is not

valid, there exists an Al such that W(A1) > c and W+(A1) < °°.

If

we set A2 = U\A1 in inequality (15) we obtain the result that W+(U\A1) _ °° .

Repeating the above reasoning with U replaced

by U\A1, we see that there exists an AE c U\A1 such that W(A2) > c and W+(A2) < °° . We obtain by induction an infinite sequence of sets A1, A2, ... , belonging to e, pairwise disjoint, and such that W(An) > c, so that

An) = n=1 i W(An) W(U n=1 which contradicts the hypothesis of the theorem. Thus for every c there exists an A such that W(A) > c and W+(A) = °° . Let us take successively c = 1 , 2, , n . . . and then apply what we have just proven to construct a sequence of sets B,,, such that for each n, W(B,) > n, W+(B,) = °° and Bn+1 c Bn. For this we first use U to find B1, then B1 to find B2 c B1, and so forth. Let us set B,,. Then W(B1\D) = lim[ W(B1) - W(B,)] = - o° so that D= W(D) = + °°, which is impossible. This contradiction completes the proof of the lemma. Theorem 4 (Hahn). Let W denote an arbitrary charge on a o'algebra e. Then U can be partitioned into two sets P and N such

that U = P U N, P n N = 0, and for every A e e, W(A n P) 0, W(AnN) <0. Proof. Since the charge can assume infinite values of only one sign, we may suppose that W does not assume the value + °° . On the basis of Lemma 1, R = W+(U) = sup W(A) < + °° AC

Let {Cn} denote a sequence of sets such that W(C,,) > ,8 - 2-n and

P= lim Cn = f 1 U Ck . n=1 k=n

If A c Cn, then W(A) = W(C,) - W(Cn\A) > Therefore, for

1

.

arbitrary A c P we obtain from the relation

MEASURE THEORY

48

A c U,-=n An Ck the result

/

W(A) _

k=n

W(A n

(c,,\'U_'cj)) j=1

i 12k > - k-,n

or taking the limit as n --> -, we find W(A) > 0 .

(16)

Let us now set N = U`P = U Dn, n=1

where D,y =

n1 (U\Ck) 1

k=n

If An c,, = o, then W(A) < 2-". (This is true because the inequality W(A) > 2-n implies that W(A U C,,) = W(A) + W(C,) > i9, which is impossible.) From this it follows that the relation

An(UCk)=0 k=n implies that W(A) < 0. Therefore if A c N, A=U (A n Dn), W(A) _ n=1

n=1

W(A n (Dn\Dn-1)) < 0 .

Thus

W(A) < 0 for every A c N. This completes the proof of the theorem.

(17)

Corollary 1. The positive, negative, and absolute variations of a charge are measures and

W+(A) = W(A n P), W-(A)

W(A n N) ,

IWI(A)=W(AnP)-W(AnN).

(18)

(19)

Corollary 2. An arbitrary charge can be represented as the difference of two measures:

W(A) = W+(A) - W-(A) .

(20)

Corollary 3. sup I W(A') I< I W I (A) < 2 sup I W(A') A'CA

A'e(g

Proof.

A'cA

A'eE5

Formula (20) follows from (18) since

W(A) = W(A n P) + W(A n N) = W+(A) - W-(A) , and inequalities (21) follow from (20) and (19).

(21)

1.

49

MEASURE

Let W denote the set of all finite charges on a a-algebra e. This set is a linear space if the sum of two charges and the

product of a charge and a number are defined in the natural manner (tW) (A) = tW(A) .

(W, + WZ) (A) = W1(A) + WZ(A) ,

We now define a norm on W:

IIWII=IWI(U) It follows from formulas (18) that

IItWII=ItIIIWII and from (15) that

Thus W becomes a normed space. Convergence in W is called , converges to W convergence in variation. If { for n = 1, 2, in variation, that is, if I W - W, I 0 as n 00 , then W, (A) I

W(A) uniformly over all sets A E C25: I

Ae(q

I W - W,

(cf. inequality (21)). Theorem S. The space W with norm II W II = W I (U) is a Banach space (that is, a complete normed linear space). We need prove only the completeness of the space. Suppose that W,,, - W, I 0 as n, n' oo . For every A G C5, the sequence of the numbers W,(A) is a fundamental sequence and it converges to a finite limit. Let us set WO(A) = WW,(A). The set function W0(A) is defined on C25 and is finite and additive. Let us show , denote that it is countably additive. Let {Ak}, for k = 1, 2, I

I

I

a sequence of disjoint sets in e. Then W0(Ul

Ak)

-

WOCkU1

W.(k=n+1

Ak/

Ak)

W \k=n+1 Ak/

= lim p-.

W,m+p1

J

A,)

+ lim 11 W.+p - W. II p-.m

-

(22)

The right-hand member of this inequality can be made arbitrarily small by the suitable choice of m and n. It follows from (21) that {W.} converges in variation to W0. This completes the proof of the theorem.

MEASURE THEORY

50

2.

MEASURABLE FUNCTIONS

From an intuitive point of view, a random variable e

is a

(variable) number that corresponds to each possible outcome of an experiment. Since the outcomes of an experiment are described

by elementary events, a random variable can be regarded as a function of an elementary event, e = f(u) for u e U. On the other

hand, in elementary probability theory a random variable c

is

completely characterized by its distribution function F(x) = p{e < x}.

Corresponding to the event { < x} is the set of elementary events {u, flu) < x}., Therefore, for it to be meaningful to speak of a

distribution function of a random variable, the set {u, f(u) < x} must for arbitrary real x belong to e. In this section we shall study the class of functions defined on a measurable space { U, Cam, p}

which enjoy this property. U.

Definition 1. Let e denote a a-algebra of sets of the space Let f(u) denote a function defined on an Cam-measurable set M

and assuming real values (and possibly the values ± o c). Such a function f(u) is said to be ce-measurable if for every real x the set {u; f(u) < x} is e5-measurable.

We note a few properties of measurable functions. Theorem 1. Let A denote an arbitrary Borel set in the ndimensional space E. and let f (u), , .(u) denote Cam-measurable functions all defined on the same set M e Then the set Cam..

{u; u e M, (fl(u), f2(u),

, f, (u)) e Al

is Cam-measurable.

Proof. Since {u; u e M, (fl(u), ... , fn(u)) e A'\A"} , f (u)) e A']\{u; u e M, _ {u; u e M, (fl(u),

(.f (u), ... , f, (u)) e A"} {u; u

M, (.f (u), ... , .f, (u)) e

k=1

A}f

= U {u; u e M, (f(u), ...,fn(u)) e A' ')I k=1

the class ( of sets A contained in En, such that the set

{u;ueM, (f(u), , fn(u))eA} is Cs-measurable constitutes a a-algebra. Furthermore, 5?.I contains the n-dimensional infinite intervals = {(x1, ... , x,m); x1 < a1, ... ,

a,y}

2.

MEASURABLE FUNCTIONS

51

since n

{u; u e M, (fl(u), ..., f,,(u)) E

n {u;; u e M, fk(u) < ak}

Consequently t contains all Borel sets in E,.. Corollary 1. If f(u) is an s-measurable function, then for every x the sets {u; u e M, f(u) < x}; {u; u e M, f(u) > x} , {u; u e M, f(u) > x}; {u; u e M, f(u) = x} , {u; u e M, a < f(x) < b}

,

etc.,

are s-measurable.

REMARK 1. As one can see from the proof of Theorem 1, the assertion in that theorem holds for an arbitrary function f(u) defined on an Ce-measurable set M and satisfying the condition {u, u e M, f(u) E K} e 0 for a certain class of sets K that generates a o'-algebra containing 01 (the o'-algebra of Borel sets in E1). In particular the function f(u) defined on Me e is s-measurable if for arbitrary real x one of the following systems of sets

{u; u e M, f(u) < x} {u; u e M, x < f(u)} {u; u e M, x < f(u)}

is Cam-measurable (x may range only over an arbitrary everywheredense set).

Theorem 2. Let { fn(u), n = 1, 2,

, u c M} denote a sequence Then the functions

of C5-measurable functions. n

n

x

f (u), lim

sup fn(u), inf n

are Cam-measurable.

The proof follows from the relations: {u; u E M, sup f (u) > x} = U {u; u e M, f (u) > x} , n

n=1

{u; u e M, inf f (u) < x} _ U {u; u e M, fn(u) < x} , n

n=1

{u;ueM,limf,,(u)<x}=U U k=1 n=1 j=n {u; u E M, linm f (u) > x} = U U n {u; u e M, fj(u) > x +

MEASURE THEORY

52

Definition 2. The characteristic function XA(u) of a set A is defined as the function that is equal to 1 for u e A and equal to 0 for u 0 A.

Note the following obvious relations: (1)

XAnB(u) = ZA(u)/6B(u);

(A n B= 0);

XAUB(u) = XA(u) + XB(u) XA(u)

(2)

(3)

1 -XA(u),

x1irA,y(u) = llm XA,y(u);

(4)

xiimA,y(u) = llm XAJU)

(5)

Definition 3. An Cs-measurable function J(u) is called a simple

function if there exists a finite collection of sets, each contained in the domain of definition of f and together covering this domain of definition, such that f assumes a constant finite value on each member of the collection (though possibly differing from member to member).

Suppose that a simple function f(u) is defined on a set Me e , an (where av a; if i :/- j, for and assumes the values al, a2,

i,j=1,...,n).

Let us set A; = {u; u e M, f(u) = a;} for j= 1,

, n.

Then

the A, are Cs-measurable and

.f(u) = 3-1 3

a,XA;(u),

ueM

(6)

where XA,(u) is the characteristic function of the set A;. On the other hand, every function that can be represented in the form (6) is a simple function defined on M. Theorem 3. For a function f(u) ( where u e Me e) to be emeasurable, it is necessary and sufficient that it

be the limit of a

sequence of simple functions that converges everywhere on M.

The sufficiency follows from Theorem 2. To prove the necessity we set Proof.

AN _2NN = {u; u e M, f(u) < -N} ; ANA; = {u; n e M, k

k

1

2N

< .f(u) <

2

j

f- 2N N + 1, -2NN+2, ,2NN,

AN,2NN+, = {u; u e M, f(u) > N}

fN(u) =

2NN+1 k k=-2NN

1

2N

y

X AN k(u), u e M .

fN(u) - f(u) I < 2-N if I .f(u) I < N, fN(u) = N if f(u) >N, andfN(u) < -N if f(u) < -N. Consequently, lim fN(u) = f(u), u e M. Then

I

2.

MEASURABLE FUNCTIONS

53

This completes the proof of the theorem. REMARK 2. If f(u) is nonnegative (or at least bounded below) and Cam-measurable, it is the limit of an everywhere-convergent nondecreasing sequence of simple functions. To see this, we note that the functions fN(u) constitute in this case a nondecreasing sequence beginning with some number N.

Let us now look at functions g(x) defined on some metric space R into the extended real line (that is, the set of real numbers

with the values ± - included). Let 0 denote the Q-algebra of Borel sets contained in R. Definition 4. A function f(x) for x e R is called a Borel function if for arbitrary real a the set {x; f(x) < a} is a Borel set. Definition 5. A Baire function is defined as a function belonging to the smallest class B of functions defined on R that satisfies the following two conditions: (a) B contains all continuous functions; (b) B is closed under passage to the limit; that is, if {fn(x)j, for n = 1 , 2, , is an arbitrary sequence of functions fn(x) e B that converges in R, then limn- fn(x) e B.

Theorem 4. The classes of Borel and Baire functions coincide. Proof. Let Q denote the class of Borel functions. Q contains all continuous functions and is closed under passage to the limit: {x; lim fn(x) < a} = {x; lim fn(x) < a}

= k=1 U n=1 U j=n n {x; .fj(x)
k

IGO,

if fn(x) a Q(for n = 1, 2, Thus Q contains all Baire functions. Let us now show that every Borel function is also a Baire function. Let us show first that the class of Baire functions is linear. Let K(g) denote the class of all functions f(x) defined on R such that for arbitrary real

a and (3 the function of + lag

is a Baire function. K(g) = { f(x); af(x) + 8g(x) e B for arbitrary a and Q}. If g and f are

continuous, then f(x) e K(g). Furthermore, the class K(g) is closed under passage to the limit: If fn(x) e K(g), for n = 1 , 2, , and if lim fn(x) = fo(x), then a(x) e K(g). Consequently B C K(g) for an arbitrary continuous function g(x). From the definition of the class K(g) it follows that the relation f e K(g) implies g e K(f).

Thus, the relation B c K(g) for g e C implies that C c K(f) for arbitrary f e B, where C is the class of continuous functions. Since K(f) is closed under passage to the limit and contains C it follows

MEASURE THEORY

54

that B c K(f) for arbitrary f e B. This means that for arbitrary

Baire functions f and g and arbitrary real numbers a and 8, af(x) + ,8g(x) e B; that is, the class B constitutes a linear space.

Let us now look at the class of sets a whose characteristic functions are Baire functions. We have the following: a. a contains all closed sets. Let F denote a closed set, let XF(x) denote the characteristic function of F, and let p(x, F) denote the distance from the point x to the set F. Since I p(x, F) - p(y, F) p(x, y), the function p(x, F) is continuous. Let us set P(x, F) if p(x, F) < s , (1

-

if p(x, F) > s . The function ff(x) is continuous and lim, f,(x) = XF(x). 0

l

Conse-

quently F e a. b. The class a is monotonic. Let A. denote a monotonic sequence of sets A. e a and define Ao = lim A. Then xA0(x) = lira XA.(x) e B.

It follows from a and b that I contains all Borel

sets in the space (Theorem 3, Section 1). c. Let f(x) denote an arbitrary Borel function. On the basis of Theorem 3, there exists a sequence of simple Borel functions fN(x) such that f(x) = lim fN(x) (7) The simple functions fN(x) admit a representation of the form (6) in which the A. are Borel sets. Since the class B constitutes a linear space, simple Borel functions belong to B. Since B is closed under passage to the limit, on the basis of equation (7) an arbitrary Borel function is a Baire function. This completes the proof of the theorem. Let us now look at the properties of measurable functions. denote a sequence of finite aTheorem 5. Let fl(u), measurable functions defined on an a -measurable set M and let

,

, denote a Borel function in n-dimensional space E. the function q (fl(u), , fa(u)) for u e M is Cs-measurable.

(p(tl,

Proof

Then

For arbitrary real a, the set

B. = {(t1, ...,

p(tl, ..., t,a) < a}

is a Borel subset of E. The set {u; u e M, p(.f (u), .f2(u), ... , f.(u)) < a} , = {u; u e M, (fi(u),

e Ba}

is a -measurable on the basis of Theorem 1. This completes the

2.

MEASURABLE FUNCTIONS

55

proof of the theorem. Corollary 1. If f and g are s-measurable finite functions, then the functions f ± g, fg, and 11g are also Cam-measurable. Here l/x must be assigned some value unique for x = 0.

This follows from the fact that the functions x ± y, xy, and l/x are Borel functions. Corollary 2. For any two Cs-measurable functions flu) and g(u), where u e M, the sets {u; u e M, f(u) < g(u)j; {u; u e M, f(u) = g(u)} are measurable.

The proof follows from the measurability of the function flu) - g(u). Definition 6.

Let p denote a measure with domain of defini-

tion e. Two functions f and g are said to be equivalent (more precisely, p-equivalent) on a set Me e if the set A = {u; u e M, flu) zt- g(u)} is Cs-measurable and p(A) = 0.

Definition 7. A o-algebra of sets C is said to be complete (or p-complete or complete with respect to the measure p) if an arbitrary subset N' of a set N of p-measure 0 is Cam-measurable; that is, if

the relations N' C N, N e C5, and p(N) = 0 imply

N'ee.

(8)

The measure p defined on a p-complete o-algebra of sets is also said to be complete. Of course relation (8) implies that )"(N') = 0. Theorem 6. If C25 is a p-complete a-algebra, if f(u), for u e M, is a 25-measurable function, and if the functions f(u) and g(u) are equivalent on M, then g(u) for u e M is also s-measurable. It follows from the hypothesis of the theorem that for arbitrary real a, {u; u e M, g(u) < a} = {u; u e M, f(u) < a}\N' U N", where N' and N" are subsets of the set N = {u; u e M, f(u) :94- g(u)} of pmeasure 0. By virtue of the completeness of the measure, the set {u; u e M, g(u) < a} is Cam-measurable. The set of all C5-measurable functions defined on M and

equivalent to a given function f(u) is obviously some complete equivalence class of functions. In many cases there is no point in distinguishing among equivalent functions. Then the word "function" actually refers to an entire class of Cam-measurable functions that are equivalent to each other. In what follows we shall often proceed from this point of view. Let us make some remarks concerning terminology. A certain property is said to hold p-almost-everywhere on M if the p-measure

MEASURE THEORY

56

of the set of points at which this property does not hold is equal to 0. For example, if two functions f and g are equivalent on M, we may say that f and g coincide u-almost-everywhere on M. A sequence of functions f (u), for n = 1, 2, , and where u e M, is said to converge fit-almost-everywhere to the function f(u) on M if the p-measure of the set of those points u E M at which f (u) does not exist or does not coincide with f(u) is equal to 0. If a property holds p-almost-everywhere on a set M, let us indicate this property with the expression (mod p) instead of the more cumbrous "p-almost-everywhere." Thus if f and g are equivalent on a set M, we can write f(u) = g(u), u e M(mod p). Similarly, if {f(u)} converges to f(u) p-almost-everywhere on M, we can write simply f (u) = f(u), u e M(mod p). 3.

CONVERGENCE IN MEASURE

A sequence of random variables n is said to converge in probability to a random variable e if for arbitrary s > 0 P{ -$ > s}--'0 as n o, and we indicate this fact by writing = P - lim s,,. Corresponding to this definition in the general theory of functions is the following: Let { U, e, p} denote a space with a measure denote a sequence of p-almostand let { f (u)} for n 1, 2, everywhere finite s-measurable functions on U. Definition 1.

A sequence {f(u)} is said to converge in p-measure

to a Cs-measurable function f(u) if for arbitrary s > 0, p{u; I fn(u) - f(u) I > s} , 0 as n We indicate this by writing flu) = p-lim f .(U) . In this section we shall consider those properties of sequences of functions that are related to convergence in measure and we

shall look at the relation between convergence in measure and ordinary convergence (convergence at each point of some set). Some measurable space { U, e, p} is considered fixed. All the functions in question will be assumed Cs-measurable and (mod ft) finite even though this may not be explicitly stated. , denote a given sequence of Let {f(u)}, for n = 1, 2, measurable finite (mod p) functions on U. Let S denote the set of points of U at which the sequence {f(u)} converges to a finite

limit and let D denote the set of points at which this sequence

3.

CONVERGENCE IN MEASURE

diverges.

57

Then

s = n UL n1 {u; I f, (u) - f.+-(u) I < k } , D = U n UL {u; I .f%(u) - .fn+m(u) I > k

I

,

(1 ) (2)

that is, the sets S and D are s-measurable, so that it is always meaningful to speak of the measure of the set on which a sequence {f(u)} converges or diverges. If p(D) = 0, the sequence of functions converges It-almost-everywhere. We set f(u) = lim f (u), u e S,

and we extend the definition of f to the set U\S = D by setting f(u) = 0 for u e D. Then lim f,(u) = f(u) (mod p), and the function flu) is the finite limit (mod ,u) of the sequence {f(u)} on U. Theorem 1. Let pt denote a finite measure. If a sequence {f(u)} of functions fn(u) converges (mod p) to a finite (mod p) function f(u) on U, then { f%(u)} converges to f(u) in ,u-measure. Proof. Let D denote the subset of U on which the sequence {f(u)} does not converge to f(u). Define

Dkm =-= U {u; I .fn(u) - f(u) I > k } , Dk = M=1 n Dkm, D = U Dk k=1

The sets Dk constitute an increasing sequence and the sets Dkm constitute for fixed k a decreasing sequence. Consequently, p(D) = limk_ p(Dk) = 0, so that ji(Dk) = 0 and p(Dk) = lim f t(Dkm) = 0. Thus for every k and s > 0 there exists an m such that p{u; I fn(u)

- f(u) I> k}< s

for all n > m, which means that the sequence { f, (u)} converges to f(u) in p-measure.

The following theorem asserts that a sequence of functions that converges in ,u-measure has no more than one limit (mod It). Theorem 2. If f(u) = It-lim f (u) and g(u) = p-lim fn(u) (for u e M), then the functions f and g are equivalent on M. This is true because p{u; u e M, f zt- g} kim It{u; u e M, I f(u) - g(u) I > k }

MEASURE THEORY

58

{u; u e M, I f(u)

- g(u) I > k

s p{u; u e M,

f(u) - f (u) I+ I f(u) - g(u) I> k 1

<- P{u; u e M, I .f(u) - .f%(u) I> k}

+ p{u; U e M, I ff(u) - g(u) I > k >0 ask,. Definition 2. A sequence {f(u)}, for n = 1, 2, be bounded with respect to a measure It on a set M if sup lt(u; u e M, I f,(u) I > L}

,

is said to

0 as L ---+ o

, converges Theorem 3. If a sequence { f (u)}, for n = 1, 2, in It-measure to a finite (mod ,u) function f(u) on U, it is bounded on U with respect to the measure lt. Proof. For arbitrary L,

If (u) - f(u) > 2

1iu; u e M, I fn(u) I > L} < It

+ p{u; u e M, I f(u) I>

J

1}

J

(3)

2 sequence The sets {u; u e M, I f(u) I > L/2} constitute a decreasing as L oc and

fl {u; If(u) I > 2

{u;

I f(u)

I=

}

Therefore the second term on the right side of formula (3) approaches 0. That the same is true of the first term follows from the convergence in measure of {fn} to f. Let us find a necessary and sufficient condition for convergence

in measure of a sequence of functions. We introduce a useful auxiliary concept:

Definition 3. A sequence {f(u)}, for n = 1, 2, , is said to be almost uniformly convergent on U if for arbitrary s > 0 there exists a set H such that p(H) < s and the sequence {f(u)} converges uniformly on U\H. The concept of an almost uniformly convergent sequence should

not be confused with the concept of a sequence that converges uniformly almost everywhere on U. for n = 1, 2, Lemma 1. If a sequence {

, of Cs-measurable

finite (mod p) functions converges almost uniformly on U, it converges

59

3. CONVERGENCE IN MEASURE

almost everywhere on U. Proof. For every integer k, there exists a set Hk such that p(Hk) < 2-11, and the sequence { fn(u)} converges uniformly on U\Hk.

Then the sequence { fn(u)} converges on an arbitrary set M, = U\ U;=, Hk and consequently converges on the set u;=1 M. The set p=1

H=U\UMp=fIUHk, p=1 k=p

where the sequence { fn(u)} (for n = 1, 2,

.

) may diverge, has

measure

lt(H) = lim P((J Hk) = 0 . k=p

/

Definition 4. A sequence { fn(u)} of Cam-measurable finite (mod

functions is said to be fundamental in p-measure if for arbitrary oc. s > 0, p{u; I fn(u) - fm(u) I > s} 0 as n, m

Theorem 4. If a sequence { fn(u)} of Cs-measurable finite (mod p) functions is fundamental in p-measure it contains an almost uniformly convergent subsequence {fnk(u)}, for k = 1, 2. Let us find an nk such that

Proof.

p{u; I f -(u) - fm(u) I > 2k} < 2k

for n, m > nk. Without loss of generality we may assume that the sequence {nk} is an increasing sequence. Suppose that Ek = {u; I J nk(u) - fnk+l(u) I >

2k }

Then if u U;=, Ej and i, j > k (for i < j), it follows that

I AM -

{{''

,/ n j (u) I {'

C IJ nd(u) - fnti+1(u) I + I Jni+1(u) 2°

2111 +

.

+

21-1

{

- A+2(u) I + ... + I l ny-1(u) - fnj(u)I

- 2-1

that is, the sequence { fnk(u)} converges uniformly on the set U\Hk, where

Hk = U Ek and u(Hk) < j fY(Ej) < 21-k j=k j=k in other words, the sequence { fnk(u)} converges almost uniformly on U. Theorem 5. For a sequence { fn(u)} of functions to converge in

MEASURE THEORY

60 t_t-measure, it is

necessary and sufficient that it be a fundamental

sequence in p-measure.

Proof of the Necessity.

If { ff(u)} converges in measure to the

function f(u), then l-p{u;

I f -(u) - fm(u) I > s} p{u; I f .(u)

as n, m

)0

- flu) I > 2 I+ t`{u' I f(u) - fm(u) I > 2 I

.

Proof of the Sufficiency. If a sequence {f(u)} is fundamental in measure, it contains by virtue of Theorem 4 a subsequence that converges almost uniformly and hence in measure to some finite (mod p) Cam-measurable function flu). Then

p{u; I f(u) - f -(u) I > s} F`{u;

I f(u) - fflk(u) I >

2

} + p{u; I fnk(u) - f -(u) I >

2

}

On the basis of the choice of sequence { the first term on the right side of the inequality approaches zero as k - and the

second term approaches zero as k, n ---, - by virtue of the fact that the sequence {

is fundamental in measure.

This completes

the proof of the theorem. Corollary. For a sequence {f(u)} to converge in p-measure to f(u), it is necessary and sufficient that every subsequence of the functions f(u) that to f(u) almost uniformly-The

necessity follows from Theorems 4 and 5. To prove the sufficiency, we note that if the sequence { f,(u)} does not converge to flu) in measure there exists a sequence of indices nk such that p{u; I .k(u) - f(u) I > e} > 8 for some s > 0 and 8 > 0. But this contradicts the assumption that the sequence { fk(u)} contains a subsequence that converges almost uniformly to J (u). If p(U) < oo, we can replace almost uniform convergence in the statement of the corollary with convergence (mod p). Theorem 6. If p(U) < 00 and if the sequences {f1k)(u)}, for k = 1, 2, , s, of finite" (mod p) functions converge (in ,u-measure) as n oo to functions gk(u) = p-lim fk)(u), and if q(tl, , t.) is an arbitrary continuous function of s variables t;, where - co < t. < 00, for j = 1, ..., s, then q(&(u), g2(u), ..., g.(u)) = p-lim p(f,"(u), f,2)(u), ..., {e)(u)).

4.

INTEGRALS

61

,

An arbitrary Proof. Let us set F (u) = q?( fnl)(u), sequence of indices n, contains a subsequence {nkT} such that the sequence { f kr(u)} converges n-almost-everywhere to g ,(u) (for j = It follows from the continuity of the function 1, , s) as n contains in turn (Atl, , t,) that an arbitrary subsequence , g,(u)) p-almosta subsequence that converges to F(u) = 9?(g1(u), everywhere as n --. oo . This, together with the above corollary,

yields the desired result. Corollary.

If p-lim ff(u) = flu) and p-lim g (u) = g(u),

then

p-lim (af (u) + /9gJu)) = a(u) + Qg(u) , p-lim ff(u)gn(u) = J (u) g(u) REMARK. It follows from the proof of Theorem 6 that the conclusion of that theorem remains in force even when the func, t,) is not necessarily continuous but has the property tion q,(t1, , g,(u)) e A} = 0 where A is the set of points of p{u; (g1(u), discontinuity of q. In particular, if p-lim g (u) = g(u) and g(u) # 0

(mod p), then the sequence {1/g,a(u)} converges in measure to 1/g(u).

4.

INTEGRALS

In probability theory, we assign to a random variable e a specific number me known as the mathematical expectation of the random variable a assumes finitely many values x1, x2, the mathematical expectation is given by the formula

me_

If

,x

(l)

x;P{C=x:} 4=1

and it enjoys the following properties: M(aE + ft) = aMC + bMrl

,

and the inequality e < 72 implies me < M77.

We now define the mathematical exception for more general For an arbitrary random variable , we construct a sequence of random variables, each assuming finitely many values, that converge to e. We set Me = lim MEG. This definition will be meaningful if (a) lim Me exists, (b) lim Me depends only on e and not on the particular choice of sequence {en} of random variables En approximating e, and (c) the extended definition of the mathematical expectation has "good" analytical properties. It has proven cases.

MEASURE THEORY

62

impossible to extend the concept of mathematical expectation to all random variables and at the same time to satisfy these three con-

However we can do this for a rather broad class of

ditions.

random variables (for example, for all random variables that are bounded either above or below). Difficulties arise here because in

taking the limit in a sequence of sums of the form (1), the parts corresponding to both positive and negative terms can approach Therefore it is expedient to consider first random variables that assume values of a single sign. When we shift from random variables to arbitrary functions defined on spaces with measure, the concept of mathematical expectation becomes the concept of an integral. The present section is devoted to carrying this out in the general case. We assume that some measurable space J U, C25, fc} is fixed and that all the ± 00 .

functions in question are Cs. -measurable.

Let us first look at simple functions. Let f(u) denote a simple function defined on U and assuming the values c1, c2, , c,,. The quantity U

f(u) 1C(du) =

,1 Ck

fC{u, f(u) = Ck}

0 ')

is called the integral of the function f(u) over U. Let us note the simpler properties of integrals of simple functions:

f(u) > 0 implies

f(u) fc(du) > 0 , JU

(2) (3)

Ukf(u)fc(du) =

k is an arbitrary constant; Uf1(u)f-i(du) + JUf2(u)f (du) .

U[.fi(u) + ff2(u)}p(du) =

(4)

Only (4) needs to be proved. Suppose that f (u), for i = 0, 1, 2, assumes the values ck°) (for k = 1, 2, , m;), that fo(u) = f (u) + f2(u) and that Ak' = {u; f'''(u) = ck )}. Then .b

UA(i)=U k=1 and either AV, l Art) (1 A;2' _ 0 or c( O) = c.1> + c;2). Consequently,

4.

INTEGRALS

63 m°

+0 MO

k=1 MO

k=1

/ / f0(u)p(du) = E ck ),u(Ak )) = L ck) E i(Ak) n An) n Ae2)) U

_

k=1 r,s

+

cr' p(Ai°) n A(1) n A,(2))

m°

c82>i(Ak)

k=1 r,s

crup(A(1)) +

_

r,8

r

n A(1) n

A82))

c82)t(A82)) a

=

JUf(l)(u)II(du)

+

f`2)(u)I-1(du)

,

which proves (4).

Let us now define the integral of a simple function f(u) over an arbitrary set Me by means of the formula

e

JM

f(u)p(du) = U f(u)xM(u)f"(du)

(5)

,

where XM(u) is the characteristic function of the set M. If M, and M2 are Cam-measurable and have no points in common,

then X-11"'2(U) = X -1(U) + X "2(U)

Consequently, ,/ M1UM2f(u),i(du) =

(f1

,/ f(u)p(du) +

,/

L2 f(u),u(du)

L

.

(6 )

It follows from (2) and (4) that the inequality f(u) > g(u) for u e M implies (7)

JMf(u)p(du) > JMg(u)li(du) .

Before generalizing the definition of an integral, let us prove: f o r n = 1 , 2, , denote a nondecreasing Lemma 1. Let sequence of nonnegative simple functions and suppose that lim g (u) > h(u) for u e M, where h(u) is a nonnegative simple function. Then, lim

JM

h(u) p(du) .

(8)

Proof. Let 0 h1 < h2 < ... < ha denote values assumed by the function h(u). To prove inequality (8), let us confine ourselves

to the case h, > 0. In the opposite case, it would suffice to prove inequality (8) for the set M\{u; h(u) = 0}.

Let s denote an arbitrary positive number and let Qm, for each m = 1, 2,

, denote the set {u; u e M,

>t h(u) - s, n > m}.

MEASURE THEORY

64

These Qm constitute a nondecreasing sequence of Cam-measurable sets,

and U:=1 Q. = M. Suppose that u(M) < oo . Then lim p(Q.) AC(M) and lim p(M\Qm) = 0. Furthermore, for n > m, 5

gn(u) p(du) > 5QM g.(u) p(du) > QM(h(u) - s),u(du)

h(u) p(du) - sp(Q.) Qm

h(u)p(du) - sp(M)

- h, fc(M\Qm)

If we let s approach 0 and m approach 00 , this inequality becomes inequality (8). If p(M) let us take s = h1/2. Then for n > m,

5g(u)fi(du) > 2 P(Qm) oo, which again yields (8).

as m

Lemma 2. Let {f(u)} and {g (u)} denote two nondecreasing sequences of nonnegative simple functions and suppose that lim f (u) _ lim g (u) for u e M e e.

Then

f(u)p(du) = lim

lim

g(u)p(du)

JM

Proof.

It follows from the hypothesis of the theorem that lim L(u) > gm(u)

On the basis of Lemma 1, lim L fn.(u)p(du) > 5gm(u)p(du).

If we now let m approach - we obtain lim

f.(u) f.C(du) > lim JM

g (u) fp(du) J

By reversing the roles of the sequences {f(u)} and {g,(u)} in this last inequality, we obtain the desired result. Let us now define an integral in the general case.

Let f(u)

denote an arbitrary s-measurable function on M. We define f+(u) = max {f(u), 0}

,

f-(u) = - min {f(u), 0}, u e M

.

Then (9) f(u) = f +(u) - .f (u) , and the functions f+ and f- are nonnegative and Cam-measurable.

4.

65

INTEGRALS

Definition 1.

The quantity f(u) p(du) = lim

p(du) ,

(10)

JM

JM

where {f(u)} is an arbitrary nondecreasing sequence of nonnegative simple functions that converges to f(u), is called the integral of the Cs-measurable nonnegative function f(u) over the set M. If f(u) is an arbitrary Cam-measurable function and if one of the integrals J Mf

+(u)p(du),

J

(11)

Mf-(u)a(du)

is finite the integral of the function f(u) is defined by f

JMf(u)li(du) = L f+(u) ft(du) JM

J'-(u)p(du) .

(12)

If both integrals (11) are finite, the integral of the function f is also finite and the function f is said to be integrable over the set M.

This definition is meaningful: On the basis of Lemma 2, the definition of an integral of a nonnegative function is independent of the particular choice of approximating sequence in formula (10). In particular, for a nonnegative simple function the definition of the integral (10) coincides with the original definition. Furthermore,

for an arbitrary simple function, the functions f+ and f- corresponding to it are simple, and on the basis of property (4), which asserts the linearity of the integral of a simple function, formula (12) for simple functions gives the originally determined value of the integral. Let us now look at the basic properties of an integral. Theorem 1. If p(M) = 0 and f is an arbitrary function defined on M, then L f(u)p(du) = 0 .

(13)

This follows immediately from formulas (12), (10), and (1'), Theorem 2. MlUM2

,/ .f(u)ft(du) = Ml f(u)p(du) +

f(u)p(du)

,

(14)

MZ

where Ml n M2 _ 0, where Ml and M2 belong to e, and where one side or the other of the equation is meaningful. It suffices to consider (14) for nonnegative functions. But for

these functions, equation (14) follows from the definition of an integral and the fact that this equation holds for simple functions.

MEASURE THEORY

66

Theorem 3.

Maf(u) p (du) = a

f(u)p(du)

(15)

Equation (15) is valid for simple functions. By taking limits, we extend this equation to arbitrary nonnegative functions, and by virtue of for a >t 0 , (af )+ = of+, (af )- = of (af)+ _ - elf-, (af)- _ - af+ for a < 0

and formula (12), the result is carried over for arbitrary Cam-measurable

functions whose integrals over M are meaningful. Theorem 4. JM

[flu) + g(u)]p(du) _ .f(u)p(du) + Lg(u)li(du) ,

(16)

if the the integrals on the right are not infinite of different signs. If the functions f(u) and g(u) are nonnegative on M, then (16) is valid because this equation holds for simple functions. The case in which f(u) and g(u) are both nonpositive reduces to this case by

means of (15) with a = - 1. Suppose now that f(u) > 0 and g(u) < 0 everywhere on M, that f(u) + g(u) > 0 on M, c M, and that f(u) + g(u) < 0 on ME c M, where M, U MQ = M. Then by setting f(u) + g(u) = h(u), we obtain f(u) = h(u) + (-g(u)), so that Miflu) Fi(du) =

JMLh(u) p(du) + Ll

- g(u) ft(du) ,

that is, Ll [flu) + g(u)]fu(du) = Ll flu) p (du) - Ll g(u)p(du) if the right-hand member of this equation is not an indeterminate form of the fo frm 00-00. Analogously, -g(u) = f(u) + (- h(u)). Therefore, J MQ -

g(u) f t(du) _ -

ME

_ _

g(u)p(du)

Mgflu) ft(du) + ME

Mg

- h(u)p(du)

J lu)ft(du) - ME h(u)p(du) ,

so that, (again, L2 (flu) + g(u))ft(du) _

LEflu) ft(du) + L2 g(u)p(du)

if the right-hand member of this equation is not an indeterminate form of the form - - 00. By combining the integrals over the

4.

67

INTEGRALS

sets M, and M2, we obtain formula (16), by virtue of (2).

Let f(u) and g(u) denote any two functions defined on M. Let us set M, = {u; f(u) > 0, g(u) > 0}, M, = {u; f(u) < 0, g(u) < 0}, M3 = {u; f(u) > 0, g(u) < 0}, M, _ {u; f(u) < 0, g(u) > 0}. Then

M= UM, where M; (1 M; = o for i :t- j, and formula (16) is proved for each of the sets M; for i = 1, 2, 3, 4. If we now replace M in formula (16) with the set M; for i = 1, 2, 3, 4 and add the equations obtained, we get, in view of (14), equation (16) in the general case.

Theorem 5. If f(u) and g(u) are equivalent on M, then L ff(u)p(du) = L g(u)p(du) Indeed,

flu) p(du) _

g(u) ft(du) +

[flu) - g(u)}p(du)

,

and mr

[flu) - g(u)]p(du) _

[flu) - g(u)}p(du) = 0

(by virtue of Theorem 1). Theorem 6. If f(u) > g(u) for u e M, we have f

JMflu)p(du) > Lg(u)p(du)

(17)

if both integrals exist. Proof.

The function cp(u) = f(u) - g(u)

is

nonnegative for

u e M. It follows from the definition of an integral that

p(u) p(du)

0 and hence

f(u)p(du) - Mg(u)fi(du) > 0 , if the difference on the left is defined. On the other hand, if the integrals are infinite but of the same sign, (17) is obvious. Theorem 7.

If JMff(u)p(du)

+

then p(u; u e M, f(u) = + oo} = 0. In particular, if the function f(u) is integrable, it is finite almost everywhere.

MEASURE THEORY

68

Proof. L f+(u)p(du) ? L h(u))Y(du) ,

and h(u) = h if f+(u) = + where h(u) = 0 if f+(u) < + where h is an arbitrary positive constant. Consequently,

CIO ,

L f+(u)p(du) > h p{u; u e M, f(u) = + 00},

which can be bounded as h

only when f(u) < + 00 almost

00

everywhere.

Theorem 8. If f(u) is integrable on M, then I f(u) I is also integrable on M. Proof.

If f(u) is integrable on M, then 1Mf +(u)1t(du) < oo,

J

f (u)ft(du) <

Since

.f(u)

I = f+(u) + f-(u) ,

it follows that JMI

f(u) I

p(du) = f+(u)p(du) +

f-(u)1.t(du) < - .

Theorem 9. If I f(u) I < h(u) for u e M and if h(u) is integrable on M, then f(u) is also integrable on M.

This follows from the fact that f+(u) < h(u) and f-(u) < h(u) and from equation (16). 5.

INTERCHANGING LIMITS AND INTEGRATIONS.

L,

SPACES

Let us now look at the possibility of taking the limit under the integral sign. What conditions do we need to impose on a sequence of functions f.(u) to ensure that lim f fn(u)p(du) = U.f(u)p(du) a J

where f(u) is the limit) (in a specified sense) of the sequence { f, (u)}? Theorem 1 (Lebesgue). Suppose that {f(u)} is a nondecreasing sequence of nonnegative Cam-measurable functions. Define f(u) _ lim f (u) (mod p). Then lim

f(u)p(du)

.

S.

INTERCHANGING LIMITS AND INTEGRATIONS.

L.

SPACES

69

Proof. For each n, let {g.k(u)} denote a nondecreasing sequence of nonnegative simple functions that {converges to f.(u), llm gnk(u) = J (u) k-.-

Since gi.(u) < fi(u), we have h.(u) < max f(u) = f.(u)

and define h.(u) = maxis. gi.(u).

in

and

lim h.(u) < lim f.(u) = f(u) . (1 ) The sequence {h.(u)} is nondecreasing, it consists of simple functions, and for all k, lim h.(u) > lim gk.(u) = fk(u) Consequently,

lim h.(u) > lim fk(u) = f(u) k-.-

n__

Comparing this with (1), we obtain f(u) = lim h.(u) n--

It follows from the definition of an integral and, formula (1) that JU

f(u)lc(du) = lim Uh.(u) ft(du) < lim

f.(u),u(du)

.

JU

Inequality in the opposite direction follows from the fact that fn (u) <_ ./ (u)

Theorem 2 (Fatou's lemma). Let {,..u)} denote a sequence of nonnegative (25-measurable functions.

Then

(2)

U lim f.(u)a(du) 5 lim U fn(u)1i(du) .

Proof. We have lim f,(u) = lim g.(u) , where

gn(u) = inf {J (u), J,, 1(u), . . .}

is a nondecreasing sequence of nonnegative functions. from Lebesgue's theorem (1) that lim f.(u) fi(du) = lim JU

JU

g.(u) p(du) < lim

U

It follows

f.(u)lt(du)

which completes the proof of inequality (2). Theorem 3 (Lebesgue). Let { f.(u)} denote a sequence of meas-

MEASURE THEORY

70

urable functions. Suppose that there exists a nonnegative integrable function s(u) such that I fn(u) I < s(u) for n = 1, 2, . Then

`U lim f (u)p(du) < lim

f (u) ft(du) U

5 lim

J U ff(u)p(du)

<

f U

TIE f. (u) P (du)

(3)

In particular, if { f (u)} converges almost everywhere in U to f(u), then

lim 5f(u)p(du) = 5f(u)P(du).

(4)

Applying Fatou's lemma to the sequence {s(u) +

Proof. f U[s(u)

+ f,(u)] p(du) > f lim [s(u) +

p(du)

u

=

[s(u) + lim f (u)] p(du) JU

_

s(u) p(du) + U

lim ff(u) p(du)

(5)

J

Furthermore,

lim `u [s(u) + f (u)] p(du) = lim IL s(u)p(du) + f f,(u) ti(du)} U

= 5s(u)(du) + lim 5f(u)p(du). U

(6)

It follows from (5) and (6), by virtue of the integrability of the function s(u), that lim

5)">

`U lim f, (u)u(du)

(7)

.

Applying this last inequality to the sequence {- f%(u)}, we obtain

lim 5U - fn(u)p(du) = - BE 5 f,(u)p(du)

>

Ulim(- fn(u)) f_t(du) _ - ` lim fn(u)lt(du) U

so that

lim JU

lim 5 U f(u)1t(du)

(8)

Inequalities (7) and (8) prove the theorem. Corollary. Under the conditions of the preceding theorem, if lim f .(u) = f(u) (mod ft), where u e U, then I f(u) - f(u) I Ft(du)

0

as n

00

(9)

5.

INTERCHANGING LIMITS AND INTEGRATIONS.

L,

SPACES

71

Proof.

lim I f(u) - fn(u) I = 0 n-.oo

and

(u) - n(u) I < 2s(u) . Applying equation (4) to the sequence {I f(u) - fn(u) I }, we obtain ./

assertion (9),

Let f(u) denote an arbitrary function for which the integral f JU

is meaningful.

f(u)p(du)

(10)

Then the formula 'P(A)

= J A f(u)[t(du),

Ae0

defines a set function on the u-algebra Cam. Theorem 4. If the integral (10) is meaningful, then the set function cp(A) is countably additive on Cam.

Since f(u) = f+(u) - f-(u) and one of the functions f+(u) or f-(u) is integrable, it will be sufficient for us to prove the theorem for a nonnegative function. Define A = Un-1 An, Proof.

where and A. e Cam, A. (1 A. = 0 for n zf- m.

Let Xo(u) and xn(u),

denote the characteristic functions of the sets A and A. for n 1 , 2,

Then

.

X.(u) = n=1 L X.(u)

and by virtue of Theorem 1, p(A)

= J f(u)p(du) _ A

Xo(u)f(u)p(du) = lim U

= lim kE A f(u)l-t(du) _

U

f(u) x

xk(u)p(du)

p(Ak)

k

Corollary. If a function f(u) is integrable on U, then for arbitrary

s > 0 there exists a a > 0 such that the inequality p(A) < a implies

L1fl1 lt(du) < E

(11)

Proof. Since integrability of the function f(u) implies inte-

grability of I f(u) 1, it will be sufficient to prove the corollary for nonnegative integrable functions. We note first of all that tc{u; f(u) = oo } = 0. Therefore, lim q {u; f(u) > N} = p{u, f(u) = -} = 0 N-.o

MEASURE THEORY

72

(cf. Theorem 1, section 4). Consequently, for some No, cp{u; f(u) > No} < s/2. Furthermore, for arbitrary N, f(u),"(du) <

cp(A) =

f(u) p(du) +

JJA

NJ

f(u) fi(du) Afl{%;f>N} .

Taking N = Np and u(A) < a = s/2Na, we obtain the desired result. Theorem S. Let {f(u)} denote a sequence of measurable functions that converges in measure to a function f(u) on U. Suppose that I f (u) I < s(u) (mod p), n = 1, 2, .. , where s(u) is an integrable function.

Then

lim J

Proof.

fn(u)p(du) =

J

l u),u(du)

In accordance with Theorem 4 of Section

3, an

arbitrary sequence { contains a subsequence { .(u)} that converges almost uniformly to f(u). It follows from Theorem 3 that lim f Uf-k;(u) (du) = `U f(u)[ (du)

.

Thus the bounded sequence of the integrals f(u)i(du) has a unique point of accumulation f(u),u(du). This completes the proof of the theorem.

We present one more theorem on limits. Here the requirement that an integrable majorant exist is replaced with another requirement whose satisfaction is in many problems easier to verify.

Theorem 6. If a sequence {f(u)} converges in measure to a function f(u) for u e U where ,uU < -, and if for some p > 1, JuI fn(u) IP[t(du) < c,

n = 1, 2 ...,

then

lim Proof.

JU

ff(u)p(du) =

f U f(u),u(du)

.

Let s denote an arbitrary positive number. {u;

Then for N > 1/s,

I f(u) - fn(u) I > e}

Define

5.

INTERCHANGING LIMITS AND INTEGRATIONS.

5flu)p(du)

-

JU

+

fn(u) p(du) I < A,y(e)\A,y(N)

LP SPACES

73

I f(u) - fn(u) I lt(du)

J

I .f(u) - .ff(u) I p(du)

+ 5 A,,(N) I f(u) - .fn(u) 11-t(du) < slt(U) I f(u) - fn(u) 11 p(du) . + Nft(An(s)) + NP-1 An(N) J 1

It follows from the inequality (a

b)P

2

< a JP

I b 11,

I

for p> 1

2

that

L(N)

I f(u) - .fn(u) 1 ' 1-t(du) < 2P-1

A,1e[I flu) IP + I.fn(u) P]lt(du) (N)

< 2f

P-1{5U [ f(u) I P + I .ff.(u) I P] lt(du)}

.

On the other hand, since f(u) = u-lim fn(u), there exists a sequence { fnk(u)} that converges to f(u) (mod lt). By virtue of Fatou's lemma and the hypothesis of the theorem, JU

I f(u) I P p(du) = f U lim I .fnk(u) JP p(du)

< lim

IP lt(du) < c

.

JU

Thus lim

f(u)1t(du) JU

- 5 U fn(u)pt(du)

5 sp(U) + N lim t(A.(s)) + f

2Pc1

= s,u(U) +

2Pc1

Since s is an arbitrary positive number and since N > 1/s, the proof is completed. Let us now look at complex-valued functions f(u) defined on U. The function f(u) = fl(u) + if2(u) where fl(u) and f2(u) are real, is said to be Cs-measurable if fl(u) and f2(u) are C5-measurable. The function f(u) is said to be integrable if fl(u) and f2(u) are integrable. The integral of the function f(u) over U is defined by the equation JUf(u)lt(du) = 5Uf1(u)1-t(du) +

iUf2(u)lt(du) .

74

MEASURE THEORY

The properties of an integral that were established for real functions are easily carried over to complex-valued functions. To the end of the present section, equivalent functions are assumed equal,

so that the word "function" actually means an entire 'class of functions that are equivalent to each other. Let LP = LP{ U, Cam, p} (for p > 1) denote the class of all emeasurable functions defined on U into the set of complex numbers and satisfying the inequality f I f(u) 11 u(du) < cc . U

Minkowski's inequality, which is proven for integrals in abstract

spaces in the same way as for ordinary spaces (see Hardy, Littlewood, and Polya; Heider & Simpson): fp(du)}11P

I f(u) + g(u) j P

l JU

(u) J P p(du)j""

I

+I

U I g(u) I P llbelonging

implies that the sum of two functions

to LP also belongs to LP. Obviously if f belongs to LP, so does af, where a is a complex number. Thus LP is a linear space. Defining the norm

of an element f of LP by fi(du)}"P

f%

IIf II =

we make LP a normed space.

IP

Obviously

IIaf1I = I 1 IIf1I , 11f +gII < IIf1I + IIg1I , so that the axioms of a normed space are satisfied. The distance between two functions f, g e LP is then defined by 1

1

PV, g) = I I f - gII =

}1/P

I f(u) - g(u) I Pd[

If a sequence { ff(u)} of functions in LP converges to f(u) in the sense of convergence in LP, that is, if j

I f(u) - fn(u) I I --> 0 as

n -* cc,

we say that {f(u)} converges in mean of order p to f(u). A sequence {f(u)} is said to be fundamental in LP if I as n, n' ---,. o o.

I

f,,,(u)I I ---p 0

It follows from the inequality

sPp{u; I f(u)

- g(u) I > s} <

J

I f(u) - g(u) I P ti(du)

that if a sequence is convergent (resp. fundamental) in LP, it is convergent (resp. fundamental) in measure. Theorem 7.

For a sequence

of functions in LP to converge

5.

INTERCHANGING LIMITS AND INTEGRATIONS.

LI, SPACES

75

in L. to some limit, it is necessary and sufficient that the sequence I f.1 be fundamental.

In other words, the space LP is complete.

The necessity of the hypothesis of the theorem

Proof.

is

general for metric spaces as a consequence of the triangle inequality: I

II + II ff -f II

fn(u) -J%'(u) I I < I I f - f % ,

Let us prove the sufficiency. Since I I ff II - I I f, I I fn - fn I I' if a sequence {fn} is a fundamental sequence, the sequence {II f, II} is bounded; thus I I fx I I < c. Furthermore, the fact that the sequence I

I

I

is fundamental in LP implies that it is fundamental in measure.

Consequently (see Theorem 4, Section 3) the sequence f f.1 contains a subsequence ff..} that converges almost uniformly in U. Using

Fatou's lemma, we obtain cP > lim UI fnk(u) 11, p(du) > Ulim I

I P p(du) _ UI f(u) I -vu(du)

from which it follows that f e L. Let s denote an arbitrary positive number. Since the sequence { that for all n >_ no,

/

s > lim

k__

>

U

is fundamental, it follows

I

U

fnk(u) - f(u) J -v p(du)

f // lim I J f.k(u) - f n(u) I Pp(du) =

(1'

JU

I

f J (u)

which proves the convergence of the sequence

- fn(u) 11, i(du) , to f in L.

In what follows, the LP space that we shall be primarily interested in is the space L2, that is, the space of functions flu) such that JU

Since I fg I

I f(u) I2 p(du) < - .

(I f 12 + I g 12)/2,

it follows that the function fg is

integrable for all f, g e L2. Consequently we can define a functional

on the set of pairs of functions f and g in L2:

(f' g) =

J

f(u) g(u)ft(du) .

This functional is known as the scalar product of the functions f and g. The following properties of the scalar product follow directly from the definition: 1. (f, f) 0 with equality holding if and only if f is equivalent to 0; 2.

(f, g) _ (g,f);

MEASURE THEORY

76

3. (af + 8f,, g) = a(f, g) + 8(.f,, g); that is, the scalar product is a positive-definite bilinear form in the two variables f and g. A metric in the space L2 is connected

with the scalar product by the formula

PV g) = I1f-gHI = 1/(f-g,.f -g) Since the space L2 is complete, it is a particular case of an abstract Hilbert space. We assume that the reader is familiar with the basic properties and the simpler theorems having to do with Hilbert-space theory (see for example Kantorovich and Akilov, Heider and Simp-

son, and also Section 1 of Chapter V of this book). We mention here only that the scalar product satisfies the Cauchy-BunyakovskiySchwarz inequality: I (f g) 12 < (.f, f) (g, g) or

f(U)g(U)fu(du) 2 < I.f(u) 1,u(du)

6.

J

I g(u)

I' (du)

ABSOLUTE CONTINUITY OF MEASURES. MAPPINGS

In the preceding section, for an arbitrary C5-measurable function f(x) such that the integral f(u) fc(du) is meaningful, we defined

a countably additive set function p(A) =

f(u)p(du) JA

on e. The function p(A) is called the indefinite integral of the function flu) (with respect to the integrating measure ft). Corresponding to this, the function f(u) in formula (1) can be regarded as the derivative of the countably additive function p with respect to the measure i. By definition we set f(u) = dp

dp How can we characterize the class of countably additive functions p having a representation of the form (1)? Is the function f(u) defined naturally by formula (1)? If p(A) is the indefinite integral of the function f(u), then by virtue of one of the simplest

properties of an integral (see Section 4, Theorem 1), cp(A) = 0 whenever u(A) = 0. We find that this condition is not only necessary but in the most important cases is also sufficient if the function p(A) is to have a representation of the form (1). In what follows we shall consistently assume that ,u and p are finite measures defined on some fixed o'-algebra e.

6.

ABSOLUTE CONTINUITY OF MEASURES. MAPPINGS

77

Definition 1. A measure g is said to be absolutely continuous with respect to a measure p (we write cp << )Y) if p(A) = 0 whenever A G C25 and p(A) = 0.

The measure p and p are said

to be singular with respect to each other (and we write p .i. p) if U can be partitioned into two disjoint subsets (that is, U = C,, U C,,, where C,,, C,, e e and C,, fl C,, _ 0) such that cp(C,z) = 0 and p(C,,) =

The singularity of the measures means that the measure cp is nonzero only on subsets of some set of p-measure 0 and vice versa. Theorem 1. Suppose that q and pt are two finite measures. The measure cp can be represented in the form 0.

(2)

p(A) = Lfu)p) + r(A) ,

where f(u) is a nonnegative pt-integrable function on U and * and pt are singular.

Define x = cp + p. We recall that L2(X) is the space of Cs-measurable functions such that I f(u) 12X(du) < -. In this JU space, let us look at the functional a(f) = f f(u)cp(du). It is Proof.

U

defined for every function f e L2(X) since I f I < 1 + I f 12 and I f 12(u)X(du) , and it is linear and bounded: I f 12 (2u)cp(du) < Ja

J

sup I a(f) I
f

c q (U). Every bounded linear functional a(f) in a Hilbert space H can be represented as a scalar product a(f) = I I

I

I

(f, g), where g is some element of H (see Kantorovich and Akilov; Heider and Simpson). Consequently, JU.f(u)p(du) = .f(u)g(u)X(du), g e L2(?)

(3)

Here, if we set f(u) = XA(u) where XA is the characteristic function of the set A e C25, we obtain T(A) _ Ag(u)? (du) < X(A) From this it follows that 0 5 g(u) S 1 (mod X). Define N = {u; g(u) _ 1}. Then p(N) = x(N) = p(N) + cp(N). Consequently )Y(N) = 0.

We set p(A) = p1(A) + */r(A) where q1(A) = p(A\N) and *(A) _ cp(A f1 N). It follows from the definition of *(A) that 2(r and p are singular. `Remembering that 1v = (p + p, we obtain from (3), JU.f(u) [1 - g(u)] q,(du) _ Uff(u)g(u)p(du)

MEASURE THEORY

78

for an arbitrary function f that is square-integrable with respect to the measure X. But this equation then holds for an arbitrary nonnegative Cam-measurable function. If we set f = X,1(1 - g) for u e N and f= 0 for u e N, we obtain cp(A`N) _ gpl(A) =

A\N

g(u)p(du) = JAg(u)p(du) ,

where g(u) = g(u)/[1 - g(u)] for u 0 N and g(u) = 0 for u e N. This completes the proof of the theorem. Corollary 1 (The Radon-Nikodym theorem). If rp and It are finite measures and a is absolutely continuous with respect to p, then is the indefinite integral off some Ft-integrable function

q (A) =

JA

(4)

f(u) it (du) .

The function f(u) is uniquely defined (mod p) by formula (4).

Formula (4) follows from (2) since #(A) = p(A 11 N) = 0 for If

p
J

Af(u)p(du) =

J

Afz(u),u(du)

for arbitrary A e e5, then g(u)p(du)

0

JA

for arbitrary A where g(u) = f(u) - f;(u). arbitrary positive s,

0=

In particular, for

g(u)ft(du) > s p{u; g(u) > e} ,

(u;g(u)>e}

that is, p{(u; g(u) > e} = 0. It follows from this that p{u; g(u) > 0} _ 0. Analogously, p{u; g(u) < 0} = 0 and g = 0 (mod p). This proves

the uniqueness (mod p) of the function f(u) in formula (4). Corollary 2. If q and p are finite measures on e, then q can be represented as a sum of the form p = Tl +'r, where p, < It and 1 It. This representation is unique. Corollary 3. For the measure It to be absolutely continuous with respect to p (where p < oo and p < co), it is necessary and sufficient that for arbitrary s > 0, there exists a 8 > 0 such that the inequality p(A) < 8 implies the inequality p(A) < s. The proof of this condition is obvious. The necessity follows from a property of an integral (cf. corollary to theorem 4 of Section 5).

6.

ABSOLUTE CONTINUITY OF MEASURES. MAPPINGS

79

Theorem 2. If fe and p are finite measures and q << p, then for every e -measurable function f(u), v

f(u)q (du) =

u

,f(u)

fi(du)

(5)

,

if one side of this equation is meaningful.

The class K of functions for which equation (5) holds enjoys the following properties: (a) It contains the characteristic functions of arbitrary Cs-measurable sets by virtue of the definition of the derivative drpldp; (b) it is linear and hence contains all simple functions; (c) it is closed under passage to the limit for an arbitrary nondecreasing sequence of nonnegative (or nonpositive) functions (see Section 5, Theorem 1). It follows from these three properties that K contains all Cs-measurable functions for which one side of equation (5) is meaningful. Let X and Y denote two sets and let y = f(x) denote a mapping of X onto Y (which we denote by X Y). This means Proof.

that to every x e X is assigned an element y = f(x) of the set Y and that for every element y e Y there exists an x e X such that y = fix). In the relation y = f(x), the element y (of the set Y) is called the image of the element x (of the set X) and the element x is called the preimage of the element y. If A is a subset of X,

the image f(A) of the set A is the subset of Y consisting of the images for all elements of the set A: f(A) = 1y; y = f(x), x e Al .

The set of all x whose images belong to B is called the preimage of the set B c Y and is denoted by f-1(B): f-1(B) = {x; f(x) e B} .

It is easy to see that for an arbitrary class /of J{sets {Ba}, J-1(U Ba) = U f-1(Ba), a a

-1(I a1 Ba) =

I

f

1

a

-,(B.)

In addition,

f-1(B) = f -'(B) .

Suppose that a or-algebra of sets

is defined on X. The mapping y = f(x) of X onto Y induces in Y a unique new ort

algebra of sets e f consisting of those sets B c Y whose preimages

are sets of the or-algebra e: e f = {B, f-1(B) e e}. We shall say that the Q-algebra e f is the image of e under the mapping f(x) and we shall let it be denoted by f(e). Suppose that a measure p is defined on e. By means of the

MEASURE THEORY

80

mapping y = f(x), we can transfer the measure from 0 to f(c

).

Obviously

To do this, we set lt'(B) = p[ J'-1(B)] for every B e f(s).

p' is a measure on f(s). Let us agree in this case to write pf-' and to call the measure p' the image of the measure p. Let Y denote a metric space. The mapping y = f(x) is said to be Cam-measurable if f(e) contains all Borel sets in Y. In this definition, it is sufficient to require that f(C5) contain some class of sets TZ that generates the a-algebra of Borel sets 0 since it

will then follow that f(c) : 0. The definition of a measurable mapping can be generalized as follows: Let X and Y denote two spaces with o'-algebras ex and defined for X and Y respectively. A mapping y = f(x) defined on X onto Y is said to be measurable (for given ex and Cam,) if c

f(ex) D %. The usefulness of this definition is related to the fact that a measurable mapping y = f(x) enables us to transfer the measure p from the o'-algebra e. on which it is defined to a given a-algebra Cry by using the device just described. We set p'(B) = The following theorem corresponds

pf-'(B) = p[ J'-'(B)] for B e C25y.

to the rule for change of variable in evaluating an integral. Theorem 3. Let y = f(x) denote a mapping of a space with measure {X, C25, p} onto Y and let g(y) denote an f(e)-measurable real function defined on Y.

Then

(6)

g(fx))p(dx) ,

JYg(y)!pf-1(dy) =

if at least one of these integrals is meaningful.

Since the class of functions g(y) for which equation 6 holds is linear and closed under passage to the limit for nonnegative nondecreasing sequences, it will be sufficient to verify this equation for the characteristic functions XB(y) of f(e)-measurable sets. Let B denote a member of f(C25). Then yXB(y)pf-1(dy) =

JBpf-'(dy) = p(f-1(B))

Jr-1cB p(dx)

= XXs[flx)]tt(dx)

,

which yields the desired result. 7.

EXTENSION OF MEASURES

At the heart of the axiomatization of probability theory is the concept of a probability space, that is, the space of elements and

7.

EXTENSIONS OF MEASURES

81

a probability defined on it. A probability space is not a simple mathematical object, however, and it is by no means always defined in elementary probability-theoretic problems and constructions. For example, in an elementary exposition of probability theory a

finite collection of random variables is characterized by a joint distribution function F(x1, x2, ..., Xn) = P{e1 < x1, $2 < x2, ..., n < xnJ .

There is no mention here of any a-algebra of sets or of countably additive probability. This section is devoted to the question: How can we construct probability spaces from more elementary objects? Definition 1. A nonnegative set function . defined on all subsets of a set U is called an outer measure on U if a.

X(0)

b.

X(A) 5 Yk X(Ak)

0, (1)

for an arbitrary finite or countably infinite collection of sets Ak covering A (that is, A C Uk Ak). The property expressed by inequality b is called the subadditivity

of the set function. We find that we can use the outer measure to define a nonempty a-algebra C25 of the sets such that the outer measure x will be a countably additive set function on C25.

Definition 2. A set E is said to be Caratheodory-measurable if for arbitrary A c U, x(A) = X,(A n E) + X,(A\E)

(2) of all

Theorem 1 (Caratheodory). The nonempty class C25 Caratheodory-measurable sets is a complete (with respect to X) a-algebra. The outer measure X on C25 is a measure. The following propositions are steps in the proof of this theorem. Let us agree throughout this section to refer to

Caratheodory-measurable sets simply as "measurable sets." Lemma 1. The sets 0 and U are measurable. Proof.

By virtue of property a of an outer measure, the

measurability condition (2) is trivially satisfied for the empty set and the entire space U since

AnO=O, AnU=A,

Lemma 2.

A\O=A,

A\U=0.

The complement of a measurable set is measurable.

MEASURE THEORY

82

Proof. If E is measurable, it follows from the equations An R= A\E and A\E = AnE, where A is arbitrary, that ) (AnE) + ) (A\E) = ,(AnE) + X(A\E) = M(A);

that is, E is measurable. Lemma 3.

If E is measurable, if A and B are subsets of U,

and if B and E are disjoint, then

x[An(EUB)]=,(AnE)+x(AnB). Proof.

(3)

Let us apply condition (2) with A represented here by

the set A

U B).

Since

[An(Eft B)]nE=AnE and [An (E U B)]\E = AnB , equation (2) reduces to equation (3). Lemma 4. If Ek, for k = 1, 2, , n, are measurable, if A and B are arbitrary subsets of U, if B n Ek = q, and if Ek n E. _ 0 ( f o r r, k = 1 , 2,

, n) whenever r z?L k, then

X[An(E1U ... UE.UB)]

=X(AnE1)+ ... +x(AnEE)+x(EnB)

(4)

The proof follows from (3) by induction. Lemma 5. The collection of all measurable sets constitutes an algebra of sets. Proof. Since E1\E2 = (E, U E2) by virtue of Lemma 2 it will be sufficient to show that measurability of E, and E2 implies measurability of E, U E2. Let us apply condition (2), twice, once with E replaced by E,: ).(A) = x(A n E1) + x(A\E1)

,

and once with E replaced by E2 and A rephlaced by A\E,: M(A\E1) = x[(A\E,) n E2] + X[(A\EJ\E2]

Since (A\E,) n E, and AnE, are disjoint and their sum is A n (E, U E2),

by using Lemma 3 we obtain M(A) = x(A n E,) + )[(A\E1) n E2] + x[(A\(E1 U E2)]

which proves the measurability of the set E, U E2.

It follows from Lemma 4 and 5 that the outer measure ) is an additive function defined on the algebra of measurable sets e. Lemma 6. If {Ek}, for k = 1, 2, , is a sequence of measurable pairwise disjoint sets (that is, Ek n E. _ 0 for k s), then

7.

83

EXTENSIONS OF MEASURES

X(A

(5) j x (A n Ek) nV k=1 Ek) = k=1 Proof. By virtue of the subadditivity of the outer measure,

X[An(UEk)]<EM(AnEk) k=1 k=1

(6)

On the other hand,

x[A

k=1Ek)] = n (tik=1Ek)] > ?[An (Li

Taking the limit as n

k=1

X(A f Ek)

in the last inequality, we obtain

(7)

x[An(VEk)]>EX(AnEk)

k=1

k=1

Inequalities (6) and (7) prove the lemma. Lemma 7. The class of all measurable sets constitutes a a-algebra. Proof. Let {Ek} denote a sequence of measurable sets. We need to show that Uk 1 Ek e Cam. Since

U Ek = E, U (EE\E1) U ... U (E\O' Ek) U .. .

k=1

k=t

/

we may confine ourselves to the case in which the sets Ek are pairwise disjoint. Suppose that U=1 Ek = Ek is measurable and therefore

(Uk=1 Ek) U

B. The set

Uk=1

X(A) =

4A n (U Ek)] + 4A\U Ek] k=1 X(AnEk) + 4A\iJ Ek] . k=1

Taking the limit as n

k=1

oc, we obtain (using Lemma 6)

Ek) %(A) >k=1j X(A n Ek) + (A\U k=1

.

Since inequality in the opposite direction always holds because of the subadditivity of the outer measure, the assertion is proved. Lemma S. Every subset N' of a set N of outer measure zero (X(N) = 0) has outer measure zero and is measurable. Proof. If N' c N and MN) the subadditivity. Furthermore,

= 0, then MN') = 0 because of

X(A\N') < MA) < x(A n N') + X(A\N')

,

X(A n N') = 0 ,

and consequently, X(A) = X(A n N') + X(A\N') = M(A\N')

which proves the measurability of the set N.

,

MEASURE THEORY

84

Theorem 1 is an immediate consequence of Lemmas 6-8.

Theorem 1 shows how we can use the outer measure of sets to construct a measure. However, the possibility is not ruled out that the 6-algebra of the sets that are measurable with respect to this measure is trivial, that is, that it consists only of the empty set and the entire space U. On the other hand, in most cases we are interested not in arbitrary measures but in measures that coincide with a given set function on some class of sets I that does not constitute a a-algebra. For example, let us look at the particular problem characterizing the statement of the question in the general case. Let F(x) denote a given distribution function of some random variable e. We wish to construct a measurable space J U, Cam, P} in which U is the real line (- Ca , oo ), 0 is the

complete a-algebra of sets containing all Borel subsets of the real line, and p is a probability measure on 0 that is consistent with the given distribution function F(x), that is, such that

P{e < x} = p{(- -, x)} = F(x) . The distribution function F(x) determines the probability that the value of the random variable 6 will fall in the left-closed rightopen interval [a, b):

P{a<$
following two conditions (we treat the empty set as belonging to VI): (a) The intersection of two left-closed right-open intervals is such an interval. (b) The difference of two such intervals is either such an interval or the union of two such intervals. Note that the interval function F[a, b), is additive; that is, if

a, < a, < .. < an, then F[ao, an) = F[ao, a.) + F(al, a.) +

.. + Flan-,, an)

The problem now is to find a measurable space {U, e, p} such that e contains all left-closed right-open intervals [a, b) and P[a, b) _ F(b) - F(a). Let us consider this problem in the general case. Definition 3. Let {m, t2} and {m', TZ'} denote two set functions such that IJ C Z' and m(.) = m'(i) for , e V. The set

function {m', V'} is then said to be an extension of the set function {m,

t}.

An extension of a given nonnegative set function as a measure can often be obtained by means of a device proposed by Lebesgue: First we construct from m an outer measure defined on all subsets

7.

EXTENSIONS OF MEASURES

85

of the space; then we use Theorem 1. Let {m, fit} denote an arbitrary nonnegative set function satisfying the conditions 0 e TZ and m(0) = 0. For arbitrary A we define

X(A) = inf {i m(Ok); Ok G R, U Ok D Al k=1

k=1

(8)

,

where the infimum is taken over all finite or countably infinite . If coverings of the set A by the sets Ok e TZ, for k = 1 , 2, A is not covered by a countable union of sets belonging to fit, we define

X(A)=-.

(9)

The function x is defined on the class U of all subsets of the set U.

The set function {X, U} is an outer measure.

Lemma 9.

Proof. Since m(0) = 0, it follows that %(Q) = 0. We must now show that the function x is subadditive. Suppose that

AcUAk. k=1 for at least one k, the subadditivity is obvious. Suppose that X(Ak) < - for every k. Then for arbitrary s > 0 we If X(Ak) = 00

construct for every k a sequence {Okr; r = 1, 2, to 2 such that

} of sets belonging

E m( kr) C ? (Ak) + Zk , U Okr :D Ak .

Then the set {Okr; k = 1, 2,

, r = 1, 2,

}

is

a finite or

countably infinite collection of sets that covers A, and we have

X (A) 5 kE E m(Okr) S kE X(A,) + s. r

Taking the limit as s --> 0, we obtain the desired result. Let us call the set function {X, U} defined by equations (8) and (9) the Lebesgue outer measure corresponding to {m,}. On the 2 of all X-measurable (in the sense of Caratheodory) sets, x is a measure. Let us agree to call it the Lebesgue measure in the present section. We now consider the question: When is {) , 2} the extension of a set function {m, 1Z}? We note that if

{m, fit} has an extension as a measure, m must be an additive function on TZ.

In many cases

Definition 4. A class of sets

t has the following structure: 1J

is said to be decomposable if

MEASURE THEORY

86

for arbitrary A, and AZ in R,

A,nA2e t, B

Al\A2 = u Ak , where A,* e qR, Ak n Ar = 0 for k = r . k=1

The concept of a decomposable class of sets is close to the concept of an algebra of sets. Specifically, a class of sets of the form REMARK 1.

B=(' Ak)nA, A;=1

where r is an arbitrary integer and the Ak are arbitrary sets in R, is an algebra of the subsets A. Deflnition S. A nonnegative additive set function not identi-

cally + - that is defined on a decomposable class of sets TZ is called an elementary measure.

Theorem 2. For a Lebesgue measure {X, 2} constructed from an elementary measure {m, R1 to be its extension, it is necessary and sufficient that for arbitrary A e fit the relations Uk=1 Ak D A, where each Ak e SR, imply

m(A) < I m(Ak) .

(10)

k=1

Proof of the Necessity. If {X, 2} is the extension of {m, Tt}, then X(A) = m(A) and inequality (10) follows from Lemma 9. Proof of the Sufficiency. It follows from the definition of outer measure that X(A) < m(A), if A e 1R. On the other hand if condition (10) is satisfied, then

%(A) = inf k=1 E m(Ak) > m(A)

,

from which we get MA) = m(A)

for every A e L2 (11) We must now show that all the A e R are measurable. From the subadditivity of the outer measure, we have for arbitrary A* e TR (12) (A*) < (A* n A) + M(A*\A) . Since A* n A= A0 and A*\A = Uk=1 Ak, where A. e TR (for j = 0, 1, , s), and since we may assume that the A. are disjoint, it .

follows that

'(A*) = m(A*) = Ek=0m(Ak) = ? (A* n A) +Ek=1m(Ak) ? x(A* n A) + ? (A*\A) .

(13)

7.

EXTENSIONS OF MEASURES

87

Comparing (12) and (13) we obtain

x(0*) = X(A* n A) + X(0*\4) (14) Let A denote an arbitrary subset of U. If X(A) _ oo we have X(A) > X(A n A) + X(A\0) .

On the other hand, if X(A) < . there exists a sequence of sets Ak e TZ such that A c u7=1 A and X(A) + s > k X(Ak), for s > 0. On the basis of (14) we have X(Ak) = %(Ak n A) + X(Ok\A) ,

from which we get x(A) + a > E x(Ak n A) + E X(Ak\A) _> x(A n,&) + x(A\A) k=1

,

k=1

since

AnOcU(AknA) and A\AcU(Ak\A) k=1 k=1

Thus since s is arbitrary, we have in all cases: X(A) > X,(A n A) + x(A\A)

.

Inequality in the opposite direction follows from the subadditivity of the outer measure. Consequently for arbitrary A c U and arbitrary A e J2, X(A) = X(A n A) + X(A\A),

that is, A is measurable. Thus lU c 9, and m(A) = X(A) on fit, that is, the Lebesgue measure {x, 2} is the extension of the elementary measure {m,

t}.

In connection with the solution obtained in Theorem 2 to the problem of the extension of an additive set function {m, Tt}, the question naturally arises: Is it possible to extend an elementary measure {m, fit} in another manner and obtain a measure distinct from the Lebesgue measure {X, 2}? In a certain sense the answer to this question is negative. We find that extension of an elementary measure to the smallest a-algebra a{fit} containing IDI is unique if it exists. We shall say that an elementary measure {m, 1J } is a-finite if U can be represented as the sum of countably many elementary sets each of finite measure. Theorem 3. The extension of an elementary a-finite measure {m, fit} as a measure {p, a{ J2}} is unique. Proof. Let A denote an arbitrary fixed set in TZ such that m(A) < 00. Let ao denote the class of sets of the form

MEASURE THEORY

88

for

B=UAkf1A, AkEV, AkflAj=0

k#-j.

k=1

of A. If { ul, o{ J2}} and elementary measure {m, t}, {u2, a{fit}} are two extensions of an then by virtue of the additivity of a measure, Then %,

is

the algebra of subsets

,uJB) _ E m(Ak fl A) =p2(B) p2(C). Obviously, Let denote the class A set 9 is a monotonic class and D a'. &nssque'ntly, A D a{%o}. Now E

let E denote an arbitrary set belonging to 6{J2}. The set E is covered by a countable disjoint collection of sets Aj c 9 each of finite m-measure. Consequently, for arbitrary i we have p,(E fl 0;)

p2(E fl 0,), so that p ,(E) _ p2(E) We shall now show the relation between the extension of an

elementary measure on a{JJ2} and the a-algebra 2 of all measurable sets.

Let W. (resp. tea) denote the class of sets that can be represented

as the union (resp. intersection) of a countable collection of sets belonging to J1 and let Was (resp. ''Q) denote the class of sets that can be represented as the intersection (resp. union) of a

countable collection of sets belonging to WQ (resp. '8). classes W., W8, W, Wba all belong to Q{fit}. Theorem 4. If (X, 2) is the Lebesgue measure that

The

is the

extension of a a-finite elementary measure {m, 9} satisfying condition (10), then for every set A E 2, there exists a measurable set E in X08 such that A c E and X(A) = X(E). Proof.

Since

X(A) = inf E m(Ok) = inf E X(Ok) UdkDA

UAkDA

there exists a sequence {Ok""} of coverings of the set A (that is, Uk-, Akn)

A, for each n) such that X(A) = lim E X(Ok') n--

k

Suppose that El') = Uk Okn> and E n) =

and

f}C c"

dbr e1r

nk=1

k=1

E. Then L, o

parvre

X(A) = Jim X(E(n)) _ X(E) ,

where E= nk=, Elk' is the limit of a decreasing sequence of sets and E e W... This completes the proof of the theorem.

7.

EXTENSIONS OF MEASURES

89

Let 9Z denote the class of all subsets of sets in 6{ JZ} that are of p-measure 0, where p is some measure defined on 6{9R}:

9Z = {N; NcE, Eec{ JZ}, p(E) = 0} , denote the class of all sets of the form A = (E U N')\N",

Let C25 where E e 6{Pt}, N' e W, and N" e 92.

In short, C is the class of

sets that differ from the sets in Q{JZ} by a subset of p-measure 0. Let us set #(A) = p(E). This definition is unambiguous: If A = (El U Ni)\Ni"= (E2 U NZ)\NZ', then E,\EZ C N1" U NZ and E2\E, c Ni U Nz'.

Since E,\E2 and EZ\E, belong to 6{9Z} we have p(E2\E,) = p(E1\E2) = 0 so that ft(E2) = ft(E). One can easily show that C25 is a v-algebra

and that ft is a measure on it. Definition 6. The measure {f, C25} is called the completion of the measure {p, o{ P }}.

Theorem 5. If a a-finite elementary measure {m, T2} satisfies condition (10), the completion {a C25} of its extension {X, orITZ}} onto the minimal or-algebra coincides with the Lebesgue measure {X, 2}.

Proof. E e 6(1JZ),

Since the a-algebra 2 is X-complete it follows that for

N', N" e 92, and A = E U N'\N", we have N', N" e 2

In other words C25 c 2. Suppose now that A is an arbitrary set of finite measure in 2. Then there exists an EDA belonging to 6{TZ} (in accordance with Theorem 4) and X(E\A) = and X(A) = M(E).

0.

Let us set E\A = F. Then there exists an E' D F belonging to

tr{9JZ} such that x(E') = X (F) = 0.

Thus

A= E\F, F c E', E' e or{ V}, (E) = 0, that is, F e W. In the general case in which A is an arbitrary set belonging to 2 we apply the last relation to the set A (1 Ok. This completes the proof of the theorem. These last theorems show that in a certain sense, measurable sets do not differ greatly from sets belonging to the minimal oralgebra 6{TZ}. Specifically, a measurable set A differs from some E e 6{ IJZ} by a subset of a set in c{TZ} that is of measure 0. The following result is a particular case, but it will be used in what follows. Theorem 6. Let U denote a complete separable metric space, let 0 denote the or-algebra of Borel subsets of U, and let p denote a finite measure on 0. Then for arbitrary A e 0 and arbitrary s > 0 there exists a compact set K, contained in A such that p(A\K,) < e. Proof.

Let us first prove the theorem for the particular case

of A = U. Let {xn} denote a countable everywhere-dense set in

MEASURE THEORY

90

, let {Sk(xn)} denote the sequence of U. For each n = 1, 2, closed spheres of radius 1/k (for k = 1, 2, ) with center at the point xn. For each k let us choose nk so that ,u(U1 sk(in)' > P(U)

- 2k

Define nk

Sk = U Sk(in) . n=1

The set Sk is closed and admits a finite (1/k)-net.

Let us set

KE=fls,. k=1

Then K. is closed and it admits a finite (1/k)-net for arbitrary integral k; that is, K, is compact. On the other hand, It(U\KE) < i I- (U\Sk) k=1

Thus the assertion of the theorem holds when A coincides with the entire space U. Furthermore, an arbitrary closed set F contained in a complete separable metric space is itself complete and separable. Consequently the theorem holds for an arbitrary closed set F = A. Let St denote the class of sets B that can be represented simultaneously in the form G, B= U Fn = n=1 n G. n=1

where the F. are closed and the G are open. The class 9 contains all closed sets and is an algebra of sets. Therefore a{ft} _ 58. Let St'* denote the class of sets for which the assertion of the theorem is valid. Let {Bn}, for n = 1, 2, . , denote a nondeer-eas-ing--sequence of sets belonging

91*.

Define

B0 =UBn, n=1 Let {K.} denote a sequence of compact sets such that K. C Bn, It(Bn\Kn) < s/2n}1 and li(B(,\Bn0) < s/2. n=1

so that B0 e S*.

Kn) G

Then

+ n=1

1i(Bn\Kn) <

In analogous fashion one can show that

B0= fl Bn e

if {Bn} is a decreasing sequence of sets. Thus S V

7.

EXTENSIONS OF MEASURES

91

is a monotonic class. Since it contains closed sets, R c W. Therefore This completes the proof of the theorem. R*

Let us prove a frequently used result dealing with the possibility of approximating an arbitrary measurable function by simple ones.

Theorem 7. Let 1fl denote a decomposable class of sets, let a denote a finite elementary measure admitting an extension ft to the complete o'-algebra 25, let f(u) denote a Cam-measurable function, and let

g,(u) denote a function that assumes only finitely many distinct values and that,, is measurable with respect to M. Then for arbitrary s > 0, (a) if f(u) is finite (mod ,u), there exists a g,(u) such that

I f(u) - g,(u) I > e} < s, and (b) if f(x) e L,{ U, C25, ft}, where r > 1, there exists a g,(x) such that uI f(u) - g,(u) I' ft(du) < E. fc{x;

Proof.

We note that the class of functions for which the

theorem is valid is linear. Since every measurable function f(x) can be approximated in the sense of convergence in measure and since every f(u) e L,{U, Cam, ft} can be approximated in the sense of the metric of the space L, by a simple Ce-measurable function, it

will be sufficient for us to prove the theorem for the characteristic functions of s-measurable sets. Let A denote a member of e and let XA(u) denote its characteristic function. Since

,u(A) = inf I fl(Ok) for UCkIA k

Ok e X12

,

there exists for given positive s a sequence {Ok} such that

t (A)k - /1(A) < 2 Let us now find an n such that

2

1

k

Let g,(u) denote the characteristic function of the set Uk-1 A k' Then {u; I XA(U) - 9E(u) I > 0}

C{U ok\A} u k=1

fl{x, I XA(u) - gE(u) I > 0} < t; and

I

JU

XA(U) - 9E(u)

I r p(du) < s,

which completes the proof of the theorem.

n }1

MEASURE THEORY

92

REMARK 2. Assertion (b) of Theorem 7 remains valid even without the assumption that the measure fl is finite. To see this, we note that iff N, = {u; I f(u) I > s}, fu(N,) < - and

I .f(u) I r p(du) = Jim U

I f(u) 1' fl(du) .

JN E

Therefore it suffices to approximate the function f(u) (in the sense

of the metric of Lr) on the set N. Let us now look at the extension of an arbitrary additive set function.

On a decomposable class I2 of sets consider' the real set function T (A) satisfying the conditions 1.

q(O)=0,

2.

qi(Ak), if A;, n A, = O (i p(U ok) = E k=1 k=1

t.

Let R(A) denote the class of all decompositions of the set A as the union of finitely many disjoint sets belonging to I and set p(A) = supRO E i q (Ak) I. The function p(A) is called the absolute variation of the set function. The absolute variation enjoys the following properties:

MA) > 0, p(O) _ 0, P(Ak), Ai n A, = O(i 2'. P(U Ak) k-1 k= 1'

j),

3'

that is, it is an elementary measure on fit. A function qp(A), where A e 93"1, is called a function of bounded

variation on 9t if ,u(A) < oo for arbitrary A e TI. Theorem 8. Let {q , 931} denote an additive real set function satisfying the following conditions: (a) it is of bounded variation on 931,

(b) the absolute variation ft(A) of the function T(A) is subadditive:

P(A) < Ek/--(Ak) f

if U kAk D A.

Then there exist two measures {v1, C25} and {iZ, C25}

such that 9 C e

and

q)(A) = vl(A) - v2(A) , II(A) = vl(A) + I (A) Proof. On the basis of the hypotheses of the theorem, the function p(A) can be extended as a complete measure {fl, C25}. We note that condition (b) stipulating the subadditivity of i is equivalent

to the condition that p is countably additive on M. From this it follows that the function q )(A) is also countably additive on fit. This is true because A = Uk=1 Ak and A. n Ar = 0 (for n :/- r). Let us set

8.

THE PRODUCT OF TWO MEASURES

93

o\Uok=UA k=1

k=1

where the AN are pairwise disjoint members of fit. Then N

'P(0) - Gi cP(Ak)

as N

-E

)y(AN)

k=1

= It

(A\U N

n=1

fi

0

-. We define on 9 the two nonnegative set functions P(0) , v2(A) _ (A) 2 'P(0)

v1(0)

It follows from the definition that p(0) = Vl(0) + VZ(A), T (A) = V1(L) - VZ(0)

Since p and T are countably additive on TZ, so are v1 and v2. Therefore v, and v, have Lebesgue extensions as measures {v1, (251} and {v2, 2JZ} respectively. Since vi(A) < p(0) we have C25 c (25, for

i = 1, 2.

This completes the proof of the theorem.

REMARK 3.

We cannot assert that satisfaction of the conditions

of Theorem 8 implies that a set function q has an extension onto the u-algebra of the sets. The function P = v1 - v2 C25

99

is

defined only on some class of sets C25o c C consisting of those A e C25 such that vi(A) < oo for i = 1, 2. However we can define the integral with respect to the set function iP for every Cam-measurable

and fl-integrable function f(u). We make this definition by means of the relation Jv.f(u)p-(du) =

Jf(u)v1(du) -

Jaf(u)v2(du)

(15)

Here we should remember that if f(u) is a fi-integrable function, it is also v;-integrable (for i = 1, 2). The integral with respect to the set function P defined by equation (15) is called the LebesgueStieltjes integral. 8.

THE PRODUCT OF TWO MEASURES

Let U and V denote two arbitrary sets. The set of ordered pairs (u, v), where u e U and v e V, is called the Cartesian product of the sets U and V and is denoted by Ux V. Here (u, v) _ (u1,v1)

if and only if u = u1 and v = v1. If V = U, instead of writing Ux U we may write U2. The simplest example of the Cartesian product of two sets is the coordinate plane, which can be regarded as the Cartesian product of the real axis with itself. If A c U and B c V then the set A x B is a subset of U x V. It is called

MEASURE THEORY

94

a rectangle with sides A and B.

If R denotes a class of subsets of U and R denotes a class of subsets of V, the product SR x R of these classes is defined as

the family of all sets of the form M x N for Me t and N e W. If 1Z = vl and R = (252 are two a-algebras of sets, their product (251 x e2 is not an algebra. However, it is a decomposable class of sets (cf. Definition 4 of Section 7). To see this, note that (M1 x N1) n (M2 x N,) = (M1 n M2) x (N. n N,.) (M1 x N1)\(M2 x N2) = [(M1\M2) x N.1 U [(M1 n M2) x (N1\N2)]

(1 ) Let p1 denote a measure defined on e1 and let p2 denote a measure defined on LJ2 Let us define on vl x e2 a set function m(0),

if A = A x B, A e 21, B e e2 . (2) In the case of an indeterminate form of the type 0- -, we define the value of m to be 0. m(0) = p1(A)p2(B)

Lemma 1. The function m(i) defined by formula (2) is an elementary measure on e1 x (252 satisfying the subadditivity condition m(Do) < E m(Ok) , k

if Do C Uk Ok ,

(3)

where

Ok e e1 x e2 f o r k = 0, 1, ... . Proof. We note that if 02 C Al where Ai = M% x Ni for i = 1, 2, by virtue of formula (1) we have

O1=02UA'UA", A'=(M1\M2) x N1,

(4)

A" = M2 x (N1\N2)

where 02, A', and A" are pairwise disjoint. Furthermore. m(Al) = p1(Mju2(Nj = p1(M2)li2(N2) + p1(M1\M2)li2(N) +

m(O2) + m(0') + m(0") .

Thus the function m(0) is additive with respect to these particular decompositions of the set Ol. In particular, if Al = A. U,&3(A2 n 03 = 0) , then

m(O) = m(p2) + m(p3) .

Additivity in the general case is obtained by induction. If

A=UAk, k=1

8.

THE PRODUCT OF TWO MEASURES

95

where the rectangles Ok are pairwise disjoint, then 0\A, = Al U o",

where 0' and 0" are determined from formula (1). From what we have shown,

m(0) = m(On) + m(0') + From the induction hypothesis,

m(0') = mCk=1 u (0' n Ok)) =

k=1

m(0' n z

)

It then follows that n-1

m(0) = m(O&) + E m(0' n Ok) + m(0" n Ok) k=1

n-1

= m(OR) + k=1 E maw u A,,.] n &k) n-L

= m(0.) +

k=1

mR+

m(Ok)

k=1

m(Ok) .

This proves the additivity of the function m(0). Let us now prove the subadditivity property (3). Suppose that Do C Uk=1 Ok where Ak = Mk x Nk for k = 0, 1, , and suppose that xk(u, v) = xk(u)xk'(v) where xk(u, v), xk(u), and xk'(v) are the

characteristic functions of the sets Ok, Mk, and Nk respectively. Then Xo(u1 v) -:5k=1E Xk(U, V) =k=1 E xk(u)xk (v)

If we integrate this inequality with respect to It, we obtain x0 (v)f l(Mo) _ Ek=1xk (v)f'"1(Mk)

Then integrating with respect to p2, we obtain m(Ao) = LL Z(No) ul(MO) k=1 G L P" (Nk) u1(Mk) = E m(Ak) k=1

which completes the proof of the lemma. Theorem 1. Let {U, el, pl} and {V, `J2, 12} denote two measurable

spaces and let e, denote the smallest 6-algebra containing 251 x %. There exists a unique measure v defined on 0, that satisfies the condition

v(M x N) = 111(M)112(N), M e e,, N e 252.

(5)

Proof. The proof follows from Theorems 2 and 3 of Section 7 and the preceding lemma. Definition 1. The measurable space {U x V, e,, v} is called the product of the measurable spaces J U, (251, p1} and IV' v2, 11J1

MEASURE THEORY

96

the o'-algebra C23 = a{251 x e2} is called the product of the o-algebras e1 and e2; and the measure v is called the product of the measures u, and u2 (and we write v = p, x u2). Let the completion of the u-algebra 25, (with respect to the measure v) and the completed measure be denoted by 252 and v respectively.

Let E denote an arbitrary set contained in U x V. Definition 2. The set of all v e V for which the point (u, v) belongs to E for fixed u e U is called the u-section of the set E and is denoted by E... We write E.. = {v; u, v) a E, u fixed}. The v-section of the set E is defined analogously. We indicate it by E.,, and we write E,,, = {u; (u, v) e E, v fixed}. Now consider a function f(u, v) defined on U x V. Definition 3. A function of a single variable (v e V) defined by the equation f.(v) = f(u, v) where u is fixed is called the u-section of the function f(u, v). We write fu.(u, v) = f,.(v).

The v-section f .(u, v) = f (u) of a function flu, v)

is defined

analogously.

Lemma 2. Let {U, e,, pl} and {V, e2, u2} denote two measurable spaces and let {U x V, (253, v} denote their product. Suppose that the measures p, and p2 are finite. Then for any (253 measurable set E a.

for arbitrary u the set E.. is COQ measurable and for arbitrary

v the set E.,, is e, -measurable, b. p2(E..) is an e,-measurable function of u and pl(E.,,) is an s. -measurable function of v, c.

v(E) =

JU

p2(E,4.)p,(du) =

up,(E.)p2(dv) J

(6)

Proof. Let 9 denote the class of sets for which the theorem is valid. Obviously, St contains rectangles and finite unions of

them and 9 is a monotonic class of sets (on the basis of Lebesgue's theorem on the integration of monotonic sequences of functions, Theorem 2 of Section 5). Since finite sums of rectangles constitute an algebra of sets (cf. Remark 1, Definition 4, Section 7), it follows

on the basis of Theorem 3 of Section 1 that R contains e,. This completes the proof of the lemma. Lemma 3. Suppose that {U, e,, pl} and {V, e2, p2} are measurable sets with complete finite measures. Let {e3, v} denote the completion of the measure {e3, v}. The assertion of Lemma 2 remains in force for p,-almost-all u and p,-almost-all v and for an arbitrary 5-measurable

set E.

S.

THE PRODUCT OF TWO MEASURES

97

By virtue of Theorems 4 and 5 of Section 7, for a there exists a set E' e e., such that E C E' and v(E) = set E e v(E'). Applying this conclusion to the set E = (U x V)\E, we obtain a set E, e e, such that E. D E and v(E1) = v(E). Setting E" = (U x V)\E1, we see that E" c E, E" e \.J3, and v(E") = v(E). Proof.

Thus

E" c E c E', E', E" c e3 v(E") = v(E) = v(E') . Let us apply equation (6) to the sets E' and E". We obtain JU}p2(E'.) - li2(Eu')}!p1(du) = 0, ft2(E'.) > 1u2(Eu") ,

from which we see that jt2(Eu:) = jt2(Eu.) for p1-almost-all u. Since

p2(Eu) <

1t2(E .) ,

this triple of functions is v(E) = v(E") =

a1-equivalent.

JU J

Thus

1t2(E')1t1(du) _

1t2(E..),u,(du) .

Reversing the roles of the measures p1 and jt2, we obtain proof of the lemma. Theorem 2 (Fubini). Suppose that {U, e,, tl} and {V, 22, p j are measurable spaces. Let J U x V, e3, v} denote their product. Suppose that the measures It, and /u2 are finite. Let f(u, v) denote an arbitrary e3 measurable v-integrable function. Then for all u(v), the section f,.(u, v) (resp. f v(u, v)) of the function f(u, v) is (252 measurable (resp. c1-measurable) and f(u, v) v(du x dv) =

=

fu..(u,v)Ii2(dv)

Jp1(du)

J vft2(dv)

f ,(u, v) p1(du) U

(7)

If the measures {e,, jtl} and {e2, ft2} are complete, the assertion of measurable function for the theorem remains valid for an arbitrary (i1-almost-all u and for jt2-almost-all v. Proof. It is sufficient to consider the case in which the function f(u, v) is nonnegative. Proofs of the general assertions of

the theorem are carried out in the same way except that in proving the first assertion we use Lemma 2 and in proving the second we use Lemma 3. Let us confine ourselves to the first assertion. Formula (7) is valid (a) for the characteristic functions of e3 measurable sets (according to Lemma 2), (b) for linear combinations

of them, and (c) for the limits of nondecreasing sequences of

MEASURE THEORY

98

nonnegative simple 03 measurable functions. On the basis of Remark

2 following Theorem 3 of Section 2, formula (7) is valid for an arbitrary e3 measurable function. This completes the proof of the theorem. REMARK 1.

Formula (7) holds for nonnegative functions f(u, v)

without the assumption of integrability when the other conditions of Theorem 2 are satisfied. REMARK 2. Theorem 2 remains valid for o'-finite measures p1

and p, that is, for all cases in which the spaces U and V are unions of a countable collection of measurable sets of finite measure.

The concepts of the product of a-algebras and measures can be generalized to finite or even countable products. Let U1, U2, denote n sets. Their Cartesian product, denoted by

, Un

n

II Uk = U1 X U2 X .

X U.

k=1

is defined as the set of all ordered n-tuples uk

(u1, u2,

is an arbitrary element of Uk for k = 1, 2,

,

u,,) where

, n.

Strictly

speaking, the formation of Cartesian products is not an associative operation. For example, (U1 x U2) x U3 is the set of all possible pairs (z, y3) where z is a pair of the form z = (y1, y2). This set is different from the product U1 x U2 x U, and from the product U, x (U, x U). However between these three spaces there exists a natural isomorphism ((y11 y2), y3) t--, (y11

y21

y3) t--, (y1, (y2; y3))

which enables us to identify them. Keeping this in mind, let us

agree always to consider products of sets as defined up to this isomorphism and to treat the Cartesian product of sets as an associative operation.

If n sets Uk = U for k = 1,

,n

coincide, the Cartesian

product n

k=1

HUk=Un

is called the nth power of the set U. {pe(k,-,n), If measures {ppk, (24} are defined on Uk, measures can be defined elk, --*,%)I and their completions inductively on the space {fi(k......

11Ur

r=k

Here elk "-1 denotes the minimal o'-algebra containing sets of the

8. THE PRODUCT OF TWO MEASURES

99

This construction is

form Ak x Ak+l x . . x A. where A, e Cam;. associative, and x ,/(k+1,k+2,...,n)

(8)

Formula (8) follows from the uniqueness of the extension of an elementary measure as a minimal 6-algebra. Let us suppose that the function f(u1, u2, , un) defined on x U. is &1'"''n)-measurable and nonnegative. On U1 x U2 x the basis of Fubini's theorem it follows that with fixed (u1, for fl(1'"''k)-almost-all (u1, ..., u,), the function ff(ul,

is &k}1,

, Uk, Uk+1,

n)-measurable.

, u,,) = JUl,u2,...,,lk(uk+l, uk+2,

This follows from the fact that n) D

(&1,---,k) x (k+l, ..., n))

and

f(ul'

, un)

/(...... .)(d(u1, u2,

{u1

U1x...XUk

. , un)

Uk+1X...XU,1

x (d`uk+l' ..., un))

, U.))

-' uk(uk+ls . . .,

Jf"(1'...,k)(d(ul,

..., Uk))

, uk)

III

AXIOMATIZATION OF PROBABILITY THEORY

1.

PROBABILITY SPACES

Let U denote a set and let C25 denote a a-algebra defined on U. Let P denote a complete measure defined on e such that p(U) = 1. Then the triple {U, Cam, P} is called a probability space. Definition 1.

Here the elements of the set U are called elementary events, the set U itself is called the space of the elementary events, and the subsets of U that belong to e are called events. The space U, treated as an event, is called the certain event; the empty set 0 is called the impossible event. If A and B are two events such that A c B, we say that event A implies event B. If A (1 B = 0, the events A and B are said to be incompatible. The event A = U\A is called the opposite of the event A. For every A e e, the quantity P(A) is called the probability of the event A. Let us enumerate a few of the simpler properties of events and their probabilities that follow immediately from the definitions and the properties of measures that we have already proven. 1. The sum and intersection of finitely or countably many events are events. 2. The probability of the certain event is 1; the probability of the impossible event is 0; the relation A c B implies P(A) < P(B). 3. If A is an event and A is the opposite event, then P(A) _ 1 - P(A). 4. If {An} is a finite or countable set of pairwise incompatible events, then P(U, E p(An). 5. If 1A.1 is an increasing (resp. decreasing) sequence of events, that is, if A. implies A,,, (resp. of An+t implies An), then

n

lim p(A%) = P(U n=1

lim p(A%) = p((j A%)) / (resp. \ nn=t

Definition 2. An arbitrary finite (mod P) Cam-measurable real 100

1.

PROBABILITY SPACES

101

Two random variables 1 = f (u)

function is called a random variable.

and 2 = f2(u) are considered equal if they are equivalent: E1 =

2

if fl(u) = f2(u) (mod p).

For every random variable e, a function F(x) of a real argument x, known as the distribution function of the random variable E, is unambiguously defined by F(x) = p(E < x} = P{u; f(u) < x}. The distribution function has the following properties: is 6. The distribution function F(x) of a random variable nonnegative, nondecreasing, and continuous from the left; also

F(- o°) = 0 and F(+ o°) = 1. Analogously, if {Ek = fk(u), k = 1, 2, , n} is an n-tuple of random variables, the function (of n real variables) F(x1, x2, ...,

n-1

r

P{S1 < x1, ..., Sn < x,} = PLn {u, fk(u) < xr} EE

k

is called the joint distribution function of the variables Ek for k = is uniquely 1 2, , n. Obviously the function F(x1, x2, , defined in n-dimensional Euclidean space En, is nondecreasing, and is continuous from the left with respect to each variable. Furthermore, F(x1, x2,

. .

. , Xk, - -I xk+2, . . . , xn) = 0

and

F(x1, ...,xk, +°°, ..., + oo) = F(k)(x1, ...,xk), , xk) denotes the distribution function of the k-tuple of random variables e1, E2. ,k where F(k)(x1,

All of the results obtained in Chapter II can now be repeated in probability-theoretic terms. We confine ourselves to a few examples. Let q(t1, t2, . . .,

denote a Borel function in n-dimensional denote n random variables. ,

Euclidean space and let e1, e2, Let us define 1 7"(E1, E2,

.

/ . ., en) = gm(f ( 1(u),

.

. ., ,f (u)) = F(u)

where Ek = fk(u) for k = 1, 2, , n. Note that if we replace the function fk(u) by an equivalent (mod P) function fk(u), the function F(u) is replaced by an equivalent function (mod P). On the basis of Theorem 5, Section 2, Chapter II, we have 7.

,

If a Borel function

q'(E1,

, En) of random variables

is finite (mod p), then it too is a random variable. In particular, the sum and product of finitely many random variables is a random variable. If a random variable E # 0

1,

AXIOMATIZATION OF PROBABILITY THEORY

102

(mod P), then l/ is also a random variable. Let {fin} denote a sequence of random variables. It is always meaningful to speak of the event S that the sequence {E%} will con-

verge to a finite limit, and to speak of the probability P(S) of that event (see Section 3, Chapter II). Definition 3. If p(S) = 1, that is, if the sequence of functions n = fn(u) converges P-almost-everywhere, we say that the sequence converges almost certainly (or that it converges with probability 1).

such that Definition 4. If there exists a random variable I > s} , 0 as n - for arbitrary s > 0, the sequence {fin}

P{I gn -

is said to converge in probability to the random variable , that is, = P-lim e, If a sequence {sn} converges almost certainly, it converges in probability (cf. Theorem 1, Section 3, Chapter II). The converse of random variables is not generally true. However a sequence that converges in probability contains a subsequence that converges almost certainly (cf. Theorem 4, Section 3, Chapter II). 8.

A necessary and sufficient condition for a sequence of

random variables to converge in probability is that for arbitrary s > 0 and 3 > 0 there exists an no = n0(e, 3) such that the inequalities n' > n. and n > no together imply the inequality P{I -,

- $.I > 6} < U

.

This condition is called the Cauchy condition with respect to the probability of the sequence {sn} (cf. Theorem 5, Section 3, Chapter II)

If P-lim C / = Y J ; for j = 1 , 2, , s, then for an arbitrary , t8) where - 00 < t j < 00 for j = continuous function (p(ti, t2, 9.

1, 2,

, s, the sequence of random variables

converges in probability to a random variable cp(721, 7),,

, 7)a) (cf.

Theorem 6, Section 3, Chapter II). Definition 5. The integral Me _

f(u)P(du) ,

e = f(u)

U

if it` is meaningful, is called the mathematical expectation of the random variable and is denoted by MC.

On the basis of Theorem 5, Section 4, Chapter II, the value of the mathematical expectation is independent of the function f(u) representing the random variable . The mathematical expectation is a functional defined on some subset of random variables -and

1.

103

PROBABILITY SPACES

enjoying the following properties: 10. M(aE + (3)7) = aME + 8M72, where a and 8 are constants and at least one of the quantities Me, my) is finite. ,

11.

is,

If xA is the characteristic function of the event A, that

if x4(u) = 1 for u e A and XA(u) = 0 for u V A, then MXA = P(A). 12. If c < )7 (mod P), then Me < Mrs. 13. For an arbitrary sequence of nonnegative random variables

Is .1, n},

MEEk=X, Mek k=1 k=1 (cf. Theorem 1, Section 5, Chapter II). 14. If I e,, 1 <)7 for n = 1, 2, , if My)< -, and if the sequence of random variables en converges in probability to C, that is, C = p-lim e., then Men, = MC (cf. Theorem 5, Section 5, Chapter II).

We shall now prove an important inequality that gives a bound for the probability of an event in terms of mathematical expectation. 15. For an arbitrary nonnegative Borel function f(x) that nondecreasing on [0, oo) and an arbitrary random variable ,

is

p{IeI > a} < MA C1) f(a) This is known "Chebyshev's inequality." Proof follows from the inequality f(I $ I) > .f(a)x[,,,,;(1 C 1), where XI.,- )(x) is the characteristic

function of the infinite left-closed interval [a, -). We now mention another inequality that is well known in analysis. 16. If g(x) is a continuous convex (downward) function for all real x and if e is a random variable with finite mathematical

expectation, then Mg(C) > g(MCC)

This is known as "Jensen's inequality." The proof can be found in the book by Hardy, Littlewood, and Polya. Let J U, C5, p} denote a probability space and let X denote an arbitrary set with fixed o-algebra of the sets B. Definition 6. Let C = f(u) denote a function defined on U into the set X and suppose that for arbitrary B e 0, {u; f(u) e B} E Cam. Then the function = f(u) is called a random element with range in X.

In other words,

= f(u)

is

a random element if under the

AXIOMATIZATION OF PROBABILITY THEORY

104

mapping u , x = f(u), we have f(Cs) i 0. In the case in which X is a metric space, we always understand by 0 the o-algebra of Borel subsets of X.

A random element with range in a finite-dimensional linear space is called a random vector. 2.

CONSTRUCTION OF PROBABILITY SPACES

Let {FB1,82,.... ,,(x1, x2, , xn)} where n = 1, 2, and 0k e O for k = 1, 2, , n, denote a family of distribution functions satisfying the compatibility conditions (cf. Section 1, Chapter I) and describing a random function in the broad sense. Is it possible to

define a probability space J U, Cam, p} and a family of random variables

e = fe(u), u e U , B e O, on it in such a way that the joint distribution function of the sequence {eBl, a2, , den} coincides with the given function F91,9.....0,a(x1,

, xn) for arbitrary n = 1, 2,

Ok e O, where k = 1, 2, .., n? The following problem is less difficult. Let F(x1, a distribution function of n random variables 1, c2,

, x%)

and denote

, e,4 and let A denote an arbitrary n-dimensional Borel set. How should we

define the probability of the event {e1, e2, - , fin} e A? A precise statement of this problem follows: Let F(x1, x2,

, x,b)

denote a distribution function of n independent variables, let E. denote n-dimensional space, and let e denote the u-algebra of all Borel subsets of E. How can one construct a probability space {E,,, C25, p} in such a way that F(a1, a2, ...,

P(Ia1,a2,...an)

for arbitrary real ak (for k = 1, , n)? (Here, Ia1,d2, the n-dimensional orthant of the form

,a

denotes

Ia1,d2,...,d,L = {(x1, . . ., x,); x1 < a1, ..., x < an} .)

We begin by solving the latter problem. Let a, b, x, , where a = (a1, a2, , denote the points , n. in the set E,z. Let us write a < b if ai S bi for i = 1, 2,

We shall refer to the set I [a, b) = {x; ai < xi < bi, i = 1,

, n},

where a < b, as an n-dimensional left-closed right-open interval or, briefly, as a left closed interval. Let a denote the class of all left-closed intervals. This class constitutes a decomposable family: I [a, b) n I [c, d) = {x; a, < xi < bi, ci < xi < di, i = 1, ... , n} = {x; max (ai, ci) < xi < min (bi, di)} = I [a', b') ,

2.

CONSTRUCTION OF PROBABILITY SPACES

105

and

I [a, b)\I[c, d) = {x; x, e [a;, b;)\[c;, d;)}

which is the sum of a finite number of left-closed intervals. Let us define the probability that a random point (el, e2, , en) will fall in a particular left-closed interval. We introduce the notation '&Lak,bk)F('x)

= F(xl, ..., xk-1, bk, xk+1, ..., x,) - F(x1,

.

, xk-1, ak, xk+i,

. , x,,)

Because of the monotonicity of F(x) with respect to each variable, of the variables x1, the function O1E xk-1, xk+1, .. xn is nonnegative and nondecreasing with respect to each variable. The probability-theoretic meaning of the quantity 0[akbk)F(x) is as follows: This quantity is the probability that the inequalities 6l < xl, . ., ek-l < xk-1, ak C ek < bk, ek+l < xk+l, ... , en < X,a Will be satisfied. We obtain by induction F(I[a, b)) = 11,b1)OC?2 b2) ... A[a;);bn)F(x) ? 0 , (1 ) where F(I[a, b)) is the probability of the event (el, e2,

, en) e I(a, b).

The quantity F(I[a, b)) can also be written Co

s!0°

F(I [a, b)) _

(ti,

n

t1,...,tn=0

(-l)'' F[b - t(b - a)]

,

(2)

where

t = (tl, ...,

t(b - a) = [t1(b1 - a), .. , tn(bn - an)]

The function F(I[a, b)) is an additive function on a, because if we partition I [a, b) into two subintervals 1[a, cl) and I [c1, b) where c1 = (c1, a2, , an) with a1 < c1 < b1, then

F(I[a, b)) =

a161)

. DEa;,,bn)F(x)

62) ...

0[a1

+ D(1) [o1,b,)

(2)

(a2,b2)

[a,)y,b,)F(x)

, , O(n)

F(x)

[a,t,bn)

= F(I[a, c1)) + F(I [c1, b1))

The same relation holds if the left-closed interval I [a, b) is partitioned

into two left-closed intervals by partitioning any one of the sides [ak, bk) into two parts. One can show by induction that the function F is additive for an arbitrary decomposition of LEa, b) into a union of left-closed intervals.

For the function F to be extendable as a measure on some a'-algebra of subsets of En, it is necessary and sufficient that it be subadditive (cf. Theorem 2, Section 7, Chapter II):

AXIOMATIZATION OF PROBABILITY THEORY

106

(3)

EF(Ik) k=i

F(I)

for an arbitrary system of left-closed intervals Ik (for k = 1, 2, .) such that Uk=1 Ik D I. Let us verify that this condition is satisfied in the present case. Let I[ak, bk) be denoted by Ik and let I[a(,, b0) be denoted by I. Since the function F(x) is continuous from the left, there exists an S. = (sk, , sk), where sk > 0, such that 0 < F(I[ak - sk, bk)) - F(I[ak, bk)) < 2-k'/

where 77 > 0 and k = 1 , 2,

The open intervals (ak - sk, bk)

.

cover the closed interval [a0, bo-s]. From the Heine-Borel theorem, the collection of these open intervals contains a finite subcovering, for example, {(ak - 8k, bk)} for k = 1, , n. Then the set of leftclosed intervals (Ca-1 - sk, bk)} for k = 1, .. , n covers the left-closed interval [a0, bo - s). Consider the collection of disjoint sets k-1

[a0,b0-e)n{[ak-sk,bk)

[ai-e,ba)},

k = 1,...,n

each of which is the union of nonintersecting left-closed intervals A;k) (for j = 1, 2, - , mk). Thus n

mk

[a0, bo - s) = U U 'V) k=1 j=1

and n

mk

F(I [ao, bo - s)) = k=1 E Ej=1F(Dik>) n

< E F(I[ak - sk, bk)) <

,

k=1

k=1

Taking the limit as s

F(I[ak -

s,, bk)) < E F(Ik) + 19 k=1

0, we obtain

F(I[ao, bo)) ; E F(Ik) + 77

,

k=1

and since )7 is arbitrary this proves inequality 3 and also: , Theorem 1. For an arbitrary distribution function F(x1, F(x), it is possible to construct a probability space {En, Cam, p} such that , x,) is the joint distribution e contains all Borel sets in En, and F(x1,

function of the random variables S1 = f (x) = x1, e2 = fz(x) = x,,,

,

n = fn(x) = X.Let us turn now to the general problem posed at the beginning of the section. Consider an arbitrary family of compatible distribu, x,,) where Bk e O for k = 1, , n. tion functions Fe1 e2 ... en(x1, As we have just shown, for arbitrary n in n-dimensional space

2.

CONSTRUCTION OF PROBABILITY SPACES

107

(which is denoted by E9...... en) we can define a a-algebra eel,,...0n

containing the Borel sets of the space and we can construct a probability space {E0 .., e", eo...... 0", P9........) such that p91 ... e"(Ix) will

coincide with the given distribution function F9...... 9 (x). Now the problem is to construct a single probability space {U, Cam, p} and a family of random variables ea = fe(u) for u e U, such that the joint distribution of the variables 61, e82, , a" coincides with the function p01,... B"(Ix) = Fe1,...,0 (x).

Let us generalize the statement of

the problem. Let X denote some metric space and let 0 denote the a-algebra of Borel subsets of X. Let X" denote the nth power of the space X and let &'") denote the a-algebra of Borel subsets of X". Fur-

thermore, let us assume that for every positive integer n and for where the 0h (for k = 1, 2, , n) are arbitrary points belonging to some set O, a probability measure arbitrary B B21

, 0,

P0.....,0 (A(n)) A(n) a 0(n)

(4)

is defined on and the family of measures (4) satisfies the following compatibility conditions (cf. Section 1, Chapter I): a. b.

0, 02,

Pe

,... e

e

e

(A n) x X(,") = Pe

.... 0 (A'")) .

(5)

Let S denote an arbitrary permutation of the elements , 0, . Let S also denote the transform in the space X"

that permutes the coordinates xk of a point (x1, x2, of the permutation S. Then P0...... 9

Definition 1.

(A(n))

, x") by means

= Ps(e...... en)(SAW) ,

(6)

Let {U, Cam, p} denote aAprobability space and let

g(0, u) denote a function defined on O x U with range in X. Sup-

pose that g(0, u) is s-measurable for every fixed 0 e O and that the finite-dimensional distributions of the random function g(B, u) coincide with the given family (4); that is, for every A(") e 58("), Piu; (g(01, u), g(62, u), ..., g(Bn, u)) a AO} = P01,...,0 (A("))

(7)

Then the probability space { U, Cam, p} and the function g(8, u) are called the representation of the family of distributions (4). We shall

show that under rather broad assumptions a compatible family of distributions (4) admits a representation. The role of the space U

will be played by the space 12 of all functions defined on O into X, and the elementary events will be the functions of 0 (that is, u = (0(0)), and the function g(0, u) = co(0). Definition 2. Let 2 denote the space of all functions u = co(B)

defined on a set O into a metric space X. Let AW denote a Borel

AXIOMATIZATION OF PROBABILITY THEORY

108

subset of X4. The set of functions w(0) e fI such that the point {w(01),

, w(0.)} in Xn belongs to A("); that is, the set Co1,...,on(A1n') = {o (0); (co(B), cv(82), .. , w(0 )) e Ac")}

is called a cylindrical set in fl with base A(n) over the coordinates , Bn, or simply a cylindrical set. Bi, 02,

Let us make some remarks concerning cylindrical sets and operations on them. If the number n and the points Bi, e2, , Bn are fixed, there exists an isomorphism between the cylindrical sets over the coordinates B1, , Bn and the Borel subsets of X": Every Borel set A(n) e Xn defines a cylindrical set Cg1,...,gn(A()) for which it serves as base. Different cylindrical sets correspond to different bases.

The sum, difference, and intersection of cylindrical sets

correspond to the sum, difference, and intersection of bases. follows immediately from the definition of a cylindrical set.

This

With regard to operations on cylindrical sets in the general case, we need to keep in mind that a fixed cylindrical set can be defined by means of different coordinate systems. x X'). Thus, obviously, Ce i"'n o Co e n'e ntv"'ntr It is easy to see that any two cylindrical sets C = Co...... 0,(A( )) and C = CBi,...,6;.(B(-)) can always be regarded as cylindrical sets over a single sequence of coordinates B , BP containing both B1, BE, . , on and 0', 0 ,1, , 0.. From this it follows that we may consider algebraic operations on a finite number of cylindrical sets as defined on a fixed coordinate sequence. Therefore we have 1

Theorem 2. algebra of sets.

The class (9 of all cylindrical sets constitutes an

To this we need to add that if O contains infinitely many points and X contains at least two points, then IS is not a Q-algebra. To

see this, note that the set U Cok({x'k})

k=1

where {xk} (for k = 1, 2, cylindrical set. We shall now prove:

) is a sequence of points in X, is not a

Theorem 3 (Kolmogorov).

Let X denote a complete separable

metric space. The family of distributions (4 satisfying compatability conditions (5) and (6) admits a representation.

Let us define a set function p'(C) on the algebra f of cylindrical subsets of the space a by setting p'(C) =poi ,, Bn(An>), Proof.

2.

CONSTRUCTION OF PROBABILITY SPACES

109

if C is a cylindrical set with {A(n)} over the coordinates Bi, 02,

, On.

The conditions of compatability ensure, uniqueness of the definition of the function p'(C). Let {Ck} for k = 1, 2, , n denote an ntuple of cylindrical sets. We may assume without loss of generality that these sets are defined by the bases A$p) over a single p-tuple of coordinates Bi, 0Z, , 0 . Algebraic operations on the sets Ck correspond exactly to the same operations on the sets AkP>. Since the measure pB...... 0,(A1P)) is countably additive in XP, it follows that

the set function p'(C) is finitely additive on E. We now extend the function P'(C) from the algebra (9 to a measure p on some a-algebra . For us to be able to apply Lebesgue's method, it will be sufficient (on the basis of Theorem 2, Section 7, Chapter II) to verify that for every C e E and an arbitrary covering {Ck} for k = 1 , , n, (where Ck e (9) of the set C (thus we are assuming CC uk=1 Ck), the inequality

p'(C) < E p'(Ck) k=1

(8)

is valid.

Let us show that if E Ck = C (C E 1.., Ck E c, k = 1, 2, ...) k=i

and Ck fl C,. = 0 for k ? r, then

(9)

p'(C) = E p'(Ck) k=1

Inequality 8 will then follow for an arbitrary covering of the cylin-

drical set C by sets in & Let us set C U Ck = D,,

.

k=1

These sets D. constitute a decreasing sequence of cylindrical sets and the intersection of all these sets is empty: n=1

D,, = CECk = QS k-1

(10)

It follows from the additivity of P' that n

p'(C) = S p'(Ck) + P'(Dn) k=1

To prove (4), it suffices to show that limn__ p'(D,,) = 0. Let us assume the opposite, that is, that

lim p'(D,) = L > 0

.

(11)

AXIOMATIZATION OF PROBABILITY THEORY

110

Let B. denote the base of the cylindrical set

D,,.

Suppose

that D. is situated over the coordinates e1, 02, , Bmn. Here we assume that with increasing n, the set of corresponding points 01, 02, .. , 0,,, does not decrease. As shown above, this assumption does not restrict generality. For every B,,, there exists a compact set K. c B. such that L

Pet ...,emn(Bn\Kn) < 2n}1

n = 1, 2, .. .

,

(cf. Theorem 6, Section 7, Chapter II). Let Q. denote a cylindrical set over the coordinates 01, 02, , 0mx with base Kn. Define

G. = r=1 n Qr Let Mn denote the base of the set Gn. Obviously Mn is compact since it is the intersection of closed sets, at least one of which, K,,,

is compact.

Since the sets G. constitute a decreasing sequence, it follows that if w(0) a Gn+,(p > 0), then (o(0) e G,,. From this in turn it follows that if {x1, x2, . . , x,,, . . , xmn+n} e Mn+,, (p > 0), then , xmn} a M,,. The sets G. are obviously nonempty. Fur{x1, x2, thermore, since n

n

D,.\Gn = r=1 U (Dn\Qr) C r=1 U (Dr\Qr)

we have P'(D,.\G,.)

C

E P'(D,\Qr) _

T=1

r

(Br\Kr) C 2L

from which it follows that from lim P'(G,) = lim P'(D,) - lim p'(D,\Gn) > L n-Ice

n-+oa

.

n-Ieo

For every set Mn, let us choose a point {xi"), , xm'}. On the basis of what was said above, for arbitrary k the sequence {x(")}, n = 1, 2, belongs to a compact set in X' and the sequence r) },

p = 0, 1, 2, , is contained in M,,. By means of a diagonal process let us find a sequence of indices n; such that (xin+'°),

,

for each k the sequence {x( n7)} converges to some limit x(°). Since the X(() } e Mn. set Mn is closed, it follows that for arbitrary n, M' Let us define a function w(0) by setting cw(O,) = xk) for k = 1, , n and then extending its definition to the remaining points in an arbitrary manner. Then for arbitrary n we have w(0) e GncD,,. Consequently, the set nn-1 D. is nonempty, which contradicts (10).

3.

INDEPENDENCE

Thus, inequality (11) is impossible and lim P'(D") = 0

Consequently, the function p' satisfies inequality (8) and admits an extension as a complete measure (p, l ) such that CE D Z. Let us define the function g(0, (o), for (o e dZ and 0 e O, by g(0, w) = (o(0). For an arbitrary Borel set Al") c X" and arbitrary n, 01, 02, .., 0,,. we have P{(g(01, w), g(0,, w),

= P{(w(01), w(0,),

, g(B", (o)) e A(")} ., w(B")) e Al")} = PO...... e"(AI"))

Thus we have constructed a representation of the family of distributions (4). This completes the proof of theorem. 3.

INDEPENDENCE

Let {U, Cam, p} denote a fixed probability space. We recall that two events A and B are said to be independent if p(A (1 B) = p(A)P(B).

This definition has the following immediate consequences: The events U and A are independent for arbitrary A; if p(N) = 0 and

A e e, then N and A are independent; A is independent of A if and only if p(A) = 0 or p(A) = 1; if A and B are independent, then the events A and B are also independent; if p(N) = 0 and the events A and B are independent, then the events A U N and B are independent, as are the events A\N and B. Definition 1.

Suppose that {Ax}, for X e A, is an arbitrary class

of events and A is an arbitrary set of indices. The events A2 are said to be independent (or mutually independent) if for an arbitrary k-tuple of distinct indices, p(Atil n Az2 n ... n A,) = p(A.)p(A;Q) ... p(A.)

.

We can make the same remarks regarding a sequence of independent events that we did for two independent events. In particu-

lar, if we replace an arbitrary finite or infinite set of events A. in a sequence of events {A"} with their complementary events, the new sequence of events will also be independent. Let {A"} denote an infinite sequence of events. The event lim A. was defined above (see Section 1, Chapter II) as the set of those elementary events contained in infinitely many of the events A". What can we say about the probability of the event lim A"? Theorem 1. If Zn=1 P(A") < oo, then the event limA" has prob-

AXIOMATIZATION OF PROBABILITY THEORY

112

ability 0.

The proof follows from the formula

limA.=n UAk, n=1 k=n on the basis of which p(lim An) = lim

n__

k=n

Ak//

< lim j P(A,,) = 0 n-m k=n

For a sequence of independent events, this theorem can be strengthened by the following remarkable fact: Theorem 2 (Borel-Cantelli). Let {An} denote a sequence of independent events. Then the probability of the event Urn A. is 0 or 1 according as the series n=k

p(A,)

converges or diverges.

It suffices to show that the relation

E p(A,) = -

n=1

implies p(lim An) = 1.

If A* = lim An, then

U\A* = n=1 U flk=n(U\Ak) and

P(U\A*) = limp( n (U\Aki) n__

k=n

/

= lim n P(U\Ak) = lim l (1 - p(Ak)) = 0 nee. k=n

n-- k=n

by virtue of the divergence of the series Ek=1 P(Ak). Let us generalize Theorem 2 and the concepts we have presented to classes of events. Definition 2. Let {P2} (for .. E A, where A is an arbitrary

set) denote a family of sets of events. The sets 1fl are said to be (mutually) independent if an arbitrary class of events {A,}, consisting, for each X of a single event A, e 7t, is a class of independent events.

Theorem 3. Suppose that 912 are decomposable classes of events, that U e 9, (cf. Section 6, Chapter II), and that the fit, (for % CA) are independent.

Then both the sets of minimal a-algebras a{922} (for

3.

INDEPENDENCE

113

X e A) generated by the classes of events Wx and their completions 6{ R } (for X e A) are independent. Proof. Let Ttzo denote the set of all events A such that the events {A, Ax; X e A, X 7t- Xo} are mutually independent. Let us list

certain simple properties of the set 0xo: a. ,, is a monotonic class of events (cf. Definition 8, Section 1, Chapter II). This assertion is a simple consequence of the definition of independence of events. b. A, U A2 e

If A, and A. belong to 0zo and if A, fl A2 = o, then This is true because if p(Ati fl A21 fl ... fl A2) = p(A;)p(A2) ... txo.

for i = 1, 2, then

P([A1UA2IflA2.fl ... flAj = P(A, fl A,, (1 ... fl A,,) + P(A2 fl A21 fl

...

fl A2 )

= P(A, U A2)p(AA1)p(A22) ... p(Azn)

As we see, property b and property a are independent of the special structure assumed for the sets Wx (for . e A, X # Xj. c.

91zo C Tt.

Let us now look at the set N of events that can be represented , n, where n is in the form B = C1 U . . U C, for C; = 1, arbitrary, and the C; are incompatible events in VZz. N is an algebra of sets. On the basis of property b we have N c 7txo. On the

basis of property a and Theorem 3, Section 1, Chapter II, the set Tjzo contains the a-algebra Q{%zo}. Since addition of events of probability zero to an event A e 7txo or removal of such events yields an event in txo, the set 6{ xo} is also contained in 9Jtzo. This completes the proof of the theorem. Theorem 4. Let ez, for X e A, denote a set of independent ualgebras and suppose that A = Al U A2 (A1 fl A2 = 0). Let i. = a'{ez, x e A%} for i = 1, 2, denote the smallest a'-algebra containing all the a-algebras e2 for X e A. Then the a-algebras 01 and 02 are independent.

Proof. We note that the events n1

n2

C'=uBk and C"=uB,', k=1

k=1

where mm

/

Bk = 1 I Akr' Bk = I I Akr r=1 r=1

and

Akr (A")

AXIOMATIZATION OF PROBABILITY THEORY

114

for some

are _ Xk, e Al (resp. X = Xk, e A2) belongs to mutually independent. The collection 0i°) (resp. 01 )) of all sets

of the form C' (resp. C") constitutes an algebra of sets. On the basis of Theorem 3, the smallest a-algebras containing 0i°> and O(G) are independent. On the other hand, 6{o;°>} = 8, for i = 1, 2.

This completes the proof of the theorem. REMARK 1. It follows from what has been shown that if A is partitioned into an arbitrary collection of disjoint subsets A. (for a e A), the 6-algebras (for a e A) uje2, ? e Aa} constitute a set of independent a-algebras. Let us now look at an arbitrary sequence of independent 6algebras e for n = 1, 2, . On the basis of the Borel-Cantelli theorem, every event A that is contained in infinitely many of the A e e. has probability either 0 or 1. This result can be generalized to arbitrary events generated by the collection of all a-algebras e., for n = 1 , 2, , independently of the n-tuple of Q-algebras el, L12, Let us make this assertion more precise. Let ,Cam,,. Q{vk, ek+1, ... , v,y, ... } = Ok

denote the smallest o'-algebra containing e for n = k, k + 1, Their in.

The Ok constitute a decreasing sequence of a-algebras. tersection

o=n o , k=1

is also a o'-algebra.

Let us define

lim k=1

Q{LJk, vk+l, . . . }

Obviously, the 6-algebra lim e. is unchanged by replacing an arbitrary finite number of Q-algebras e1, , e. with other 6-algebras. 1, 2,

Theorem 5 (Kolmogorov's general 0-or-1 law). If e., for n = , are mutually independent o'-algebras, then every event in

lim e. is of probability 0 or 1. Proof Let A denote a member of lim Ce,,. Then A e 58k for arbitrary k. Consequently, A and o'{e1, , e,_1} are independent. Hence A and c{l.J1, , en, } are also independent. Since , e,,, }, A is independent of A. But this is possible A e o'{251, only when p(A) = 0 or 1. Let us now turn to the question of independent random variables. -

3.

INDEPENDENCE

115

Definition 3. Random variables C, (for X e A) are said to be (mutually) independent if the sets D? (for x e A) are mutually inde-

pendent, where fit, consists of all events of the form {u; C, < a},

where - co < a < oc. The definition of independence of a set of random variables is equivalent to the following: Random variables C, (for X e A) are independent if for arbitrary n and arbitrary X; e A (for j = 1, n),

, ,n is the joint distribution function of the variables C,1, C,Q, equal to the product of the distribution functions of the variables Caj

P{bat
II P{C,.
The definition of independence of a set of classes of random variables is analogous. Let us consider a set of random variables {Cx, . e A,,} where It is a fixed index and x ranges over a set A,, depending

on the index p. For convenience, let us call this set a class and let us look at the set of such classes indexed by p where p ranges over a set M. Definition 4. The classes of random variables {C-, x e A,,} for It e M are said to be (mutually) independent if the sets of events MP

(for It e M) are mutually independent, where 0,, consists of all events of the form

{u; Ca, < a, ..., z <

n= 1,2,...,X;eA,,,-oc
1

)

Definition S. A a-algebra of events a{C,, X e Al generated by events of the form {u; Cal < a1, ... , Ca,, < a.} ,

n = 1, 2, , X, G A, - oo < a; < oo , is called the a-algebra generated by the class of random variables for X e A. Let 6{C,, X e Al denote the completion of the a-algebra a{C ), e Al. Obviously, o {C,, X E Al is the minimal a-algebra of events with respect to which all the C, are random variables (that is, the minimal a-algebra of sets with respect to which all the functions C, = f2(u) are measurable).

We note in particular that the a-algebra of events generated by a single random variable C is a minimal a-algebra containing events of the form {u; C < a} where - oo < a < oo . Theorem 6. If the classes of random variables {C12', x e A,,}, for

AXIOMATIZATION OF PROBABILITY THEORY

116

p e M, are independent, then the sets of a-algebras a{(a, ? e A,,}, for It G M, and their completions Proof.

X e A,,} are also independent.

Let L1t,, denote the set of events of the form

G Ija, b) , where Ija, b) is an arbitrary n-dimensional left-closed interval (cf. Gz`l, cp22 , ... ,

Section 2), where the components a, and b; of the vectors a and b can assume the values ± -, where n is a natural number, and where the X; are arbitrary members of A. It follows from Definition 4 that P

k y / [(k, ... 1

I

=i

y , Sink) a

7

bk)J

11 P{(Cal, ..., Sink) C Ink) [ak, bk)} k=1

This means that the fit,, are independent sets of events.

Since these

are decomposable families of events, Theorem 3 can be applied, and the Theorem is proved. Corollary. Let {g,,(t1, t2, , t8)}, for p = 1, 2, , denote a sequence of arbitrary finite Borel functions of s real variables. If the sequences of random vectors C,, = {C , , are , C; 'J, for p = 1, 2, mutually independent, then the random variables = g, (l , ,B) are also mutually independent.

The concept of independence of random variables and the theorems proved are easily generalized to random elements in an arbitrary measurable space X. We shall not repeat all the preceding

formulations and theorems. We note only that when the Z1 are random elements, we should consider in the preceding exposition not events of the form {u; C21 < a, ... , Ca < but events of the form {u; C21 e Al, ... ,

xe

where the Ak are sets in the a-algebra B of subsets of X (cf. Definition 6, Section 1). Thus, for example, the random elements bk = fk(u), for k = 1, 2,

, n, with values in Xk are said to be independent if for arbitrary Ak e 0k (where the 0k are the given a-algebras of subsets of Xk) we have Pl k 1 {bk e Ak}I = 11 P{Ck e Ak} k=1 l

Each of the random elements

bk

defines a measure on Ok:

3.

INDEPENDENCE

117

ftk(A) = Pfk 1(A) = P{Ck e Al ,

A e 8k

,

k= 1,

.

,n.

Let us now look at the sequence of random elements (el, e2, It is easy to see that this sequence can be regarded as a random element with values in H n=, Xk. Specifically, let 01n' denote the product of the 6-algebras :LJl, , 0,,. If C = A, x A, x x An, for A; e B;, where i = 1, , n, then .

, en).

{u; (f (u) ...

(2)

e Cl = n {u; i(u) e Ail , i=l

that is, the preimage of C is C25-measurable.

From this it follows

that the preimage of an arbitrary set in the minimal a-algebra containing all C (cf. Section 6, Chapter II), that is, the preimage of an arbitrary set in 0(%', is ce-measurable. Let f denote the measure in rjk=l Xk induced by the n-tuple (e1, f,/" l........ n(C) = P{(e1,

, ell):

.., ,s) e C} .

Let us assume that the elements ei, for i = 1, Formula (2) shows that

, n, are independent.

pl,2....,n(A, x A2 x ... x An) = ftl(A1)p2(A2) ... pn(An)

Since the extension of the measure from a decomposable family of sets (cf. Theorem 3, Section 7, Chapter II) to the minimal v-algebra is unique, the measure P1,2..,..n is the product of measures Ill, P29 , p,,. The converse is true: if the measure f-t1,2,...,,coincides with the product of the measures P1, p, , fpn, then the random elements ell e2, .. , en are independent. From this we get: Theorem 7. The random elements el, e2, , $n are independent if and only if the measure P1,2,...,n induced by the n-tuple (el, e2, , fin)

on the 6-algebra 0cn' = {01 x ... x 0.1 is the product of measures

p; (for i = 1,

, n) induced by the elements ej on O;.

Theorem B. Let g(xl, x2) denote a &&21-measurable finite function, and let e, and e2 denote independent random elements. Suppose that M I g(el, e2) 1 < - . Then T(x) = Mg(x, e2) is a 01-measurable function of x and Mg(e1, e2) = MT(e1), or Mg(e1, S2) =

J X1

pl(dxl)L g(x1, x2)p2(dx2) . X2

The same formula remains valid for an arbitrary !&') -measurable

function, where the tilde - denotes the completion of the a-algebra (measure) if the measures 11, and P2 are assumed complete.

Theorem 8 is a direct consequence of Theorem 3, Section 6, Chapter II. On the basis of Theorem 8 and Fubini's theorem,

AXIOMATIZATION OF PROBABILITY THEORY

118

Mg(e1, S2) =

g(x1, x2)1-t,,,.(d(x,, x2))

x1 x x2

Corollary. If e1 and e2 are independent random variables with finite expectation, then Me1e2 = Me1Mee2

Let X denote an arbitrary space. Suppose that an infinite sequence of measures 1.tn, where

defined on some Is it possible to construct a 1, is

or-algebra 0 of subsets of X. probability space {U, e, P} and to define on it a sequence of independent random variables C,, with values in X such that the When measure induced on 0 by the element C. coincides with X is a complete separable metric space, a positive answer to this question is contained in Kolmogorov's theorem (Theorem 3, Section 2). We note that in this case it is easy to prove the corresponding theorem with no restrictions on the set X. We shall now give some consequences of the general 0-or-1 law (Theorem 5) for random variables.

Theorem 9. Let {C,.} denote a sequence of independent random variables and let e = a{C.} denote the or-algebra generated by

Define e- = o'{c, n+l,

}.

Then

exists either with probability 1 the limit of the sequence or with probability 0; b. the limit of the sequence {Q is constant with probability 1; , X. ) is a function of an infinite number c. if z = f(x1, of variables x1, , x,,, , and if f(C1, , C,,,, ) is Can-measurable for arbitrary n, then it is constant with probability 1. a.

Proof.

(a)

The set of points of convergence of the sequence

{Q (cf. equation (1), Section 3, Chapter II) belongs to lim e.. (b) Similarly, for arbitrary a, the set {u; lim C. < a} belongs to lim e.; that is, for arbitrary a, we have P{lim C. < a} = 0 or 1. (c)

This result is obtained from the same reasoning as in (b).

4.

SERIES OF INDEPENDENT RANDOM VARIABLES

In this section we consider the conditions under which a series of independent random variables converges. We begin by giving some important inequalities that indicate bounds for the probability that the maximum partial sum of a series of independent random variables assumes a great value. The idea of obtaining these inequalities is Kolmogorov's. In what follows we always assume that for n = 1, 2, , is a sequence of independent random variables,

4. SERIES OF INDEPENDENT RANDOM VARIABLES

119

we define n++

Sn = G ei yy

ro=1

where n = 1, 2, . , and we assume o = 0. We let X(A) denote the characteristic function of the event A. Theorem 1 (Kolmogorov's inequality). Suppose that M. = 0 for n = 1, 2, Let g(x) denote an arbitrary continuous nonnegative convex-downward function that increases for x > 0. Let t denote a

.

positive number.

Then

p{mas I bk I > t} <

I)

(1 )

Mggl(C)

Proof.

Let Ak, for k = 1, 2,

, n, denote the event

yy Ak = { I S1 I < t, ..., bk-1 I < t, I

I bk I

t}

.

The events Ak are pairwise incompatible, and A = Uk=1 Ak is the event I Ck I > t}. We have y

n

Mg(I C. I) > M k=1 E X(Ak)g(I C.

_ Ek=1MX(Ak)g(I Ck + G. - bk) 1)

Since the variables ek are independent, we have (from Theorem 8, Section 3) MX(4k)9(I Ck + G. - bk) I) = MX(Ak)gk(Ck)

where gk(x) = Mg(I x + bn - Ck 1). Applying Jensen's inequality (inequality (17), Section 1) and remembering that the function I x is convex //downward, we obtain. yy

((

yy

gh(x) = Mg(I x + yy bn - yy Sk I) \f- g(M I x + bn - Sk I)

>g(I x+MGn - Ck)I) =g(Ix1) Thus n

n

Mg(I C. I) > M Ek=1X(Ak)gk(Sk) >_ M E X(Ak)g(I Ck I) k=1

M E X009(t) = g(t)p{ma x I Ck I> t}

(here we use the fact that if the event Ak occurs, then I Ck I > t), which completes the proof. Let us note a special case of Kolmogorov's inequality. Let us set g(x) = x2.

Then n

P 15ksn max ICkI?t < a E6k k=1 where o'2k = Mek = Dek

(2)

AXIOMATIZATION OF PROBABILITY THEORY

120

In the case in which we do not assume the existence of any mathematical expectations instead of Kolmogorov's inequality we can use:

If

Theorem 2.

P{ICn-bkIa fork=0, 1, ,n, then {max I Ck I> 2t} <

1

a

a

(3)

Proof. We define Ak = {I C, I < 2t, ...,

yy 2t, bk I > 2t}

yy I

bk-1 I

I

and

Bk = {I C. - Ck I <- t}

for k = 0, 1, ...,n. Then

C={I C.I > t}:U(AknBk), k=1 , n, are pairwise incompatible, and , n) are independent.

the events Ak, for k = 0, 1,

the events Ak and Bk (for n fixed, k = 0, 1, 2, Therefore,

1 - a> P{ICnI > t} > P{U (AknBk)} k-1

p(Ak)p(Bk) > a E P(Ak) = aP{max I Ck I > 2t} 15ksn J k=1

k=1

which implies inequality (3). The following inequality sometimes enables us to find a lower bound for the probability of the event {maxl t}: Theorem 3. If Mek = 0 and if I Ck I < c with probability 1, then p{max I lsksn

bk I

(cn+

< t} <

(4)

t)2 2

Uk k=1

where Qk = M k = DSk Proof. Let En denote the events {maxosk S. I Ck I _< t} for n This is a decreasing sequence of events. We have . 0, 1,

=

y

y

MX(EE)C2 = k=1 Z M{X(Ek)bk - X(Ek-1)Sk-1}

+

fy

n

y

yy

_ !k=1 MX(Ek-1)(bk - bk-1) - k=1 E M{X(Ek-1\Ek)Ck}

(5)

Furthermore, MX(Ek-1\Ek)yyk

n k=1

= MX(Ek-1\Ek)(yy bk-1 + Sk)2 1y n+ EE

(t + C)2Mx(Ek-1\Ek)

MX(Ek-1\Ek)bk c (t + C)2 L MX(Ek-1\Ek) k=1

= (t + c)2[1 - p(E,)]

(6)

4.

SERIES OF INDEPENDENT RANDOM VARIABLES

121

Also, MX(Ek-1)(bk - Sk-1) = MX(E,-1)(2Sk-1ek + Ck) y = 2MX(Ek-1)bk-1MCk + MX(Ek-1)MCk = akMX(Ek-1)

(7)

Relations (5) to (7) yield t2P(En)

MX(En)C'. ? E akMX (Ek-1) - (t + c)2(1 - P(E )) k=1

P(En)

Qk + (t + C)2} _ (t + C)2

or

(t + C)2

p(EJ E ak + C2 + 2ct } fk'=l

JJJ

from which inequality (4) follows.

We turn now to the question of the convergence of a series of independent random variables. We know from the preceding section (Theorem 9) that such a series converges either with probability 1 or with probability 0. Theorem 4. If Wk = 0, Dcck = ak, and

Z ak < k=1

(8)

then the series of independent random variables (9)

k k k=1

converges with probability 1.

Proof. Define C. = D=1 Ck. series (9) diverges. We define

Let N denote the event that the

EM.r=1SUp Ibk - Sk,I>r 1} k,k'?m Then

N = r=1 U Mn E.., Em,r From Kolmogorov's inequality we obtain p(N) = lim lim p(Em,r) < lim lim r2 r-boo M-,m

r-.oo m-boo

k=m

ak = 0.

In the general case, the question of convergence of the series (9)

is completely answered by:

(9)

Theorem 5 (Kolmogorov's three-series theorem). For the series of independent random variables to converge, it is necessary (resp.

AXIOMATIZATION OF PROBABILITY THEORY

122

sufficient) that for every (resp. some) c > 0, the three series

In=1P{Ienl > c} ,

(10) (11)

Men ,

(12) n=1

converge, where n = en for I n I < c and n = 0 for I n I > c. Proof of the Sufficiency. According to Theorem 4, the series

E (en - W1.) n=1 converges with probability 1. Since the series (11) converges, it follows that the series Jn=1 n converges. The series

En=E n=1

I

n=1

where;;

has only finitely many nonzero terms on the basis of condition (10) and the Borel-Cantelli theorem (Theorem 2, Section 3). Therefore the series (9) converges with probability 1. Suppose that the the series (9) converges.

Proof of the Necessity.

Then the sequence of its terms converges to zero with probability 1 since only finitely many terms in the series exceed the number c in absolute value. Therefore the series Jn=1 n converges with probability 1. Let {77n}, for n = 1 , 2, , denote a sequence of independent random variables that do not depend on the sequence {fin} and that have the same distributions as do the Let us setn =n - /n Then the series converges with probability 1; also, Din = 2Den .

I ek I < 2c ,

Men = 0 ,

The convergence of the series E;6=1 n implies that P1 max

15n5

n+ _ k=1

k

Therefore, for some t, P{ max 115nS

k=1

k

0.

It follows from inequality (4) that for arbitrary n,

2E DEk = k=1 E k=1

(2c

t)z

+ a

which proves the convergence of the series (12). It then follows

5.

ERGODIC THEOREMS

123

from Theorem 4 that the series E7=1 ;, -

coverges with probability 1. This in turn implies convergence of the series (11). On the basis of Theorem 2 of Section 3, the series (10) must converge, can exceed c only for a since when the series (9) converges, I

I

finite number of values of n. This completes the proof of the theorem.

Corollary. For the series (9) of independent nonnegative random variables to converge, it is necessary (resp. sufficient) that for every (resp. some) c > 0 the series E P{ek > c}

,

n=1

E Wk n=1

converge.

Proof. For nonnegative random variables n, we have M < Therefore convergence of the series (11) implies convergence

cMe'..

and hence of the series (12).

of the series

5.

ERGODIC THEOREMS

Let e(t) denote a stationary sequence, that is, a stationary random function that assumes real values and that is defined on the set Z of all integers (t = 0, ±1, ±2, - ). Let RZ denote the space of all real sequences w = {a(t), t e Z} and let 11 denote the a-algebra generated by the cylindrical sets R. The sequence e(t) defines a measure pF(C) on (9: Pe(C) = P({e(t); t o Z} c C), C G E

Let {PF, ,} denote the completion of the measure pF. In RZ we define an operation S representing time displacement: w' = So) if al(t) = a(t + 1) for t e Z, where w = {a(t); t e Z} and w' = {a'(t); t e Z}. The operation S has an inverse S-1:

S-'w = w"

,

w" = {a"(t); t e Z} ,

a"(t) = a(t - 1)

The condition for stationarity of the sequence e(t) means (cf. Section

1, Chapter I) that for an arbitrary cylindrical set C, PE(C) = Pe(SC) .

(1 ) Since a measure on cylindrical sets uniquely defines a measure on 1 and on its completion C, equation (1) remains valid for arbitrary AE(ge: PF(A) = Pe(SA) ,

Ae$

.

(2)

AXIOMATIZATION OF PROBABILITY THEORY

124

Definition 1. Let in, e, Ii} denote a space with a measure p and let T denote a measurable mapping of In, Cam} into itself. The mapping T is said to be measure preserving if for arbitrary A e Cam, p(T1A) = ,u(A) where T-'A, is the complete preimage of the set A. A mapping T is said to be invertible if there exists a measurable transformation T-1 such that TT 1 = T -IT = I, where I is the identity mapping. In this case the mapping T-1 is called the inverse of T. The definition of a stationary sequence is equivalent to saying that a sequence {E(t); t e Z} is stationary if the time-displacement operator S preserves the measure PF in Rz. Thus the problem of studying stationary sequences is included in the problem of studying measure-preserving invertible transformations (automorphisms) of some space with a measure.

Let us now look at the question of the asymptotic behavior of the mean _1 1

n k=o

-Ef(Tk

(o),

n-->°°

(3)

where Tk is the kth power of the mapping T, where f(w) is an arbitrary Cs-measurable function, and where {S2, e, p} is a space with measure p such that I.t(&I) = 1. To understand the meaning

of this problem, let us look at the case in which in, e, ti} is our Pf} and T = S. Suppose that f((o) is XA(e(0)), where XA(x) is the characteristic function of the Borel set A on the real space {Rz, line.

Then f(Tk(o) = XA(Sk(o) = XA(e(k)) and

-1 -nEk=o .f(Tkw) =

yn( ' w n

,

where v,,(A, (o) is the number of terms in the sequence e(0), e(1),

(4) ,

e(n - 1) with values in the set A; that is v (A, w)ln is the frequency with which the first n terms in the sequence e(t) (for t = 0, 1, fall in the set A. Thus the question we must consider is the question regarding the frequency with which the random variable E(t) assumes values in an arbitrary set A. Let us show first of all that the limit as n -p - of the mean (3) exists with probability 1. This

)

assumption constitutes the substance of the famous Birkhoff-Khinchin theorem. Lemma 1. If T preserves the measure It, if D e e, and if f(w) is a Ce-measurable nonnegative (p-integrable) function, then f

JT-1Df(Tw)tt(dw) = Df(w)li(dw)

(5)

5.

ERGODIC THEOREMS

Proof.

125

If we set f((o) = XA((o), formula (5) becomes the equa-

tion ,u(T-1(A n D)) = P(A n D)

,

which is valid for arbitrary A and D e e.

From this it follows that it is valid for arbitrary Cam-measurable nonnegative and p-integrable functions.

The following lemma is of an elementary arithmetic nature. , an denote a sequence of real numbers and let p denote an integer. We shall say that a term ak in a sequence {ak} is p-distinguished if at least one of the terms

Let al, a2,

ak, ak + ak+l, ... , ak + ak+l + ... + ak+P-1 is nonnegative. Thus ak is 1-distinguished if and only if it is nonnegative.

Lemma 2. Proof.

Let

The sum of all p-distinguished elements is nonnegative. ak1

denote the smallest p-distinguished element of

+ ak1+r, where r < p - 1, ak1 + ak1+l + denote the nonnegative sum with the smallest number of terms. For h < r, we have

a sequence, and let

ak, + ak1+1 + ... + ak1+k < 0

+ akl+r > 0, that is, all terms of the sequence ak1, ak1+l + ... + ak1+r are p-distinguished, and their sum is nonnegative. We can extend this reasoning by considering the sequence beginning with the term ak1+r+1 Thus every sequence is Consequently, akl+k+l +

broken into finite sums of p-distinguished terms, and each such sum is nonnegative. The set of p-distinguished elements in the entire

sequence coincides with the union of the sets of p-distinguished elements in the parts chosen. This completes the proof of the lemma.

The following lemma constitutes a basic step in the proof of the Birkhoff-Khinchin theorem. Lemma 3. Let f((o) denote a p-integrable function and define n1

E = (o; 3n for which 0 f(Tkco) > 0, n = 1, 2, k=o

Then

Ef(oj)p(d(o) > 0 Proof.

Consider the

(6)

.

sequence f(m), f(T(o),

,

f(TN+P-1(o).

Let s((o) denote the sum of all p-distinguished elements of this sequence. On the basis of Lemma 2 we have s((o) > 0. Define

AXIOMATIZATION OF PROBABILITY THEORY

126

Dk = {w; f(Tkco) is a p-distinguished element}, for k = 0, 1,

N + p - 1. Let Xk((o) denote the characteristic function of the set Dk. We note that n-1

Do = {co; sup Z f(Tkw) > 0 and Dk = T-ID,_1 for k < N . k=0

9V

Therefore Dk = T -'D0 for k< N. Therefore

0<

N+p-1

s(w)ft(dw) =

Z f(Tw)Xk(w),u(dw)

n

N+p-1

E

D

f(Tkw)p(dc)) .

On the basis of Lemma 1, (1n'

J Dk

Do

o

N+p-1

N

D

f(w)fi(dw) + k

Z

1

D

/

II .f(w)f"(dw) ,

,I f(Tkw)p(dw) = T_kDf(Tkw)p(dw)

k

N.

.f(Tcw)p(dw) >_ 0.

(7)

k

Since f(Tkw)Ft(dw)

JD k

I f(Tkw) I l,i(dw) _ QI fw) I fu(dw) < 0

by dividing inequality (7) by N and then letting N approach

oc,

we obtain

(8)

f(w)p(dw) > 0 . Do

The sets D. = Da(p), for p = 1, 2,

, n,

constitute an increasing

sequence:

lim D0(p) = p=1 U D0(p) = E P--

.

Taking the limit in (8) as p - o, we obtain (6), which completes the proof of the lemma. (Cf. also, A. Garsia, A simple proof of E. Hopf's maximal ergodic theorem, J. of Math. and Mech., 14, 381-382 (1965). Ed.) Lemma 4 (The maximal ergodic theorem). Suppose that f(w) is a ,a-integrable function, that ). is a real number, and that 1 n-1

E2 = co; 3n such that -) f(Tk(o) > n k=o

Then

L f(w)p(d(o) > X,"(E2)

(9)

5.

ERGODIC THEOREMS

127

This is proved by applying Lemma 3 to an appropriate choice of the function f. Theorem 1 (Birkhoff-Khinchin). Let {12, Cam, ft} denote a space with a measure, let T denote a measurable p-measure-preserving mapping of {12, C25} into itself and let fl w) denote an arbitrary i-integrable function. Then the limit 1 n-1

lim -E f(Tk(o) = f* (w) (mod ,u)

(10)

n-- n k=o exists ,u-almost-everywhere in E2. The function f*((o) is T-invariant; that is, f *(Tw) = f *(w) (mod ,u) , and it is integrable. Also, if t(&I) < -, then

(12)

Proof.

Without loss of generality we may assume that the Let us set

function f(w) is finite and nonnegative. n-

g*(w) = lim 1 EL.f(TkoJ) n k=o

n-1 ,

f(Tkc)) g*(w) = Jim 1n E k=o

We need to show that g*(w) = g*(w) (mod ft).

.

Suppose that

0
Kap={w;g*(w)> G,g*(w)
It will be sufficient to show that jt(Kp) = 0 since {(1); g*(w) > g*(a))}

= Ua<#Kap , a,19 R

where R is the set of nonnegative rational numbers. 1 %-1

g*(Tw) = lim n o f(Tkw) + f(T'o n_ f(w)

I

Note that = g*(w)

A;=

and, analogously, g*(Tw) = g*(w). This means that T 'Kap =Kap. Therefore we can apply Lemma 4 to a space with measure {Kap, e (1 Kap, ft}

From this it follows that J Kap

.

a,if

(w),u(dw) > 8p(K.p)

,

(13)

* Lemma 4 does not seem to be directly applicable. Instead (13) seems to follow most simply by choosing 0 < e < (3 and noting that 1n-1

f x a#

f fw) - (Q - e)lk(d.) = Jx aA n is chosen appropriately. This implies that

{

n

Z ft Tkw) - (3 - E)}

Jf(w)u(dw) > (Q - e)u(Kap) ; xaA

(13) now follows by letting e J, 0.

0

AXIOMATIZATION OF PROBABILITY THEORY

128

and applying (9) to the function - f(w), we obtain 5 Kip

- ap(Ka) ,

(-f(w))p(dw)

f(w )p(dw) <

Lp

Since ,8 > 0, it follows from (13) that ut(K p) < -, but then (14) is possible if and only if p(Kap) = 0. Thus the existence (mod ft) of the limit (10) is proven. Let us set f*(w) = g*(w). Then (10) is satisfied and the function f*(w) is T-invariant everywhere in 2. To prove formula (12), we set

k2 1

Akn = {w; k <.f*(w) < 21

We have fl = Uk

}

A,,,, and

T'Akn={w;2 0 we

Let us apply Lemma 4 to the set Akn. have

.f(w)p(d(o) > (-- -

5 Akn

By lettings approach 0, we obtain the inequality Lnfl(o)

p(d(o) >

2 P(Akn)

Analogously,

l w)!i(d(o) < k

+ 1

f-t(Akn)

Akn

From this it follows that f*(w)p(dw) <

f(w)I-t(dw) Ak5n

Ln

Zn

MAkn)

Summing these inequalities over all k, we obtain

5f(w)(doi)

-

f*(w))e(d(o) < Zn p( 2) .

Since n is arbitrary and p(&2) < -, we obtain formula (12). This completes the proof of the theorem. Corollary 1. Convergence of the means almost everywhere implies convergence with respect to the metric of L, if p(f1) < -. Suppose that p(2) < co and f(co) e LP (for p > 1). Then

E.f(Tkw) - f* ((0)

n k=O

P

-) 0

as n- oo

,

(15)

where 11fP is the norm of the function f in LP (cf. Section 5,

S.

ERGODIC THEOREMS

129

Chapter II). To prove this, let us take some bounded function fo(w) and let us set II f(w) - jo(o)) 11, 8. We have

Ef(Tkw) - f*(w)

n k=o

< 1n k=o [f(Tkw) - fo(Tkw)]

P

1n Lfo(Tk(()) - Jo*(oi) k=o

P

P

+ II.fo*(w) - f* ((0) IIP

From Jensen's inequality and Lemma 1, I

n

II

-I Eo [f(Tkw) - fo(Tkw)] I

-U lJ

n

k io (f(Tkw) 1

1n

k =-0

ca

P

I f(Tkw) - fo(Tkw) I

I f(w) - o(w) I 1n E1 k=o 0

P

p(dw)1 1'P

=8

Using Fatou's lemma (Theorem 2, Section 5, Chapter II), we obtain 1

I I.fo*(w) - f*((O) IIP = f Q lim

n-1

n k=O

P

[f(Tk(()) - Jo*(Tkw)]

< lim - E -1[f(Tkw) - fo(Tkw)] n k=o

,u(dw)}

1!P

<8.

1

P

Furthermore, since the function fo(w) is bounded, all its means are bounded by a single constant. Therefore if we let n approach -o in the expression P

n

ZOfo(Tkw) -.fo*(w)

1/P

= JQ n ofo(Tkw) -.fo*(w) p(dw)} I 1P

P

we can, by virtue of Lebesgue's theorem, take the limit under the integral sign. Consequently this expression approaches 0 and for sufficiently large n is less than 8. Hence

1n Ef(Tkw) - f*(w) P < k=o

38

n

no = no(8) ,

where the number 8 > 0 can be chosen arbitrarily small. Thus (15) is proved. Let us return now to stationary sequences. for n = 0, ±1, , denote a stationary Corollary 2. Let sequence and suppose that M I o I < .. Then with probability 1 the limit 1

limLek=e n-.m n k=m exists, and

M* = M.

130

AXIOMATIZATION OF PROBABILITY THEORY

Let us look at an arbitrary event A e 19 and the sequence of events

obtained from A by a "time displacement": A, St'A, St2A,

.

Let X. denote the characteristic function of the event S%A. Then the x% (for n = 0, ± 1, ) constitute a stationary sequence of random variables, and 1

n

k=O

is the frequency of the occurrence of the event A calculated from a single sample function of the sequence for n = 0, 1, 2, and for n consecutive displacements of the origin of the time scale

,

1 E Xk = vn(A) n

n k=O

On the basis of the Birkhoff-Khinchin theorem, the limit lim yn(A) = 7r(A) n

n--

exists with probability 1 and M7r(A) = Mxo = p(A). The quantity 7r(A) can be called the empirical probability of the event A. It is a random variable and is determined by a single sample function of the infinite sequence {en}, for n = 0, 1, 2, . The question naturally arises: When is an empirical probability 7r(A) independent of chance and when does it coincide with the probability p(A)? Definition 2. A stationary sequence {fin}, for n = 0, 1, , is

said to be ergodic if for arbitrary A (9 lim yn(A) = P(A) (mod P) . n-..

n

An event A e 0-F is said to be invariant (mod p) if the events S-1A and A coincide with probability 1. A sequence {en}, for n = 0, ± 1 , , is said to be transitive if an arbitrary invariable event has probability 1 or 0. Theorem 2. For an ergodic stationary sequence, it is necessary and sufficient that one or the other of the following two conditions be satisfied: a. The sequence is transitive; b. for an arbitrary Vie-measurable function f(w) such that M I fits) I < oo , the function f* (w) is constant with probability 1, where

f*((O) = lim Proof.

1n k=0 E f(Skw)

Let A denote an invariant event. Then, the events

S.

ERGODIC THEOREMS

131

coincide with probability Consequently, the limit A, SA, S2A,

1,

and

Z,((o).

lira v (A)ln

cannot be a constant (mod P) if 0 < p(A) < 1. Thus ergodicity implies transitivity. Suppose now that the sequence {fin} is transitive.

Since the function f*(w) is S-invariant (mod P), the events S-1{w; f*(w) < x} _ {w; f*(So)) < x} and

{w; f* (w) < x}

coincide with probability 1. It follows from the transitivity that for arbitrary real x the quantity P{(r); f * (w) < x} is either 0 or 1; that is, f*((o) is constant (mod P). Thus condition a implies condition b. Finally, the condition for ergodicity is a special case of

condition b, namely the case in which f(w) is the characteristic function of some event. Let us indicate a few consequences of ergodicity. Let f(w) and g((o) denote arbitrary functions belonging to the set.

Lz=L2(1, ,P). It follows from Corollary 1 to Theorem 1 that lim.

Jn

Eo f(Skw)g(w)P(dw) = n.f*((o)g((o)P(dc)) .

Let us set g(w) = )7 and f(Sk(o)

Let us suppose that the

original stationary sequence {fin}, for n = 0, ± 1, Equation (16) takes the form lim. M 1 E1 bk) = MCA)7 . %__

(16)

n k=0

,

is ergodic.

(17)

Suppose that rJ = XB((o), o = f(w) = XA(o)), where A and B belong

to F. It follows from (17) that lim. 1 E1 p(S-kA (1 B) = p(A)p(B)

(18)

lim. 1 E P(S-kA I B) = P(A) , n-- n k-o

(19)

n-+- n k=0

or (if p(B) =/-- 0)

where p(A I B) is the conditional probability of the event A, given B.

Equation (18) (or equation (19)) is for A, B e equivalent to ergodicity.

Corollary 1.

AXIOMATIZATION OF PROBABILITY THEORY

132

To prove this, it suffices to show that (18) implies ergodicity. Let C denote an arbitrary S-invariant set. In (18) let us set A = B = C. Then equation (18) becomes p(C) = P2(C), so that P(C) is equal to either 0 or 1, and our assertion follows from Theorem 2. Equation (19) has the following probability-theoretic meaning. Let A and B denote two events in C. If the event A is displaced by an infinite amount in time, the events S-1A and B become independent in mean for every event B. Condition (19) is a special case of the more stringent requirement lim P(S-"`A I B) = p(A)

,

(20)

which is called the displacement condition. Let us look, for example, at a sequence {fin}, for n = 0, ±1, of independent identically distributed random variables and let us assume that M I n I < co This sequence is a stationary sequence. On the basis of the Birkhoff-Khinchin theorem,

,

1

n-1

lim - 1 ek = c* (mod P), n-.m n k=0

Me* = Me

The random variable e* is obviously independent of any finite number of variables e0, 1, , P. Therefore C* is measurable with respect

to lim and on the basis of the 0-or-1 law (Theorem 5, Section 3), it is constant: * = c (mod P), where c = MC. Thus we obtain: Corollary 2 (A strengthened form of the law of large numbers). , denote a sequence of independent identically distributed random variables and suppose that M I ek I < co . Then with probability 1,

Let {fin}, f o r n = 0, ± 1 ,

I

n-1

lim - E ek = MC0 n-.m n k=0

.

(21)

In the present case we can say more than this. Let fE(co) and g,(c)) denote arbitrary 11e-measurable functions that depend on a finite fixed number of coordinates of the point co. For sufficiently large n, the functions f,(S' w) and gE(co) will be functions of two pairs of finite sequences of independent random variables and hence will themselves be independent. Thus lim MJ (Snw)gE(w) = MfE(w)MgE(w)

n--

(22)

Now let f and g denote arbitrary members of L2(S2, C,, P). On the basis of Theorem 7, Section 7, Chapter II, for arbitrary s > 0 there exist functions f (a)) and g,(c)) depending on finitely many

S.

ERGODIC THEOREMS

133

coordinates of the points (o and satisfying the inequalities M I f(w) - ff(w)

I2

<

62

,

M I g(w) - gf(w) I2 < E2

Then I Mf(Snw)g(w) - Mf((0)g((0) I

< I Mff(Snw)gf(w) - Mff(w)gf(w) I + 2s(I I g

I + I

ff11)

where I f I I2 = M I f(w) I2. Taking (22) into account, we then obtain lim Mf(Snw)g(w) = Mf((0)g((0) (23) I

n-.-

for arbitrary f, g e L2(SL, (9, p). This proves:

Theorem 3. A sequence of independent identically distributed In particular, it

random variables satisfies the displacement condition. is ergodic.

Another example of a process satisfying the displacement condition is a stationary Gaussian sequence whose covariance approaches 0. Thus let {En}, for n = 0, ±1, ±2, , denote a stationary Gaussian sequence and suppose that Min = m and m) = R,,. Suppose that f(w) = f(a(0), a(l), , a(p)) M(en and gf((o) = g(a(0), a(1), . , a(p)) are bounded sufficiently smooth

functions of p + 1 variables and have absolutely integrable Fourier transforms f* (X, Mff(Snw)gf(w) = M

, XP) and g* (Xo, +

-m

-oo

exp ji(

, XP).

k =0

Then,

k) +

I

k=0

J

X ff* N, ... , X P)ge ( 0, ... , ti )dXo ... dX pd/ u/0 ... ddP

+ ... + exp P

It

Y Rk-r(XkX, + 1tk/--r)

2 k,r=o

// ll

+ 2 LJ Rn+k-rX k(-i .} k,r=0

X f*(X0, ..., XP)g*(tt0, ..., pP)d20 ... dXPdito ... dpP .

If limn__ R. = 0, then by taking the limit as n -- - in the preceding equation, we obtain limn__ Mff(Sn(o)gf(w) = Mff(w)Mgf(w). By

following the same reasoning we used in the proof of Theorem 2,

that is, by approximating the arbitrary functions ff, we see that equation (23) is also valid for arbitrary functions f, g e L2((l, ,, p). Thus we have proved:

Theorem 4. A stationary Gaussian sequence whose covariance R. = m) ($0 - m) (where m = Me.) approaches 0 as n --+ oo satisfies the displacement condition (23).

AXIOMATIZATION OF PROBABILITY THEORY

134

6.

CONDITIONAL PROBABILITIES AND CONDITIONAL MATHEMATICAL EXPECTATIONS

Let us first recall the definition of conditional probability and conditional mathematical expectation in the elementary case. The conditional probability p(A B) of an arbitrary event A conditioned by the event B is defined for p(B) 0, by

P(A B) = p(A (1 B) p(B)

For fixed B, the conditional probability p(A I B) is a normed measure defined on the same a-algebra of sets as is the "unconditional" probability p(A). The conditional mathematical expectation of a random variable = f(u) conditioned by B is defined by f(u)p(du I B) .

M{e I B} = U

Keeping in mind the definition of conditional probability, we can rewrite this equation in the form (1 )

P(B)M{e I B} = 1cB f(u)p(du) .

If we are to be able to define the conditional mathematical expectations and conditional probabilities relative to events with probability 0, we need to modify these concepts somewhat. Let us note first of all that if e is the characteristic function of the Thus the conditional probabilities constitute a special case of conditional mathematical expectations, event A, then M{e I B} = P(A I B).

and we can for the moment confine ourselves to these. Let t denote a countable class of incompatible events {B;} where B; e 0 and uro=1 B; = U. We define the random variable for i = 1, 2, M{e 11Ul} by setting it equal to M{e I B;} if u e B; and we call it the conditional mathematical expectation of the random variable e with respect to the given class Tt of sets. We are unable to define it only for those values of u that belong to those B; with zero probability; that is, it is defined with probability 1. On a set B; a 1J for which P(BS) # 0, it is constant and equal to the conditional mathematical expectation e under the hypothesis B. Note that if we know the function M{e t}, we can define not only M{e I B;} when B; e Tt and p(B;) 0 but also the conditional mathematical expectation of the random variable with respect to an arbitrary set

B belonging to a{It} and such that p(B) B = U7=1 B;1, then

0.

Specifically, if

6.

CONDITIONAL PROBABILITIES AND MATHEMATICAL EXPECTATIONS

135

(2)

P(B)M{e I B} = E P(B;k)M{e I B;,} . k=1

This formula shows that if we know the conditional mathematical expectations for given Bti (for i = 1 , 2, ), we can calculate the conditional mathematical expectations for given countable sums of these sets. Hence we can calculate the conditional probability with respect to an arbitrary set in the smallest o'-algebra containing all the B. Therefore it is meaningful to speak of the conditional mathematical expectations with regard to some a-algebra of subsets. We note that equation (2) can be rewritten M{ I RIP(du) =

JB f(u)p(du) ,

Be art

.

JB

This formula is obtained on the basis of the definition of the conditional mathematical expectation with respect to an arbitrary oralgebra of events. Deflnition 1. Let 0 denote an arbitrary 6-algebra of events

that is contained in 0 and let 6 = f(u) denote an arbitrary random variable with finite mathematical expectation. Suppose that the random variable M(EE 10) is measurable with respect to 0 and satisfies

the equation M{e I jB}p(du) = JB

f(u)p(du)

(3)

B

for arbitrary Be 0. Then M{e 10} is called the conditional mathematical expectation of the random variable e with respect to the a-algebra 1. The existence and uniqueness (up to equivalence) of the random

variable M{g (0} follows immediately from the Radon-Nikodym theorem (Chapter II, Section 6). Specifically, the right-hand member of formula (3) is a finite countably additive set function defined on 0 and absolutely continuous with respect to the measure P. Therefore there exists a 0-measurable function /r(u) such that JBf(u)P(du) = B k(u)p(du) .

Here the function 'r(u) is unique (up to equivalence). It is the conditional mathematical expectation of the random variable e = f(u)

with respect to the a-algebra 0, by definition. REMARK 1.

Let q3 denote the completion of 0 with respect One can easily show that

to the probability P.

M{e 101 = M{e I

} (mod p) .

AXIOMATIZATION OF PROBABILITY THEORY

136

Since the class of -measurable functions is broader than the class of 0-measurable functions, it is sometimes expedient to consider the conditional mathematical expectation with respect to completed a-algebras.

It follows from the definition that if e >_ 0,

if c = act + /9

then M{e 1 0} > 0

2

(mod p)

,

MM{e10}=Me, and Mei(i = 1, 2) are finite, then

M{e 1 0} =

1 0} + 8M{e2 1 0} (mod P) .

(4) (5)

(6)

We define the conditional probabilities P{A 10} with respect to the a-algebra 0 as the special case of conditional mathematical expectations obtained by setting e = XA(u). In other words, for fixed A the conditional probability p{A 10} is a 0-measurable random variable satisfying the equation PI-4 o}p(du) = p(A f1 B) B

(7)

for arbitrary B e 0. We now define the conditional mathematical expectations and the conditional probabilities with respect to a random variable. Let V denote a measurable space with u-algebra 0 and let v = g(u) denote a measurable mapping of U into V. The conditional mathematical expectation (resp. conditional probability)

with respect to the function v = g(u) is defined as the conditional mathematical expectation (resp. conditional probability) with respect to the 6-algebra g-10. These are denoted by M{e I v} and p{e < x I v} respectively.

Theorem 1. The conditional mathematical expectation with respect to the function v = g(u) is a function of v: M{e I v) = 'Jr(g(u))

where *(y) is 0-measurable. Proof. Suppose that e = f(u) has a finite mathematical expectation. Let B denote an arbitrary set belonging to g-1%L7. Then

JBf(u)p(du) = p(B)

is a finite countably additive function on g-10. We have T(B) _ Tg-1(B'), where B'( a 0) is the image of B. The function Tg-1 is absolutely continuous with respect to the measure p' = pg-1. On the basis of the Radon-Nikodym theorem there exists a 0-measurable

function 'r(v) such that

6.

CONDITIONAL PROBABILITIES AND MATHEMATICAL EXPECTATIONS

q(B) =

qg_'(B')

Ja,'ilr(v)p'(dv)

=

137

.

If we now apply the rule for change of variables (Theorem 3, Section

6, Chapter II), we obtain

from which it follows that M[E I v} = i/r[g(u)] = *(v). This theorem shows that the conditional mathematical expectation with respect to a function v = g(u) can be regarded as a function of the independent variable v in the space (with measure) IV, 0, Pg-1),

whereas in the original definition the conditional mathematical expectation is regarded as a function of the variable u. Accordingly, the conditional mathematical expectation can be defined by means

of the formula L #(v)pg-'(dv)

(8)

for arbitrary B' e 0, where *(v) = M{e I v}. Let us now look at certain properties of conditional mathematical expectations. It follows immediately from the definition of conditional mathematical expectation that if a random variable = flu) is 8-measurable, its conditional mathematical expectation with respect to 0 coincides with itself: M{e 101 _ e. From this it follows that if 0 c 0', then M{M($ 10) 10'} = M{$ 10} . (9) Theorem 2. Let and rJ denote random variables having finite mathematical expectations. If $ is 0-measurable, then

M{$ 1 0} = SM{ 1} . Proof. Since, for every set Be 0, JBM{ 18}P(du) = JBg(u)p(du)

it follows that the equation

(10)

7) = g(u)

J

Bf(u)M{7) I O}P(du)

_

Bff(u)g(u)p(du)

(11)

is satisfied for every function f(u) that is simple with respect to 8. Taking the limit in equation (11) with respect to a nondecreasing sequence of 0-measurable functions, we see that (11) is valid for an arbitrary 0-measurable nonnegative function. It follows from (6) that the theorem holds for arbitrary integrable and 0-measurable functions.

AXIOMATIZATION OF PROBABILITY THEORY

138

Theorem 3. If 0 c 0', and Me is finite, then M{M{E 10'} 10} = M{E 10} .

Proof.

If B e 58, then B e 8', and consequently ')

M{M{E I

I

8}P(du) =

M{E I'}P(du) J

B

=

f(u)P(du) = 1 M{e I O}P(du) . JB

In view of the extreme members of this chain of equalities, we see that

M{M(E10')101=M{E10}. Therefore, we have: Corollary. If c 0' and rJ is a 8'-measurable random variable, and e and i have finite expectations, then (12)

M{Ey) 1581 = M{7M(E 158') 158} .

The next theorem corresponds to the elementary fact that if A and B are independent events, the conditional probability of an event A with respect to B coincides with its unconditional probability.

Let o'{E} denote the a-algebra of events that is generated by the random variable E.

Theorem 4. If the a-algebras 58 and 6{E} are independent and if Me is finite, then M{E 10} = Me (mod P) .

(13)

Proof. For arbitrary Be 0, the random variables a and X, where XB is the characteristic function of the set B, are independent. Therefore JBf(u)Pdu = MXBE = MXBME

= B(Me)P(du)

,

and since Me = const. is 58-measurable, we may take Ml as M{E 1 0}.

Corollary. If a random variable a and a random element 17 are independent and if Me is finite, then M{$ I '} = Me

(14)

Theorem 5. If en = fn(u) > 0 and e. < en+1, then M172 101 = lim M{En 58} (mod P) n-,oo

where )7 = limn_ E

.

In the equations

(15)

6.

CONDITIONAL PROBABILITIES AND MATHEMATICAL EXPECTATIONS

M{$. 18}p(du) = JB

139

.

B

where B e 0, let us take the limit as n --> 00 , keeping in mind (4), (6) and Lebesgue's theorem (Chapter II). We obtain

lim Q. i) P(du) = lim fn(u)p(du) BB n-Since the random variable limn__ m{$, 10} is 0-measurable, we

can take it as the conditional mathematical expectation of the variable rJ = limn--fn(u). Let us now look at conditional probabilities. From the properties of conditional mathematical expectations that we have proved, we get:

Theorem 6.

With probability 1,

P{A10}>0, A

= Jk=1 A, where Ak n A,. = 0 for k

r, Ak e e, then

P{A I 581 = E P{Ak 10} ; k=1

(18)

if {An}, for n = 1, 2, , is a decreasing or increasing sequence of events Ak e e and B= n;=, Ak (or B = U7=1 Ak), then P{B 101 = lim P(Ak 10) .

(19)

We recall that the conditional probabilities are defined only up to equivalence. Therefore relations (16) to (19) are only p-almostcertain. But can we regard the conditional probabilities p{A 101 as measures? In other words, the conditional probability p{A 101 = cp(A, u) is a function of u e U and A e C25 that is defined for every

fixed A only up to a set of probability 0; we now ask whether there exists a function *(A, u) (for u e U and A e C25) such that a. cp(A, u) = Vr(A, u) almost certainly for arbitrary fixed A and b. on fixed u, the function *(A, u) is a probability on the v-algebra C25.

We find that such a function *(A, u) does not always exist. Definition 2. Suppose that a function *(A, u) satisfying requirements (a) and (b) does exist. The family of conditional probabilities P{A 10} is then said to be regular. In this case we identify p{A 01 with 1/r(A, u).

In the regular case the conditional mathematical expectations are, as we would expect, expressed by integrals with conditional

AXIOMATIZATION OF PROBABILITY THEORY

140

probabilities used for measures.

Theorem 7. If P{A I F8} is a regular conditional probability and e = flu) has finite expectation, then M{e 1581 = f(u)p(du 0) (mod P) . Proof of this assertion presents no difficulty.

(20)

It is true by defini-

tion in the case in which a is the characteristic function of some event A e e. It follows from the linearity of the two sides of equation (20) as functionals of f that the assertion holds for simple functions. If we take the limit with respect to an increasing sequence

of simple functions (Theorem 5), we see that (20) holds for an arbitrary nonnegative random variable e. Repeated use of the linearity of the two sides of equation (20) completes the proof. In certain cases we need to emphasize that the conditional probability is a function of an elementary event. In such a case we shall write p{A I 01 = p (u, A) or simply p(u, A) if the a-algebra 0 is fixed. Analogously, pF(u, A) is another notation for p{A I e}. If 0 and W'(c 0) are two u-algebras contained in e and if p{A 1581 and p{A 10'} are regular, then equation (12) can be rewritten ff(v)g(v)pz(u, dv)

= (ftv')P'(v, dv'))g(v)p (u, dv) ,

(21)

where g(v) is 0'-measurable, f(v) is s-measurable, and both are p-integrable.

Since the property of regularity of conditional probabilities does

not always obtain, it is convenient to extend this property. Let us now introduce the concept of conditional distribution of a family of random variables. Let {e} denote a family of random variables and let C5{el denote the a-algebra of sets generated by the

it follows from the regularity of the conditional probability with respect to some Q-algebra 0 in the family {e}.

Since 0{el c C5,

probability space J U, C5, p} that this conditional probability is regular in {U, C5{e), P}.

Definition 3. A function Ps8{u, Al is called the conditional distribution of the family of random variables {e} if a.

for fixed A e C5{E},

it is 0-measurable as a function of u

and for fixed u it is a probability on b.

for arbitrary A e C5{el and B e 0,

6.

CONDITIONAL PROBABILITIES AND MATHEMATICAL EXPECTATIONS

141

PT(u, A)P(du) = P(A n B) ; JR

Thus, the conditional distribution of a family of random variables

is a regular representation (if it exists) of conditional probabilities of the probability space {U, P} with respect to the o-algebra 0. , ,a denote a sequence of random variables for Let 1, e2, which there exists a conditional distribution with respect to 8 and let S denote an arbitrary Borel set in the n-dimensional space E. We define (22) Qz(u, S) = Pu{u; [e1, E2, ..., e ] e S} . Then Qu(u, S) is a 0-measurable function of u and is a measure on

the a-algebra of Borel sets S for fixed u. We note that if the function Qz(u, S) is given, it still does not define the conditional distribution pu(u, A) since it is possible that two distinct subsets Sl that is, and S2 of E. correspond to a single set A E

i = 1, 2. Let Qz(u, S) denote a function that is 0-measurable on U that is a probability on the a-algebra of Borel subsets of the space E for fixed u, and that satisfies equation (22) for p-almost-all u with fixed S. We shall call this function the conditional distribution of A=

a sequence of random variables e1, e2, . . , ,a with respect to the a-algebra 58 in the broad sense. It is easy to prove the validity of:

on a subset of E.

,

,

denote a Borel function defined Suppose that its mathematical expectation Mg($1,

Theorem 8. Let g(x1, x2,

is finite and that the sequence {$;} of random variables $;, , n, has conditional distribution Qu(u, S) with respect to B in the broad sense. Then

for i = 1, 2,

M{g(se1, ... , W 1 0} _

g(x,, x2, ... , x,,)Qu(u, dx)

p-almost-everywhere in U.

Theorem 9. For an arbitrary n-tuple of random variables {E$}, f o r i = 1 , 2, , n, and an arbitrary u-algebra 0, there exists a conditional distribution Qz(u, S) with respect to l3 in the broad sense. , yn), denote the set of points contained in E and satisfying the inequalities

Let I., where y = (y1, y2, x = (x1,

, x,,,)

, x," < y.. rational coordinates. We set xl < y1, xQ < y2,

Let r denote a point in E. with

F(r) = QF8(u, Ir)

.

(23)

AXIOMATIZATION OF PROBABILITY THEORY

142

The set N of points u, such that Q(u, I,.) does not satisfy, for arbitrary

rational r, at least one of the relations F(- co) = 0, F(+ 00) = 1, 0 < F(r) < F(r') < 1 for r G r'; F(r') -, F(r), r' G r, r' -> r , is a set of 8-measure zero. For arbitrary x, we set F(x) = Qj8(u, Ix) = lim F(r) , r-,x,r<x

if u R N ,

F(x) = Q(u, Ix) = Fo(x) , if u E N , where Fa(x) is a fixed n-dimensional distribution function. The function F(x) is for u 0 N a distribution function, and it defines the probability measure Qsg(u, S) unambiguously in E. on the a-algebra of Borel sets for u 14 N. Furthermore, the equation Qz(u, S) = Po{u; [e1, ...,

E S}

which is valid for S = Ir (equation (23)), remains valid for every set S in the minimal u-algebra containing all the Ir (that is, for all Borel S) and for all u except possibly a set of points N'(S) of probability 0. Furthermore, the function Q(u, S) as a function of u is 0-measurable by construction. This completes the proof of the theorem. REMARK 2. The definition of conditional distribution in the broad sense can be applied to an arbitrary mapping g = g(u) of a probability space into an arbitrary space X. Here Theorem 9 is generalized to the case of an arbitrary separable metric space. To prove the theorem, we take not the sets Ir but spheres with rational radii and centers belonging to a countable everywhere-dense set. Everything else remains unchanged. Let e1, e2, , e denote a sequence of random variables. Define

Just as in Theorem 1, we can show that , ek}. the conditional probability (in the broad sense) Q k(u, S) can be 8k = a'{e1, e2,

regarded as a function/ of e1, . . . , ek: Q;k(u, S) = Qk(e1, ... , ek; S) The function Qk(e1, , ek; S) is determined by the equation MQk(el, ..., ek, S)xBk(el, . . ., ek) = P{S n Bk} ,

where Bk is the cylindrical set in E. defined by {e1, B$ > being a Borel k-dimensional set.

It follows from equation (24) that Qk\e1, ... , ek; S) = Qk(sell ... , e k; k, S n Bk)

(24) , ek} e Bk >,

6.

CONDITIONAL PROBABILITIES AND MATHEMATICAL EXPECTATIONS

143

almost certainly for arbitrary Bk having the same structure as Bk. This means that ek; Rk x SF......k) (mod P) Qk(61, ..., ek; S) = where Sfl,...Ilk is the (c1, .. ., ek)-section of the set S. Thus we may set

Qk(Elf ..., k, S) = p(E1, ..., ek; Sp1,...Ilk) . , xk; C) is a Borel function of (x1, x2, ... , Xk) for fixed C and it is a probability in (n - k)-dimensional space for fixed where P(x1, x2,

X1, X2, ... , X.

By successive application of Theorem 8, we obtain the following equations for an arbitrary Borel function g(x1, x2, , xn) such that exists: Mg(e1, ... Mg(e1, ..., fin)

=

P(dx1) `p(x1; dx2)

...

p(x1, ... ,

dxn)g(x1, ... ,

(25)

J

and almost certainly, M{g(

1,

... , i,,) I 1} =

P(e1; dx2) W1, x2; dx3) J

... Wl, x2, ... , xn-1; dx'n)g(x1, x2, ... , xn) .

(26)

Iv

RANDOM FUNCTIONS

1.

DEFINITION OF A RANDOM FUNCTION

In Chapter I a random function was defined as a family of random variables depending on a parameter. There we pointed out the difficulties associated with this concept in the broad sense, that is, as a set of finite-dimensional distribution functions satisfying compatibility conditions. The axiomatization of probability theory immediately suggests that it is natural to regard a random function as an arbitrary family of random variables, all defined on the same probability space. Definition 1. Let {U, C5, P} denote a probability space.

Let

g(0, u) = (0) denote a function of two variables defined on ® x U into a metric space X. If g is C5-measurable as a function of u for every 0 E O, it is called a random function. The set O is called the domain of definition of the random function and X is called its range.

Let us look at a particular case of the general definition. Let U denote a functional space. Suppose that u = u(0) for 0 E O and

suppose that the o'-algebra 0 contains all sets of the space U of the form {u; u(00) e Al for every Bo e ® and every Borel set AC X. Let p denote an arbitrary probability measure on 0. It is natural

to put the random variable g(0, u) = u(0) in correspondence with such a probability space. In some problems it is convenient to identify the random function g(0, u) = u(O) with the probability space {U, CS, P} of the type described.

It is easy to see that the general definition of a random function can be reduced to the particular case just described. Specifi-

cally, if a random function (0) is defined as a function of two variables, that is, (0) = g(0, u), then by setting u' = g(B, u) where u is a fixed member of U and letting U' denote the set of all functions u' = g(8, u) obtained by letting u range over U, we obtain a mapping T of the set U onto U'. Here the or-algebra e of sets U 144

1.

DEFINITION OF A RANDOM FUNCTION

145

is mapped into some 6-algebra e' of sets U', and the probability measure p on e is mapped into a probability measure P' on (cf. Section 6, Chapter II). For arbitrary fixed 0, the set {u'; u' _

Cam'

g(O, u)e A, where A is a Borel set in x} belongs to e' since T-1{u'; u' = g(O, u) a A, where A is a Bore] set in xJ

_ {u; g(O, u) e A, where A is a Borel set in x} e e . Thus we have obtained a probability space {U', Cam', P'} where U' is a set of functions u = u'(B). Here, for arbitrary n, B1, 02,

, Bn

(where 0k e O for k = 1, , n), the distribution of the sequence of random variables on { U, e, p} _ g(01, u), g(02, u), ... , g(0,, u)

coincides with the distribution of the sequence u'(01), u'(02), ... , u'(0,.) of random variables defined on the probability space { U', Cam', P'}.

It is important to note certain facts regarding the equivalence of random functions. In the solution of many problems we have no basis for distinguishing between random functions that are obtained from each other by transformations of probability space. We can go even farther in this direction. From a practical point of view, experiment enables us to distinguish only between hypotheses dealing with finite-dimensional distributions of a random function. Therefore, it is convenient to assume that experimental data do not enable us to distinguish between two random functions (0) and '(0) whose finite-dimensional distributions coincide, that is, the joint distributions of the sequences E(01)1 M(02), ... , W.)

(1 )

VA), x'(02), ... , V(0,.)

(2)

and

for arbitrary n = 1, 2, and Bk c O for k = 1, , n, coincide. Accordingly we take the following. Definition 2. Two random functions (0) and e'(O) defined on the same set O are said to be stochastically equivalent in the broad , n, sense if for arbitrary n = 1 , 2, and 0k e O, for k = 1 , 2, the joint distributions of the sequences (1) and (2) of random variables coincide.

In what follows we shall often use the concept of stochastic equivalence of random functions in the narrow sense: Definition 3. Two random functions g1(0, u) and g2(0, u), for 0 E 0 and u e U, defined on the same probability space { U, Cam, P}

RANDOM FUNCTIONS

146

are said to be stochastically equivalent if for arbitrary 0 e O, P{g1(6, u) # g2(0, u)} = 0 .

Obviously if g,(O, u) and g2(O, u) are stochastically equivalent

(in this narrow sense), they are stochastically equivalent in the broad sense.

Let us look at a few examples of random functions. a. The random oscillation E(t) for - co < t < co (considered in Section 4 of Chapter I), n E

S (t) _

k=1

7ke41k1,

7k = ak + iRk

where ak and Rk (for k = 1, 2, , n) are random variables, can be represented in the form e(t) = g(t, u), where u = (a1, a2, , an, R1, R2, . 8. ) is a point in 2n-dimensional Euclidean space U and g(t, u) is for fixed t a linear function of u and for fixed u is -the sum of trigonometric functions of t. The probability P in U is defined by the common distribution of the random variables a1, a2, , an, R1, R2, , Rn. On the other hand, the process c(t) can be regarded as a probability space { U', C5', P'} where U' is the 1

4

space of all complex-valued functions of the form u = Ek=17ke;2kt with given bases of exponents (X1, X2,

., Xn)

The measure P' in U' is induced by the measure P in U by means of the mapping u u' = g(t, 0). b. Consider a process e(t) whose sample functions are constant on the time intervals [k - 1, k) and assume on these intervals the values 0 and 1 with equal probabilities independent of the value of E(t) on the preceding time intervals. We can set up a one-to-one correspondence between the sample functions of the process E(t) and the nonterminating binary fractions in accordance with the scheme e(t) .---> u = 0. x1x2 ... x,n ...

,

(3)

where xn is the value of e(t) on the time interval [n - 1, n). To each nonterminating binary fraction u there corresponds some point in the interval [0, 1]. The one-to-one nature of this correspondence is violated only for fractions, all binary digits of which from some point on are the same. If a point in this interval has a representation with all 0's from some digit on, it also has a representation with all l's and vice versa. (Exceptions are the end points 0 and 1 which have only the representations 0 = 0.00 0 and 1 = 0.11 . . . 1 .. ..) The sample functions corresponding to such nonterminating binary fractions become constant from some

1.

DEFINITION OF A RANDOM FUNCTION

147

instant of time onward. Let ?I denote this event or set of events and let W. denote the event that the random function e(t) will be constant beginning at the instant t and thereafter. Then Gl = U 1l . n=1

Also, let 52C,,. mean that the function fi(t) is constant on the time interval [n, n + m). The events (for m = 1, 2, ) constitute a decreasing sequence VI,, = l;,,_, T{,,,., and P(Wn) = lim P(tn,m)

Since p(52tn.) = 2', it follows that p(?I,) = 0 and p(W) = 0. Consequently, if we neglect the random functions e(t) in the set W, which is of probability 0, there exists a one-to-one correspondence

between points of the interval [0, 1] and the random functions e(t). Let A denote an interval of length 2-n with dyadic endpoints. Since a dyadic fraction always has a terminating binary representa-

tion, A is either of the form [0. 71 ... In_101

0. jl ... In-1l)

or of the form

[0. j1 ... 011 .. 1, where each jk is either 0 or 1.

0. j1

100 ... 0)

,

The set W of random functions e(t) corresponding to the interval A consists of functions satisfying the conditions e(v) = j2, .. , e((n - 1)T) = jn The probability that fi(t) E W is equal to 2-1; that is, p(?X) = 2-n, which coincides with the length of the interval A. It follows from e(0) = j1,

this that if A is an arbitrary Borel set of points belonging to the interval [0, 11 and a is a set of random functions e(t) corresponding on the basis of (3) to the numerical set A, then p(W) coincides with the Lebesgue measure of A. Thus, the choice of random function e(t)

is equivalent to the random choice of a point u belonging to the interval [0, 1] with given Lebesgue measure on it. More precisely this means that the random process is completely described as fol-

lows: A point in the interval [0, 11 is chosen at random. Here the probability that the point u belongs to A, where A is a Borel subset of the interval [0, 11, is equal to the Lebesgue measure of A. The coordinate of the point u is described in the binary system by u = 0.x1x2 . . . xn Then the value of the function e(t) on the interval [(n - 1), n) is equal to x,,. Accordingly, the random function E(t) can be written in the form e(t) = f(t, u) where fit, u) .

RANDOM FUNCTIONS

148

is a completely defined nonrandom function of two variables t and

u (with 0
, u(8,,,)}

in X n for arbitrary natural n and 0, c O for k = 1, , n, coincides with the corresponding distribution for the given random function in the broad sense. In other words, for an arbitrary random function in the broad sense it is possible to construct a stochastically equivalent (in the broad sense) random function (representation in the terminology of Section 3, Chapter III) in the sense of Definition 1. Unfortunately, the representation obtained does not always fully serve our purpose. Arbitrary functions u = u(O) of a variable B are elementary events in the resulting probability space, and we

can no longer speak of such properties of the functions u(8) as their continuity, integrability, differentiability, and so forth. (Of course we are assuming that O and X are such that the corresponding concepts are meaningful.) Furthermore, in this probability space we cannot consider events of the form

{u(B)EA for all 8eQ,Ae8}

(4)

where Q is an unusual subset of O. Such events are important in solving many particular problems. For example, corresponding to the event {u; u = u(O) E A for all 8 E Q} = n {u; u(B) e A, 8 fixed} OBQ

in U is a set representing the intersection of an uncountable collechence, it does not necessarily belong tion of sets belonging to to the a-algebra t . These considerations suggest that in the construction of the representation of the family of distributions (1) it is desirable for U to be as narrow as possible, that is, for functions belonging to U to have the best possible analytic properties.

For example, to be able to solve the problem of calculating the probability of the event (4), we should like the random function g(O, u), which is the representation of the family of distributions (1), to have the following property:

1.

DEFINITION OF A RANDOM FUNCTION

149

c. For sufficiently broad classes of sets W in X and C, in O, there exists a countable set A of points 0; e 0 such that for arbitrary A e W and Q e C, the set of points

{u;g(0,u)eA for all 0eQ},AeW,QeC,

(5)

differs from the set {u: g(0,, u) e A for all 0; e A f1 Q}

(6)

only on a subset of some fixed set N of p-measure 0 that is independent of A and Q. Since the set (6) is the intersection of no more than countably many measurable sets, it is itself measurable and so is the set (5) (because of the completeness of the measure p).

Here the probabilities of the events (5) and (6) are equal to

each other.

A random variable that satisfies condition c is said to be separable (with respect to the class of sets SRI).

Suppose that O is a separable metric space, that C, is a class of open sets, that S?t is a class of closed subsets of the space X,

and that the function g(0, u) is a continuous function of the argument 0 for almost all fixed u. Note that under these conditions the random function g(0, u) is separable with respect to the class of closed sets W. In problems dealing with integration of random variables with

respect to a particular measure {p, fit} on O, it is desirable for the function g(0, u) to be , -measurable as a function of 0 for almost all fixed u.

In other cases, it is necessary to find a representation of a given random function in the broad sense under which almost all sample functions are continuous or have discontinuities of the first kind only, or are k times differentiable, and so forth. Finally, we should expect the possibility of obtaining a representation with special properties to be determined by finite-dimensional distributions of the random function. Analogous problems can arise for functions other than random functions in the broad sense. A random function g(0, u) can have "pathological" properties and yet there may exist a "smooth" function g*(0, u) that is equivalent to it, p{g(0, u) # g*(B, u)} = 0 ,

not "pathological." According to our definition of stochastically equivalent functions, in solving the problems with which we are concerned it is permissible to replace such a function with a stochastically equivalent regular function g*(0, u).

but that

is

RANDOM FUNCTIONS

150

Let us consider an example. Suppose that A is the set of rational numbers on the real line (- o o, co), that x(t) is the characteristic function of the set A, and that u is a random variable uniformly distributed on the interval [0, 1]. Let us set g(t, u) = x(t + u). For arbitrary fixed u, the function g(t, u) is everywhere discontinuous. On the other hand, for fixed t the function g(t, u) is equal to 0 with probability 1. Thus, with probability 1, the everywhere-discontinuous random function g(t, u) is stochastically equivalent to the function g*(t, u) = 0. In this chapter we shall consider different conditions under which there exists, for a given random function, a stochastically equivalent (or stochastically equivalent in the broad sense) function possessing specified regularity properties. 2.

SEPARABLE RANDOM FUNCTIONS

We mentioned the concept of a separable random function in the preceding section. The property of separability, however,

not a

on a random function. Under rather broad hypotheses having to do with the nature of the is

stringent restriction

domain of definition O and the range X of the random function, there exists a separable random function stochastically equivalent to the given one. However it should be noted that in constructing the equivalent separable random function it is some times necessary to broaden the range of the function in order to make it a compact set.

Let us assume that O and X are metric spaces with metrics r(01, 02) and p(xl, x2) respectively, and that O is separable. The role of the classes of sets t and C in the definition of separability is played by the closed subsets of X and the open subsets of O. Thus separability of a random function will be understood in the sense of: Definition 1. A random function g(0, u) is said to be separable if O contains an everywhere dense countable set of points {0;}, for

j = 1, 2, , and if U contains a subset N of probability 0 such that for an arbitrary open set G c O and an arbitrary closed set F c X, the two sets {u;g(05,j)EF,0 eG}, {u; g(0, u) e F for all 0 e G}

differ from each other only by a subset of N.

2.

SEPARABLE RANDOM FUNCTIONS

151

The countable set of points 0, in this definition is called the set of separability of the random function. Theorem 1. Let X and O denote metric spaces. Suppose that X is compact and that O is separable. Then an arbitrary random function g(O, u) defined on O into X is stochastically equivalent to some separable random function.

Suppose that g(O, u) is a separable random function, that I is its set of separability, and that N is the corresponding Proof.

exceptional set of points u. Let V denote the class of all open spheres in the space O with rational radius and center at a point belonging to a fixed countable everywhere-dense subset of O. The class V is countable. On the other hand, an arbitrary open subset G of O can be represented as the union of countably many spheres contained in V. Let A(G, u) denote the closure of the range of the function g(O, u) as 0 ranges over the set I(1 G.

Then

A(0, u) = n A(S, u) SEC

is the intersection of all A(S, u) as S ranges over the collection of spheres in V to which the point 0 belongs. The family of closed sets A(G, u), where 0 e G, is centered; an arbitrary finite number of sets belonging to this family have common points, and because of the compactness of X their intersection A(0, u) is nonempty. It follows from the separability of the function g(0, u) that g(0, u) a A(O, u), u 0 N. (1 ) Conversely, if (1) is satisfied for every u N such that P(N) = 0, then g(0, u) is a separable random function. To see this, note that if g(0, u) e F for all 0 e in s where F is a closed subset of X and S e V, then A(O, u) c A(S, u) c F for every 0 e S and consequently g(0, u) e F for every 0 in S. Let G denote an arbitrary open set in O. Then we may represent it as a sum G = Uk Sk of subsets of V in order to see, on the basis of what has just been said, that the relation

g(O,u)eFfor all 0eInG,u N, implies

g(O, u) e F for arbitrary 0 e G. Let us formulate the result just obtained as: Lemma 1. For a random function g(0, u) to be separable, it is necessary and sufficient that there exist a set N such that p(N) = 0

RANDOM FUNCTIONS

152

and (1) is satisfied for every u iZ N.

Thus, to construct a separable function stochastically equivalent to g(O, u), it suffices to find a function g(0, u) that satisfies (1) and coincides with the function g(O, u) with probability 1, P{g(0, u) = g(0, u)} = 0 . Lemma 2. Let B denote an arbitrary Borel subset of X. Then there exists a finite or countably infinite set of points 01, 0, , such that the set

N(O, B) = {u;g(0k,u)eB,k= 1, 2, ,g(0,u)(Z B} has probability 0 for arbitrary 0 e O. Proof.

Let 01 be arbitrary. If B1, 0,

let us set Mk = supP{g(01,u)aB,

Be0

, Bk are constructed,

,g(Bk,u)aB;g(0,u)0BI

The sequence {mk} decreases monotonically. If mk = 0, the corresponding sequence is constructed. If Mk > 0, let 0k+1 be a point

such that P{g(01, u) a B, ..., g(0k, u) a B, g(0 +1, u) 0 B) ? 2k Since the sets

Lk = {u; g(0;, u) a B, i = 1, 2, ..., k, g(ek+l, u) 0 B}

have no points in common, we have

I > E P(Lj > Consequently, Mk

0 as k ----> o o.

I

E Mk

Thus for arbitrary 0,

p{g(0,u)aB,k= 1, 2, ,g(0,u)0BI
{u;g(0 ,u)aB,n= 1,2, ,g(9,u)0B}cN(6) for every B e fit. Proof.

Let I denote the countable set of points in 0 that is

2.

SEPARABLE RANDOM FUNCTIONS

153

the union of the sets {O,, , n = 1, 2, .1 constructed for each B e U in accordance with Lemma 2. Define

O

N(6) = U N(O, B) BEST

0

If B' e 1Z and if B' c Be Ito, then {u; g(O., u) e B', O,,, e I, g(6, u)

B} c {u; g(6, u) e B,

O eI,g(O,u)eBI cN(O,B)cN(B) Furthermore, if B' = lk=1 Bk for Bk e Uto, then I

{u; g(O, u) e B', O,,, e I, g(O, u) V B'}

c U {u; g(O, u) e B', O, e I, g(O, u) q Bk} c U N(O, Bk) cN(O) , k=1

k=1

which completes the proof of the lemma. Now we can easily conclude the proof of Theorem 1. Let us choose a countable everywhere-dense set of points L e X. Let lfl, denote the class of complements of spheres of rational radius with centers at the points L. Then l2 denotes the class of intersections

of sets belonging to Ito, and this class contains all closed sets. Furthermore, for every S e V, let us consider the random function g(O, u) as defined only for 0 e S and let us construct a sequence I = I(S) and sets N(B) = Ns(0) in accordance with Lemma 3. We define

J = U I(S), sev

No= UNs(0) sev

we also define g(0, u) = g(0, u) ,

J or

if u e NB and B e J, we define &0,u) in an arbitrary manner so that g(0, u) e A(0, u). Since the values of the functions g(0, u) and g(0, u) coincide for 0 e J, the sets A(B,u) constructed for the functions g(0, u) and g(0, u) also coincide. It follows from the definition that g(0, u) e A(0, u) for arbitrary 0 and u. Since {u; g(0, u) # g(0, u)} c NB, we have if 0

u V N0;

P(g(B, u) = g(0, u)} = 1

,

which completes the proof of the theorem. Theorem 1 can be immediately generalized for random functions

with range in a separable locally compact space. Theorem 2. Let X denote a separable locally compact space and let O denote an arbitrary separable metric space. For an arbitrary random function g(0, u) defined on J with range in X, there exists a stochastically equivalent separable random function g(0, u) with range

RANDOM FUNCTIONS

154

in some compact extension X of the space X(X D X). Proof follows from the fact that every locally compact separable space X can be regarded as a subset of some compact set X. For

example, if g(0, u) is a random function with range in a finitedimensional space X, then by completing X with an "infinitely distant" point-, we easily obtain the compact space X = X U { -I with a new metric such that every closed set Fc X (in the topology of the space X) is also closed in X (with respect to the new metric). In construction of the separable sample function of a random function, it may be convenient to assign to it the additional value "oo," but obviously for fixed 0 the probability of this value is 0. In many questions it is important to know what set J can play the role of the set of separability. Before answering this question, we introduce the important concept of stochastic continuity of a random function, and present some simple theorems involving it. (A special case of this concept was encountered in Section 4, Chapter I).

Definition 2. A random function g(0, u) with range in X is said to be stochastically continuous at a point 0o e O if P{p(g(00, u), g(0, u)) > s} - 0 as r(0, 0) 0 . (2) If g(0, u) is stochastically continuous at every point of a set Bc®, it is said to be stochastically continuous on B. We note that the condition of stochastic continuity is the con-

dition imposed on a "two-dimensional" distribution of a random variable, that is, on the joint distribution of random elements g(01, u) and g(02, u), where 0, and 02 belong to O. This concept can be applied to random functions in the broad sense. Definition 3. If there exists a point x e X such that sup P{ p[g(0, u), x] > K} , 0 OeB

as K(> 0) -'

,

then the random function g(0, u)

is

(3)

said to be

bounded in probability on the set B. Theorem 3. If a random function g(0, u) is stochastically continuous on a compact set O, it is bounded in probability on O. Proof. Lets denote an arbitrary positive number. For every

point 0 let us construct a sphere Sa with center at 0 such that P{p(g(0, u), g(0', u)) > l} < 2

for arbitrary 0' e S0. From the set of spheres SB let us choose an n-tuple So,, SB2, , Se. constituting a finite covering of O. Let a denote their greatest radius. Then for arbitrary x,

2.

SEPARABLE RANDOM FUNCTIONS

155

p(g(O, u), x) < p(g(01 u), x) + max p(g(01, u), g(o,, u)) + p(g(oi, u), g(o, u))

where 0, denotes the center of the sphere containing 0 in its interior. The terms on the right side of this inequality are finite random variables.

Therefore, for sufficiently large N,

P{p(g(01, u), x) + max p(g(01, u), g(0 j, u)) > N} < >=2,...,n

2

If we assume that N > 1, then sup P{p(g(0, u), x) > 2N} < sup P{p(g(B1, u), x)

+ max p(g(01, u), g(0,, u)) > N} + P{p(g(B;, u), g(0, u)) > 1} < which completes the proof.

Definition 4. A random function g(O, u) is said to be uniformly stochastically continuous on O if for arbitrarily small positive numbers

s and s, there exists a a > 0 such that

(4)

P{p(g(0, u), g(0', u)) > s} < s,

whenever r(0, 0') < a. Theorem 4. If g(0, u) is stochastically continuous on a compact set O, then g(0, u) is uniformly stochastically continuous.

If this is not the case, there is a pair of positive numbers s and sl, and for arbitrary 3 > 0 a pair of points B and Proof.

0n, for which r(0., 0;,) < 8,,, and P{p(g(0, u),

e} > e, .

We may assume that an ---> 0 and 0 --> 0, as n as n --> oo, and

oa .

Then 0;, -

0o

sl < P{p(g(0, u), g(Bn, u)) > s}

< P{p(g(8., u), g(0,, u)) > 2 + P{p(g(B0, u), g(0 , u)) > 21

This inequality contradicts the hypothesis of stochastic continuity. Theorem 5. Let O denote a separable space and let g(0, u) denote a separable stochastically continuous random function. Then an arbitrary countable everywhere-dense set of points in O can serve as set of separability for the random function g(0, u).

RANDOM FUNCTIONS

156

Proof. Let V = {S} denote the countable set of spheres in O , n, } mentioned in the proof of Theorem 1, let J = {Bk, k = 1, 2,

denote a set of separability of the random function g(0, u), let N denote the exceptional set of values of u that appears in the definition of separability, and let A denote an arbitrary countable everywhere-dense set of points in O. Let B(S, u) denote the closure of the set of values g(-I,, u) as the point 7k ranges over A n S and let N(S, k) denote the event that g(Bk, u) o B(S, u) if 6k e S. The events N(S, k) have probability 0. To see this, let {'y,} for r = 1, 2, ,n,

denote an arbitrary sequence of points in A (1 S that converges to 0k.

Then p{g(Ok, u) 0 B(S, u)} < p{lim p(g(ek, u), g(7 u)) > 0} r-,00

lim p{lim p(g(ek, u), g(7r, u)) > n lim lim P{p(g(Bk, u), g(7r, u)) > n } = 0 .

Suppose that

N' = U U N(S, k) . S BkeS

Then P(N') = 0. If u NUN' and g(-y, u) e F for all -1 e A (1 G, where G is some open set and FcX is closed, then for every 0k e G and S such that Bk e S c G, we have g(Bk, u) e B(S, u) c F .

From the definition of the set {0k}, it then follows that g(O, u) e F for all 0 e G and u 0 NUN'. Thus the set A satisfies the condition in the definition of a set of separability of a random function. 3.

MEASURABLE RANDOM FUNCTIONS

Let O and X denote metric spaces with distance r(01, 02) and p(x,, x2) respectively, let g(0, u) denote a random function with range

in X and domain of definition O, and let u denote an elementary event of the probability space {U, e, p}. Let us suppose that a o'-algebra of sets

containing Borel sets

is defined on O and that a complete measure It is defined on 9. Let oo{SI x Cam} denote the smallest or-algebra generated in O x U by the product of the or-algebras R and a and let v{SI x Cam} denote

its completion with respect to the measure p x P (cf. Chapter II, Section 8).

3.

MEASURABLE RANDOM FUNCTIONS

157

Definition 1. A random function g(O, u) is said to be measurable

if it is measurable with respect to 6{.Q x 01. By definition a random function g(5, u) is Cs-measurable for arbitrary B e O. On the other hand, if a random function is measurable, then on the basis of Fubini's theorem, g(8, u) is .Q-measurable as a function of 0 for p-almost-all u. In other words its sample functions are Q-measurable with probability 1.

Let us now look at conditions that ensure the existence of a measurable separable function stochastically equivalent to a given random function. Theorem 1. Suppose that O and X are compact. If for )ualmost-all 0, a random function g(8, u) is stochastically continuous, then

there exists a measurable separable random function g*(B, u) that is stochastically equivalent to the function g(8, u).

On the basis of Theorem 1 of Section 2, corresponding to the function g(O, u) there is a stochastically equivalent separable random function g(8, u). Let I denote the set of separability of the function g(8, u). As in Section 2, A(G, u) denotes the closure of the range

of g(B, u) as 0 ranges over the set G f1 I and A(O, u) denotes the intersection of all sets of the form A(S, u) where S is an arbitrary open sphere in V that contains 0. By virtue of the separability, g(8, u) e A(O, u) almost certainly (that is, for u 0 N, where P(N) = 0).

On the other hand, if P{g'(B, u) = g(B, u), 0 e I} = 1

and g'(0, u) e A(8, u) (where u 0 N), then g'(B, u) is also a separable random function (cf. Lemma 1, Section 2). Let us construct a function g*(0, u) that is stochastically equivalent to the function

g(B, u) and measurable with respect to the u-algebra v{ft x e}. For arbitrary n, let us cover O with a finite number of spheres Sk") e V, for k = 1, 2, . , m of diameter not exceeding IIn. In each Sk ), let us choose a point 01) e I and let us set g".(B, u) = 9(o), u) for k-1

OeS7) U S j=1

where k=1, 2, measurable.

, m". Obviously the functions g"(6, u) are v{ft x e}Furthermore, P[g"(B, u), g(8, u)] = P[g(Bk"), u), g(O, u)]

if

(l )

RANDOM FUNCTIONS

158 k-1

0 e Sk

I

1

7==1

Also, r(0$ ), 0) 5 1/n. If we set G.,m(B) = P{u; p[gn(0, u), gn+m(O, u)] > b"}

then by virtue of the hypothesis of the theorem, the function G,,m(B) approaches 0 as n --, - for 1.t-almost-all 0. Therefore 1-i x p[0, u; p{gn(0, u),

g.+.(0, u)} > s] =

Gn,m(0)1i(d0) -, 0 J

as n , oo ; that is, the sequence {gn(0, u)} is fundamental with respect to the measure It x P. It contains a subsequence {gnk(B, u)} that converges (p x P)-almost-everywhere to some v{ft x Cam}-measurable function g(O, u). Let M1 denote the set of points (0, u)

at which this convergence does not take place.

For (0, u)

M1,

we have g(0, u) a A(0, u). Since the set M1 has measure 0, it follows that 1t-almost-all its 0-sections have P-measure 0 (cf. Section 8, Chapter II). Let K, denote the set of values of 0 whose corre-

sponding sections have nonzero P-measure. We set 1g(0,u)

for OeIUK1UK,

9(0,u)

for

g*(B,u)-

where K is the set of all 0 at which the limit relation (2) of Section 2 is not satisfied. Then g*(0, u) e A(0, u) (where u 0 N); that is, g*(B, u) is separable. Furthermore it is 6{Q x e}-measurable since it coincides with a measurable function almost everywhere in e x U (exclusive of points 0 e K1 U K and u e N). Furthermore, if 0 (Z K, U K, then by virtue of (1) and the condition of stochastic continuity, P{9(0, u) = g(0, u)} = 1

,

from which it follows that the random functions g*(B, u) and g(0, u) are stochastically equivalent. This completes the proof of the theorem.

We can make a number of statements generalizing Theorem 1. REMARK 1. In Theorem 1 the requirement that the spaces O and X be compact can be replaced with the requirement that they be locally compact and separable. The compactness of the space

X was necessary only so that we might refer to Theorem 1 of Section 2. Now, however, we can refer to Theorem 2 of Section 2. Here the separable and measurable representation g*(B, u) of the function g(0, u) assumes values, generally speaking, in some compact topological extension of the space X. Furthermore, if the space 0

4.

CONDITIONS FOR NONEXISTENCE OF DISCONTINUITIES

159

is locally compact and separable it can be represented as the sum of countably many compact sets.

The reasoning can be applied to each

such compact set in particular. The assertion of the theorem then follows for their union also. Furthermore, the measure p need not be finite; it is sufficient that it be o'-finite. From this we get: REMARK 2. The assertion of Theorem 1 holds for the case in which O and X are finite-dimensional Euclidean spaces and the

measure f p, .Q} is Lebesgue measure in O. Now we note that the proof of Theorem 1 would be simplified if we did not require separability of the measurable representation of the given random function. The set I would not come into the

picture and the points Oj,' could be chosen arbitrarily from the corresponding sets. Of the properties of the space X, we should need to use only its completeness. Thus we have: REMARK 3. If X is a complete metric space, if O is a locally

compact separable space, and if p is a o-finite measure on a aalgebra containing Borel subsets of 0, then the random function g(O, u), B e O, u e U, with range in X, which is stochastically continuous for p-almost-all 0, is stochastically equivalent to the measurable random function. The following result, which has great significance, follows immediately from Fubini's theorem (Theorem 2, Section 8, Chapter II):

Theorem 2.

Let (0) = g(0, u) denote a measurable random

function with real range.

If

J8 I e(o) Ift(de) < then for an arbitrary set B G St,

Me(B)p(dO) =

a

e(B)p(d0) .

The last equation indicates the commutativity of the symbols representing the mathematical expectation and integration with respect to a parameter. 4.

CONDITIONS FOR NONEXISTENCE OF DISCONTINUITIES OF THE SECOND KIND

Let e(t) where t e Z = [a, b] denote a random process with range in a complete metric space X. Definition 1. If sample functions of a process have, with prob-

RANDOM FUNCTIONS

160

ability 1, left- and right-hand limits at every t e (a, b) and have right-hand (resp. left-hand) limits at the point a (resp. the point b), the process is said not to have discontinuities of the second kind. Throughout the present section, we assume that the process g(t) is separable. We let J denote the set of separability of the process.

Definition 2. Let s denote a positive number. A function y = f(t) with range in X is said to have no fewer than m a-oscillations on a closed interval [a, b] if there exist points to, , t,,, where a < to < t1 < .. < t, < b, such that p(f(tk_1), f(tk)) > s

for k= 1, ,m.

For a function y = f(t) not to have discontinuities of the second kind on a closed interval [a, b], it is necessary and sufficient that for arbitrary a > 0 it have only finitely many a-oscillations on [a, b]. Proof of the Sufficiency. Let us prove the existence of the limit f(t - 0) for arbitrary t e (a, b]. Let {tn} denote an arbitrary sequence such that t t t. There exist only finitely many numbers (where Lemma 1.

nk < nk+1) such that p(f(tnk),' `tnk+1)) > a .

Consequently from some m onward, the inequality p(f(tn), f(tn+k))
where k > 0, holds for all n > m, that is, the sequence { f(tn)} converges. This implies the existence of

f(t - 0) =

lim f(r)

.

rtt

Proof of the existence of f(t + 0) on [a, b) is analogous. Proof of the Necessity.

Suppose that one of the one-sided limits

(for example, the left-hand limit) does not exist at a point to. Then there exists a sequence t t to such that for arbitrary n, sup p(f(tm), f(tn)) > a , m>n

that is, there are infinitely many a-oscillations. REMARK 1. Definition 2 can be carried over in a trivial manner to random functions defined on an arbitrary set of real values of t.

If the sample function of a separable process e(t) has no fewer than m a-oscillations on [a, b], it also has no fewer than m aoscillations on the set of separability J except possibly for a set of sample functions N of probability 0. Theorem 1. Suppose that a separable random process e(t), for

4. CONDITIONS FOR NONEXISTENCE OF DISCONTINUITIES

161

t e [a, b], with range in X satisfies the following conditions: a. There exist numbers p, q, r, and C (where p > 0, q > 0, r > 0,

and C > 0) such that, for arbitrary t1, t2, and t3 such that a < t1 < t2 < t3 < b, we have MpP[e(t1),

/

/

1n

1 e(t3)] < C I t3 - t1

I1+1'

/ (1 )

the process e(t) is stochastically continuous on [a, b]. Then c(t) has no discontinuities of the second kind. b.

Proof It follows from (1) and Chebyshev's inequality that P[{p[e(t1), e(t2)] ? sl} n {p[eC(t2), e(t3)] > F'2}]] M pP[e(t1), e(t2)] p9[S(t2), e(t3)] SP

Ea

=

C I t3

- t1

I1+r

6PSE

(2 )

In proving the theorem, we actually use inequality (2) instead of condition (1). It follows from (b) that for the set of separability we can choose an arbitrary countable set that is everywhere-dense on [a, b]. For such a set, let us take the set J of all dyadic numbers belonging to [a, b].

For the sake of convenience let us take [a, b] = [0, 1]. We shall break down the proof into several steps. 1. Let Ak,,, denote the event

{p[e(k 2"

1 ), (2n)] < En}

'

where sn = C(1/P+Q)(2)r(n - 1)/2(p + q) = La- , a = 2-r/2(P+9) < 1

L = (2r/2C)(11P+9),

and

Bkn = Ak,% U Ak+1.n = fPK k 2n 1 )' e(2 )] U

{+(Z ),

e(k 2n

< En}

1 )] <

On the basis of inequality (2), P{Bkn} <

k = 1, 2, ..., 2% - 1

2-(1+r/2)(n-1)'

where Bkn is the event complementary to the event Bkn. define

D. = m=n n nk=1Bkm Then for the complementary event Dn we have P(Dn) < L L P(Bkm) m=n k=1

<

2,n-1

E

m=n k=1

2-(1+r/2)(m-1)

(3)

Let us

RANDOM FUNCTIONS

162 or

p(Dn)

2_(m_1)2

<2

1-

m=n

2-r/1 < 1

(4)

N

From this it follows that for D =

Dn,

p(D) = lim P(Dn) = I 2.

Let us fix k, m, and n and let us set pa,m =

P,,m = p

[e(-)

k 2%

e(. +

'

+

2m+n

i 2'"+%

,

mm = omaxm pa,m

)]'

k+1 2n

mm = max pa!"" 0:9i:92'"

)],

(m=0,1,2,...). Let us choose a sample function e(t) for which the event Ak+l,n n Dn+, occurs. Then

po,o=p1,o=p[e(--) e(k

2 1)]
Since they event Dn+1 occurs, at least one of the two quantities

e(2

+ 21}1)]' [e(.2 + 21}1)' e(

k

2 1 )]

(5 )

less than Lan}1. If this holds for the first of them, then pl,l < Lan+1 and a fortiori p1,1 < L(a" + an+') is

On the other hand,

p1,1 -

P[e(2k

2nI

e(k

+

2+1

< po,° + p1,1 < L(an + an}1) .

In the case in which the second of the quantities (5) is less than Lan+l, the same reasoning can be applied with the roles of the quantities pl,l and pi,, reversed. In both cases, w < L(a" + an}1) (01 < L(an + an+'), Analogous reasoning leads inductively to the inequalities mm < L(an + an+' +

+ an+m) <

Lan

1-a

(6)

,

(0'. < L(an + an}1 + ... + an+m) < Lan

(7)

1-a

Let us suppose that (6) and (7) are satisfied. Since (/k

i

k

pea+S,m+1 5 pi,,n + p 1 2n + 2n+' )' e(2n

+

22i%++=+I')] '

4.

CONDITIONS FOR NONEXISTENCE OF DISCONTINUITIES

2m++1 ),(2

pc+1,m +

pea+I,m+1

163

+ 2± 1)]

and since occurrence of the event Dn+1 implies satisfaction of one of the inequalities p[[F(k

i

2n +

[ (k

+

Ps2n

1

i + 11

2i + l )]

(k

2,n +

2m+n

/k

2n+m

2n

2n+ m}1 J

2i + l )]

+

2n+m+1

)]

<

La-+n+' ,

L n+m+l

a

,

we have P2a+1,m+1 < wm + Lan+m+1. Furthermore, P2i,m+1 = p;,m These relationships prove inequality (6). Inequality (7) is proved in the same way.

Let us show that if the event D. occurs for a sample function e(t), then for arbitrary m there exists a number j,,,m such that -1\ e(k- 1 + j 1] < Lan (8) max P[K k 2n 22n+m (1 - a)2 3.

and

max

9 n, m+15 j 52 m+1

p[c(kn 1 + L\

2

2-+m

)/,

e(k2 111 < l\ /1

Laa)2 (I

(9) Let us note that occurrence of D. implies satisfaction of one of the inequalities

(kl l ( 11 Lan, P[e\2 /' e\ k 2 1 )] < Lan 1 )' e\2 )] < If both these inequalities hold, it follows from (6) and (7) that we can take the number 2m for j,,.. Suppose that only the first of these inequalities is satisfied. On the basis of (6), inequality (8) holds for j < 2m. Consider now the interval [k/21, (k + 1)/2n]. For this interval at least one of the inequalities PrLe\

k

.

2n

[(k) 2-

PPR2k is satisfied.

2k

e\ 2 n+ 1 )] < Lan+1 +1 ), c(k 1 )] < Lan+1

-

(10)

If the first of these is satisfied, then on the basis of (6), 1

PLC(2 )'

\2 +

Lan}1 2n+m

)] <

Consequently max

<

2m-1

L

p[e(k \ 2n

1-a (an + an+1)

e(k

1

2n

1-a

RANDOM FUNCTIONS

164

On the other hand, if the second of inequalities (10) is satisfied, it follows from (7) that +

max 2'm-15i

2'm

52'mfl

p[e(k 2 n 1 +

2 nfm Z

/ e(k

n

1

/J < 1 ana

We can continue this process inductively, and after a finite number of steps we arrive at proof of inequalities (8) and (9). 4. Suppose that the event D occurs. Then beginning with some n0, all the events D. for n > n0 occur for the sample function of the process. For arbitrary s > 0 let us find an n > n0 such that 2Lan/(1 - a)2 < s. We note that on an arbitrary set of the form rk-

JnL

1

2-

'

k+ 1 2n

there cannot be more than a single s-oscillation. To see this, suppose that such a set contains three points to, t1, and t2 such that

2nt0=k- 1 + 10 2nt1 = k - 1 + 2 2nt2 = k

-1+

Zm

where 0 <-j0 < Jl < J2 < 2m}1, and P(e(t1), e(t0)) > s P(e(t2), e(tl)) > s

Then at least two of the three numbers j, J1, j2 lie on one side of the number j.,n (cf. paragraph 3 of the proof). Suppose for example that jl < jn,n. Then I e(t1) - e(t0) I > s, which contradicts the inequality 1

e(t0)) <

e( k 2n

_ + P(S(

2n

n

1 ), (t1)

1

2L a)2

<

Thus the set

J n 1k - 1 L

2n

k+1 2n

cannot contain more than one s-oscillation. From this it follows that the function e(t) has no more than 2n s-oscillations on J. Thus the sample functions of the process e(t) have, with probability 1, finitely many s-oscillations; that is, they do not have discontinuities of the second kind.

4.

CONDITIONS FOR NONEXISTENCE OF DISCONTINUITIES

165

Corollary. Suppose that a random process in the broad sense with range in a complete separable locally compact space is defined on the interval [a, b] and that the "three-dimensional" distributions of this process satisfy condition (1) (or condition (2)). Then there exists a stochastically equivalent random process without discontinuities of the second kind.

We now give a different test for nonexistence of discontinuities

of the second kind. Suppose that 2 8 = a{e(z); z < s} is the ualgebra generated by random elements e(z) where r < s. We introduce the quantity a(s, 6) = inf

sup P{p(e(s); e(t)) >_ e 1 r' ads;tss+asb

8} ,

(11)

where P' is an arbitrary set in a. such that P(P') = 1.

One can easily see that there exists a F, e a{e(z); r 5 s} such that p(P0) = 1 and the greatest lower bound in (11) is attained on the set Po: a(s, 6) =

sup

5e5t$a+85b

P{p(e(s); e(t)) > s I a81

E(r)os

We shall show that the limit condition a(s, 6) 0 as 6 0 for arbitrary positives ensures nonexistence of discontinuities of the second kind for separable processes. Suppose that [c, d] is a fixed interval contained in [a, b] and that I is an arbitrary n-tuple of instants of time tl, t2, , t., where s S c < t, < t2 < ... < t,,, < d. Let A(e, Z) denote the event that a sample function of the random process e(t) on [c, d] n Z has at least one e-oscillation. Lemma 2.

With probability 1, P{A(e, I) I g$} <

Proof.

2a(4 d

- cl

(12)

.

Let us note first of all that since a8 c

for s < t, on

t

the basis of the properties of conditional mathematical expectations (cf. Theorem 3, Section 6, Chapter III) it follows that for s < t < u, P{p(e(t); e(u)) >_ s 19B} = M{P{p(e(t); e(u)) >- e 1 zst} I is.}

a(e, u - t)(mod P)

(13)

.

We now define Bk = {p(e(c), e(t:)) <

2

,

i = 1, 2, ..., k - 1 p(e(c), e(tk))

,

2

RANDOM FUNCTIONS

166

Ck = {p(e(tk), e(d)) ? 4 },

Dk = Bk (1 Ck, k= 1, ..., n ,

Co = {P(ec), (d )) > 4 }

The events Bk are incompatible and A(s, I) c C(, U D where D = Uk=L Dk. To see this, note that if A(s, I) holds, then there exists a k at which the quantity p(e(c), e(tk)) first becomes equal to or greater than s/2; that is, one of the events Bk (for k = 1, , n) occurs. If Ck occurs, that is, if p(e(tk), e(d )) < s/4, then p(e(c), e(d)) ? P(e(c), e(tk)) - P(e(tk),

that is, the event Co occurs.

&)) > 4

Thus A(s, I) c Co U D.

probability 1,

/

P(Dk 1!U.} = M{XD, 2`)B} = M{M{XBkXCk 1

tk} I

;

Now with

91}

/

x = M{XBkP{Ck I 9tk} 19-1 < a`\\4 d - C /M{hBk `2)a}

where X., denotes, as usual, the characteristic function of the event A.

From this we get

a(4 , d - c)M{ E XB,

P{D 12`7S8} = E P{Dk I

a(4 d On the basis of (13), p(Co p(A(s, I) I

$} < P{D I

`258}

cl (mod p)

8} < a(s/4, d - c). Thus

8} + P{C. I a.1 < 2a(S , d - cl (mod P)

\

,

which completes the proof of the lemma. Lemma 3. Suppose that Ak(e, I) denotes the event that e(t) has at least k s-oscillations on I. Then P{Ak(s, I) I

8} <

12a ( 4 , d- C)]k (mod p) .

(14)

Let Br(s, I) denote the event that the sample function of the process e(t) has at least (k - 1) s-oscillations on the set Proof.

(t1, , t,.) but has fewer than (k - 1) s-oscillations on (t1, . , tr_1). The events Br(s, I) (for r = 1, 2, , n) are incompatible and U Br(s, I) = Ak-1(f, I) D Ak(s, I)

r=1

On the other hand, if Ak(s, I) (1 BB(s, I) q, there is at least one s-oscillation on the set (tr, tr+1, , tn). Consequently

4. CONDITIONS FOR NONEXISTENCE OF DISCONTINUITIES

167

Ak(s, I) C r=1 U (Br(s, I) n Cr(E, I ))

where Cr(s, I) means that e(t) has at least one s-oscillation on , tn). Therefore (t,., tr+1, P{Ak(s, I) I ge} < E P{Br(s, I) n Cr(s, I)$} (mod P) .

(15)

r=1

Using Theorem 3, Section 6, Chapter /III, we obtain P{Br(s, I) n Cr(s, I) I

z`

s} = M{M{X 8r(e1I)XCr(e,I) I £tr} I

z`

e}

M{XBr(e,I)P[Cr(E1 I) I "qtr} 12`7Sa}

< 2af 4 , Substituting this inequality P{Ak(s, I) I

d - C)P{Br(s1 I) fig} (mod p)

into` (15), we/

obtain

$} < 2a(4, d - c)E P{Br(s, I) 19e1

=

2a(6

,

d - c)P{Ak-1(s, I) I 8} (mod p)

,

from which the desired result follows. Theorem 2. If e(t) is a separable process and lim a(s, 3) = 0

(16)

s-o

for arbitrary s > 0, the process e(t) has no discontinuities of the second kind.

It will be sufficient to show that with probability

1

every

sample function e(t) has only finitely many s-oscillations. Let J denote the set of separability of the process g(t). Let us represent

it in the form J= U In n=1

where I. is an increasing sequence of sets each consisting of finitely many elements. Let s denote any positive number. We partition

the interval [a, b] into m subintervals Or for r = 1, equal length, so that

, m, all of

2a(4, bma)=''<1 Then

p{A°°(s, in or)

18} < p{Ak(s, J n Or) I Jim P{Ak(s, In n Or) I z s} ; Rk

RANDOM FUNCTIONS

168

so that P{A°°(s, I n A,) I

a} = 0 (mod P)

Consequently, P{A`°(s, J)} = 0. theorem. REMARK 1.

and

p{A°°(s, in OT)} = 0

This completes the proof of the

Under the hypotheses of Theorem 2 the process

e(t) is stochastically continuous. If s < t, then P{p(e(s), e(t)) > e} _

MP{p(e(s), g(t)) > s Ie} < a(s, 8) for t - s < S. Corollary. A separable stochastically continuous process with independent increments has no discontinuities of the second kind.

On the basis of the definition of processes with independent increments and uniform stochastic continuity (cf. Theorem 5, Section 2), we have Proof.

P{p(e(s), e(t)) >> s I

z`

8} = p(p(e(s), fi(t)) > e} < p (s, d) (mod P)

where t - s I < 3, p (s, 8) is independent of t and s, and p (s, d) as a --. 0. Thus the hypotheses of Theorem 2 are satisfied. I

0

We recall that if a process is separable the values of the sample functions e(t) are, with probability 1, the limiting values of sequences {e(t;)} as t; -+ t, where each tti belongs to the set of separability.

Here if a process has no discontinuities of the second kind, then

e(t) will, with probability 1, be equal to (t - 0) or (t + 0) for every t.

Theorem 3. If e(t) is a stochastically continuous process without discontinuities of the second kind, there exists a process e'(t) equivalent to it whose sample functions are continuous from the right (mod P). Proof.

Let A denote the event that the limit lim E(t + 1 ) n-. \ nf

exists for each t c [a, b]. The probability of this event is 1. Let us set e'(t) = e(t + 1/n) for the outcome A and e'(t) = g(t)

for the outcome A. We then have {R'(t) # e(t)} = mUI {P(et), e'(t)) > -M1 n A , PW(t) # g(t)} = lim P({P((t),'(t)) > m } n A) /

On the other hand

S.

CONTINUOUS RANDOM FUNCTIONS

1

> m}

P{P(e(t), c'(t))

= P{U1

169

n {p(w), e(t +

n

> m I}

n

= limp lnn

m}}

lim P{p(e(t)' e(t

+ n )/ > m} Thus p{e'(t) =t- c(t)} = 0. We note that with outcome A the function e'(t) is continuous from the right. This completes the proof of the theorem. .

One can prove by an analogous method the existence of a stochastically equivalent process that is continuous from the left. 5.

CONTINUOUS RANDOM FUNCTIONS

Let Z denote the interval [a, b], let X denote a complete metric space, and let e(t) denote a random process defined on Z into X. Definition 1. A process e(t) for t e Z is said to be continuous if almost all sample functions of the process are continuous on Z. For processes without discontinuities of the second kind we can establish a rather simple sufficient condition for continuity. and k = 0, 1, , mn Theorem 1. Let {tn,k} for n = 1, 2, denote a sequence of partitions of the interval [a, b]:

a = t, ,0 < tn,1 < ... < tn,'mn = b , define

n=

max (tn,k - to k-1) '-' 0

as

15ksm

n , oo

If a separable process e(t) has no discontinuities of the second kind, then the random process is continuous if for every s > 0, m,n

E P{p[e(tn,k), ee(tn,k-)] > 6} --' 0

as

n

k=1

Let v, (where 0 < v, < oo) denote the number of values at which p[e(t + 0), e(t - 0)] > 2s and let vEn) denote the number of indices k for which p[e,(tn,k), S(tn,k-1)] > s. Obviously Proof.

of t

vE S limn-. vEn).

On the other hand mn

MvE

= Lk=1P(o[e(tn,k), e(tn,k-1)] > 6}

RANDOM FUNCTIONS

170

By virtue of Fatou's lemma (Theorem 2, Section 5, Chapter II), MvE < M lim v, < lim MvE"' "__

"-)00

Thus Mv, = 0; that is, v, = 0 with probability 1 for arbitrary s > 0. Consequently, for arbitrary t we have (t - 0) = e(t + 0) with probability 1. By virtue of the separability of the process,

fi(t) = e(t - 0) = (t + 0); that is, the process is continuous. Theorem 2. Suppose that there exist three positive constants for arbitrary e > 0, C, r, and G such P{P[W(t1), PAW), c(t2)] > e} < CI t2

-

tl

Ir}1

(2)

/ If the process g(t) is separable, it is continuous. Proof. Condition (2) is a special case of condition (2) of Section

4 (with q = 0 and ,8 = p). In addition, condition (2) ensures stochastic continuity of the process. Therefore the process $(t) has no discontinuities of the second kind. If {t,,,,} for n = 1, 2,

and k = 0, 1, . , m" is a sequence of partitions of the interval [a, b], then E P{P[e(t.,k), e(t",k-1)] > e} G k=1

CI tk - t",k-1 k=1

I"'

ea

< C(b - a) max I 4,k - tn,k-1 '-' 0 F,

as

X" = max I tn,k - t",k-1 I -- 0 15k;%="

By virtue of Theorem 1, the process e(t) is continuous. REMARK 1. Condition (2) of Theorem 2 can be replaced with the somewhat more stringent but, from a practical point of view, more convenient inequality MPp[e(t), e(t2)] < C I t2 - t, I1+r .

(3)

Applying Chebyshev's inequality to the left-hand member of inequality (2) and keeping (3) in mind, we obtain the right-hand member of (2). These two theorems give only sufficient conditions for continuity of a random process. For the particular case of processes with independent increments, the conditions of Theorem 1 are also necessary.

Theorem 3. If a process E(t) for t e [a, b] with independent increments is continuous, then condition (1) is satisfied for an arbitrary

S.

CONTINUOUS RANDOM FUNCTIONS

171

and k = 0,

sequence {tn,k} for n = 1, 2, the interval [a, b] such that

, mn of partitions of

max (tn,k - tn,k-1) --' 0

15k:g.,

Proof. Let us set Ok = sup, tl_t2i5k p,e(t1), e(t2)]. Since the process e(t) is continuous on [a, b], it follows that A, ,O as h ,O with probability 1. Therefore limk..o p IAk > E} = 0. On the other hand, if X. < h we have e(tn,k-1)]7 > E}

P[O, > E} > P{Sup

p{p[e(tn,1), EE(tn,0)] >E}

+ P{oetn,l), e(tn,)7 < E}p{p[S(tn,2)' S(tn,l)] > E} m,b-1

... + H PIP, (tn. k), e(tn,k-1)] k=1

E}

X p{p[e(t,.,mn), W.,.,,-1)] > E} m +n

e P p{Ok :-< E}L P{p[e(tn,k), S(tn,k-1)] > E} ,

k=1

from which it follows that for arbitrary s > 0, I p{p[e(tn,k), e(tn,k-)] > e} C k=1

p{Ok > E} P{Qk 5 E}

- .0

This completes the proof of the theorem. From Theorem 2 of Section 2 and Theorem 1 of the present section we get another test for continuity of a process. Theorem 4. If a process e(t) is separable and as h --> 0.

lima(') =0

(4)

a-.0

for arbitrary s > 0 where a(E, S) is determined by formula (11) of Section 4, then the process e(t) is continuous.

Since satisfaction of condition 4 implies that the process c(t) has no discontinuities of the second kind, it will be sufficient to verify relation (1). Remembering that p{p[e(tn,k), e(tn,k-1)] > E}

a(E, Otn,k)

where Otn,k = tn,k - tn,k-1, we obtain the result that E p{p[e(tn,k), S(tn,k-1)1 > E} < (b - a) max a(E, Otn,k) 15k6n

k=1

as X.

0.

__, 0

Otn,k

This completes the proof of the theorem.

Let us look at the condition for continuity of a Gaussian process e(t) that assumes real values. For the characteristic function of the random variable e(t) we have the expression

RANDOM FUNCTIONS

172

'`p'(?, t) = Meaaect> = exp (im(tp.

- 2 U2(t) X2

If a process e(t) is continuous, it follows from Lebesgue's theorem on taking the limit under the integral sign that T(x, t) is continuous with respect to t for arbitrary X. This in turn implies continuity of the function

In T(t, X) = im(t)?. - a2(t) and hence continuity of the functions

with respect to t

o,2(t) =In NP(?, t) + In Y'(- X, t) and

ix

m(t)

[1n P(X, t) +

2

E(t)]

.

Thus continuity of the functions m(t) and c2(t) is a necessary condition for continuity of a Gaussian process. Turning to sufficient conditions for continuity, let us suppose that m(t) = Mc(t) = 0. (If this is not the case, we may consider the process e'(t) = c(t) - m(t) instead of e(t).) Let R(t1, t2) denote the correlation function of the process e(t). Then M(e(t2) - e(t1))z = R(t2, tE) - 2R(t1, t2) + R(t1, t1) = OR, M(e(t2) - ee(t1))Em _ (2m - 1)! I OR I-

.

Using Remark 1, we obtain the following result: Theorem 5. If there exist C > 0 and a > 0 such that m(t) is continuous and I R(t2, tE) - 2R(t1, t2) + R(t1, t1) I < C I tE - t1 I mo, t; a [a, b] ,

then a separable Gaussian process with mathematical expectation m(t) and correlation function R(t1, tE) is continuous. For Gaussian processes with independent increments we can go even farther. In such a case, R(t1, t2) = R(t1, t1) = c2(t1) for t1 < tE, OR = a2(t2) - a2(t1), and E1 PJI Wk) - \tk-1) I > E} < 64 L MI Wk) - Wk-1) 14 k= E [QE(tk) E4 k=1

<

- 6E(tk-1)]E

U22 (a)]

E

3[Q (b) 64

max

[o2(tk) - 6Eltk-1)]

5.

CONTINUOUS RANDOM FUNCTIONS

173

Thus if the function c2(t) is continuous, then by virtue of the corollary to Theorem 2 of Section 4 and Theorem 1 of the present section a separable Gaussian process with independent increments is continuous.

Theorem 6. For a separable Gaussian process e(t) with independent increments to be continuous for (a < t < b), it is necessary and sufficient that the functions m(t) = Me(t) , o2(t) = M[ee(t) - m(t)]2

a
V LINEAR TRANSFORMATION OF RANDOM PROCESSES

In this chapter, we shall consider linear operations on random Examples of such operations are differentiation and

processes.

integration of processes, and transformations by means of differential and integral equations. The problem of prediction of the value of

a random function or the problem of filtering a useful component from the observed values of a random function that is the sum of a transmitted signal and a distorting "noise" is actually solved, at

the present time, only in the framework of the theory of linear transformations of random processes. To study the problems that are presented here we shall consider

a Hilbert space of random variables, which in a number of cases enables us to obtain final solutions that are suitable for applications. It is assumed that the reader is familiar with the elementary concepts

of the theory of Hilbert spaces, a few facts of which are recalled in Section 1.

1.

HILBERT SPACES

Let H denote a linear space over the field of complex numbers. Let (x, y) denote a complex-valued function defined for x, y c H with the following properties: a. (x, x) > 0 with equality holding only for the zero element of the space H; that is, (x, x) = 0 implies x = 0 and conversely; b.

(x, y) = (y, x);

c.

(Xx1 + px2, y) = X(x1, y) + 1 (X2, y).

We call (x, y) the scalar product of the elements x and y of the

space H. We use this product to introduce the concepts of the length of a vector x, the distance between two vectors, and orthogonality of two vectors. Specifically, the length of a vector x is defined 174

1.

175

HILBERT SPACES

as 7/(x, x) and is denoted by 11 x I I. The distance between two vectors x and y is defined by p(x, y) = ll x - y = V(x - y, x - y). We have the following inequalities: I (x, y) 12 < (x, x)(y, y)

or

I (x, y) I< I I x I

I

I

lyl

l

Ilx+yll < Ilxll + Ilyll

(1 )

(2)

The first of these is called the "Cauchy-Schwarz inequality" and the second is called the "triangle inequality."

Two vectors x and y are said to be orthogonal if (x, y) = 0. We indicate their orthogonality by writing x 1 y. Two sets A and B contained in H are said to be orthogonal (and we write A 1 B) if every vector x e A is orthogonal to every vector y e B. We note that for a pair of orthogonal vectors x and y, llx +y112 = 11x1

2+lyll2-

A linear space X with a scalar product defined on it is called a Hilbert space if it is complete, that is, if every sequence of elements of H such that I x 0 as n , 00 uniformly with respect to m > 0 converges to some limit in H. An example of a Hilbert space is the space of functions L. that was introduced in Section 5 of Chapter II. The elements of this space are the complex-valued functions f(u) defined on some I I

I

space U with measure { U, C5, fc} that are C5-measurable and squareintegrable: JU

l f(u)12 ,u(du) < oo

.

The scalar product of two functions f and g in L2 is defined by

(f' g) _

f(u)g(u),u(du)

In particular, f and g are orthogonal in L2 if f(u)g(u)p(du) = 0. Completeness of the space L2 follows from Riesz's theorem (Theorem 7, Section 5, Chapter II). Let x(8) denote a function defined on an arbitrary set O into H and let *(0) denote a nonnegative function defined on © that assumes arbitrarily small positive values. An element h of H is called the limit of the function x(O) as *(0) 0 (and we write either

h = lim x(O) or x(O) , h as *(0)

0), if for arbitrary positive s

there exists a positive 3 such that 1 x(B) - h I I < s for 0 < 1r(B) < 6 . The scalar product is a continuous function of both variables: 1

If

LINEAR TRANSFORMATION OF RANDOM PROCESSES

176

0 (for i = 1, 2), then (x,(O), x2(O)) , (hl, h2) as xi(B) ---p hi as *(0) 0. We have the following condition for the existence of a limit: Lemma 1. For the limit lim x(B)

to exist, it is necessary and sufficient that the limit lim

(x(O'), x(O")) = A

(3)

'1"(8')4-'(0")-0

exist.

Proof. The necessity follows from the continuity of the scalar product. Here if h = lim x(O), then urn

)yo (x(8'),

(4)

x(B" )) _ (h, h)

To prove the sufficiency we note that x(O') - x(O") I = (x(O') - x(O"), x(O') - x(O") = (x(6'), x(O')) -2 Re (x(O'), x(O")) + (x(O"), x(B") I2

In equation (3) the number A is nonnegative. Therefore, for arbitrary positive s there exists a positive a such that I I x(O') - x(O") I I <62 whenever 0 < *(0') + *(O") < a. Now there exists a sequence 10.1

is a fundamental sequence. such that 0 < 1/r(OJ < 1/n and By virtue of the completeness of the space, there is a limit lim x(O) = h. Then 11 x(O) - h 11 < 11 h - x(O) 1 + 11 x(O) - x(8)

as n --> oo if

0

*(0) , 0 as n , oo. This completes the proof

of the lemma. A subset H' of H is called a subspace if it is linear and closed in H. One of the basic elementary facts from the theory of Hilbert spaces is the following: For a fixed but arbitrary subspace F of a

Hilbert space H and for an arbitrary vector x e H, there exists a unique decomposition of the form

x=xF+xN where xF e F and xN 1 F. The vector XF is called the projection of the vector x onto the subspace F. The projection xF of the vector

x onto the subspace F enjoys the elementary property that the distance between x and a vector x' belonging to F attains a minimum value if and only if x' = xF. One can easily see that the set of all vectors y that are orthogonal to F constitutes a subspace. It is called the orthogonal complement of F and is denoted by N. The

1.

177

HILBERT SPACES

fact that N is the orthogonal complement of F is indicated by N = H G F or H = F ® N. If this last relation is satisfied we say that the space H is the orthogonal sum of the subspaces F and N. The equation

Hl = F, ®F2 ® ... ®F.® . ..

,

) are subspaces of H, means that the subspaces F; and F; are for arbitrary i and j (with i j) orthogonal and that an arbitrary vector x e H can be represented in where H 1 and Fk(for k = 1 , 2,

the form of a series x = x1 + x2 + ... + x +

.

, where x,e F

In such a case, we say that H1 is the orthogonal for n = 1, 2, sum of the subspaces F.

Let G denote a subset of H. Let us consider all possible sums of the form Ek=1 Xkgk where n is an arbitrary integer, the Xk are arbitrary complex numbers, and gk E G (for k = 1, , n). The set of all such sums is called the linear hull of G. The closure of the linear hull of G is called the closed linear hull of G. This closed linear hull of the set G is the minimal subspace containing G.

Let {la, a e Al denote some set of mutually orthogonal normalized vectors, that is, such that (Ia., lp) = 0 for a = Q, (la, la) _ I 2 = 1. Such a system of vectors is said to be orthonormal. An orthonormal system {la, a c Al is said to be a basis in H if the closed linear hull of the set {la, a e Al coincides with H. For an arbitrary orthonormal system of vectors {la, a e Al and 11 la

I

an arbitrary vector x e H we have the inequality EI(x,la)I2< IIx112

aEA

The left-hand member of this inequality has only countably many nonzero terms. This inequality is called Bessel's inequality. For an orthonormal system to be a basis for a space, it is necessary and sufficient that equality hold in Bessel's inequality for an arbitrary vector x in that space, that is, it is necessary and sufficient that for arbitrary x c H,

IIXI12=E(x,la)I2 aEA

(5)

Equation (5) is called the "equation of closure." If H has a basis with countably many vectors, every other basis is also countable. This will be the case if and only if H is a separable metric space. Let A denote a linear operator mapping H into H. 11 A I I= sup J1 A(x) 11 11-1 1<1

The quantity

178

LINEAR TRANSFORMATION OF RANDOM PROCESSES

is called the norm of the operator A. Finiteness of the norm of an operator is a necessary and sufficient condition for continuity of that operator. Operators having a finite norm are said to be bounded. For an arbitrary bounded operator A there exists a bounded operator A* satisfying the relation (Ax, y) _ (x, A* y)

for every pair of elements x, y in H. The operator A* is called the adjoint of A. If A = A*, we say that A is a bounded self-adjoint operator. An example of a bounded self-adjoint operator is the operator PF indicating the projection of P onto a subspace F. This operator is defined by the equation PF(x) = XF

where xF is the projection of x onto F. The following relations are characteristic of the projection operator:

PF = 1

PF=PF

.

An operator S mapping H onto H is called a unitary operator if it preserves the scalar product, that is, if (Sx, SY) = (x, Y)

A unitary operator is always linear and always has an inverse that coincides with its adjoint: S-1 = S*. An example of a unitary operator is the Fourier transform (D in the space L2 of square-integrable functions defined on (- oo , oc ) with respect to Lebesgue measure: If f e L2 then

g(Y) = V f) = 1/ - J_-e `--f(x)dx where the improper integral is to be understood to mean -o

(6)

= l.i.m.

A,B-.}oe -B

The inverse (adjoint) operator to the Fourier transformation operator is given by the formula (in L2).

*(g) = 1/Tc

_

g(Y)e'--dy

.

The unitariness of the operator 1 means that Jm f (x)f2(x)dx = _g1(Y)g2(Y)dy

where gk = 4)(fk) for k = 1, 2.

(7)

The Fourier transform gt(y) = e-it&go(y) = e-'tvg(y) corresponds to the function ft(x) = f(t - x);

1.

179

HILBERT SPACES

hence it follows from formula 7 that

fit - x)f(x)dx =

e`ivg2(y)dy

(8)

Relationship (7) is called "Parseval's equality."

Let S denote a linear operator. A subspace H, of H is said If in addition the space H2 = H 0 Hl is also S-invariant, we also say that H1 reduces S. to be S-invariant if SH, C H1.

Lemma 2. If S is a unitary operator mapping H1 onto H1, then H1 reduces S. Proof.

We need to show that (Sy, x) = 0 for all x e H1 and

y e H2 = H e H1. From the hypothesis of the lemma it follows that S-'H1 C H1. Therefore S-1x = x' e H1. Consequently (Sy, x) _ (y, S-'x) = (y, x') = 0 for all x e H1 and y e H2.

A generalization of the concept of a unitary operator is the concept of an isometric mapping. Let H1 and H2 denote two Hilbert spaces with scalar products (x, y), and (x, y)2. A mapping V of the space H1 onto H2 is said to be isometric if (Vx, Vy)2 = (x, y)1 for arbitrary x, y e H1. An isometric mapping is always linear and has an inverse isometric mapping. If there exists an isometric mapping of H1 onto H2, the spaces H, and H2 are said to be isometrically isomorphic.

Let Gi denote a subset of Hi (for i = 1, 2). Let L(G;,) denote the linear hull of the set Gi and let H(Gi) denote its closure in H. Let us suppose that there exists a one-to-one mapping y = '/r(x) (for x e G, and y e G2) between G1 and G2 such that (x1, x2)1 = (y1, Y2)2, yi = *(xi)(i = 1, 2), xi a G1, yi e G2. Theorem 1. A one-to-one scalar-product-preserving mapping y = a/r(x) on G1 onto G2 can be extended to an isometric mapping of H(G1) onto H(G2). Proof. Let us extend the correspondence y = *(x) between G, and G2 in two steps. First let us establish a one-to-one correspondence

between L(G) and L(G2) and then between H(G) and H(G2). We define

*(i\k=1 X"xkl/ =

(9)

k=1

for arbitrary n, xk a G1, and complex numbers Xk (for k = 1, This is an unambiguous definition. Indeed, if Z

Z X kxk = E f'"kxk , k=1 k=1

xk e Gl ,

, n).

xk e G1 ,

then without loss of generality we can assume that n = m and xk = xk

LINEAR TRANSFORMATION OF RANDOM PROCESSES

180

(for k = 1 , , n) (since some of the Xk and p; may be equal to 0 we may arrange the sets as required). Then n

n

X k'*(xk) k=1

k=1

2

f'"k*(xk)

2

n

E (xk - Pk)(X, - f r)(Y' (xk), *(xr))2

k,r=1

L (Xk - t^k)(Xr - P?)(xk, xr)1

k,r=1

2

H

Xkxk k=1

k=1

Pkxk

=0,

that is, *(E' Xkxk I = *(E' 1 L PkxkJ

These same considerations also indicate that the converse is true, that is, that for every y e L(G2), equation 9 puts exactly one element x e L(G1) in correspondence with It also follows from (9) that this icorrespondence is linear and it follows from the formula Y.

11CL1 akYk, k=1 L RkYk 12 = Ek,r=1 ak9r(W (x11)1 J (xr))2 =k,r=1 E

xr)1

+

L akxkl E 8kxk k=1 k=1

Yk =

,,I,,

Y' (xk)

)11

that it preserves the scalar product. Thus we have established an isometric correspondence between L(G1) and L(G2). Suppose now that x e H(G1). Then x = lim x (where each x e L(G1)) as n -- oo. If yn =

for yn e L(G2), then I Yn - Yn+m 1 I

12

= I I xn - xn+m 112 -. 0

where m > 0, as n -- oo; that is, l.i.m. yn = yo exists. Let us set yo = i/r(x). To prove the single-valuedness of this definition let us take an arbitrary sequence {x'}, where each x;, e L(G1), such that lim x;, = l.i.m. x,,. We-need to show that l.i.m. y;, = urn 'r(x) = yo. If x2, = x" and 4_1 = xn (for n = 1, 2, ), then l.i.m. x,'; = x and on the basis of what we have shown, l.i.m. y" exists. From this it follows that l.i.m. *(x2n) = l.i.m. *(xn) = y" and

y" = l.i.m. *(4'n-1) = l.i.m. *(xn) = Yo Now it is easy to show that the extended mapping y = *(x) is a oneto-one mapping of H(G1) onto H(G2) and that it preserves the scalar product. This completes the proof of the theorem.

2.

2.

HILBERT RANDOM FUNCTIONS

181

HILBERT RANDOM FUNCTIONS

Let {U, e, p} denote a probability space. Definition 1. The set of complex-valued random variables f(u) for u e U such that M I C 2 < - is called the Hilbert space L2 = L2(U, Cam, P) of random variables of the probability space {U, Cam, P}.

The scalar product in L2 is defined by (C, '2) = MCP,

C,

2

Corresponding to this definition, we have the norm random variable C

of the

:

C

I = {M I C

I2}1f 2

.

Two random variables 5 and r) are said to be orthogonal if

(C, I) = Kf = 0 . The square of the norm 11 C I I2 of a real random variable C coincides I2 with the second-order moment = M I C I2, and if MC = 0 it coincides with the variance. If C and - are both real and MC = M, = 0, their orthogonality implies that they are uncorrelated. 11

I

Definition 2. A complex-valued random function C(B) for 0 e O is called a Hilbert random function if

MIC(B)I2<-,

0e0-

A Hilbert random function can be regarded as a function defined

on 0 into the Hilbert space L2: B---'C(O) = f(B, u)eL2

In particular, if 0 is a real interval (a, b) the Hilbert random function should be regarded as some curve in the Hilbert space L2. The representation C = C(O) for B e (a, b) is a parametric equation for that curve. In the present chapter we consider only Hilbert random functions. For this reason we shall often drop the word "Hilbert." Suppose that the nonnegative function *(0) defined on 0 assumes arbitrarily small positive values. Definition 3. A random variable 77 e L2 is called the mean-square

limit (abbreviated m. s. limit) of a Hilbert random function C(O) as *(O) --+ 0 if C(9) - 77 as *(0) 0 in the sense of convergence in

the Hilbert space L2, that is, if for arbitrary s > 0 there exists a 8 > 0 such that MI)-VO)I2<62

for all 0 such that 0 < *(0) < 8.

LINEAR TRANSFORMATION OF RANDOM PROCESSES

182

In particular if p is a metric space with metric r(01, B2), the function C(0) is said to be mean-square continuous at a point 00 e O if m I C(B) - C(0) 2 --, 0 for r(01, 00) 0 . (1) Definition 4. The covariance B(01, 02), where 01, 02 e O, Hilbert random function c(B) is defined as the quantity

of a

(2) B(01, 02) = Mb(B)b(02) = (5(0), C(02)) . Lemma 1. For a random function C(0) to have a mean-square limit as *(0)

0, it is necessary and sufficient that the limit lim B(01, 02)

If this condition is satisfied, then

as i/r(01) + Lr(02) --> 0 exist. M

172

2

=

lim

B(01, 0)

+"*(02)-'0

(3)

where

77 = 1.i.m. V0) .

This lemma is a rephrasing of Lemma 1 of Section 1. Corollary 1. For C(B) to be mean-square continuous at a point 00 it is necessary and sufficient that the covariance B(01, 0Z) be continuous at the point (00, 0o).

The necessity follows from Lemma 1 and the sufficiency can be proven by simple calculation. REMARK 1. Mean-square continuity of C(B) at the point 0, implies stochastic continuity of C(B) at that point. This is true because, by virtue of Chebyshev's inequality, P{I W) - V0.) I > e} < M I (B) - V0.) 11 2

If l(0) is mean-square continuous on O (that is, at every point of (9), the sample functions are not necessarily with probability 1 continuous on O. Indeed, for a Poisson process we have M I C(t + h) - C(t) = xh + (xh)2 , but the sample functions C(t) are, with positive probability, discon12

tinuous.

Let us examine successively differentiation, integration, and orthogonal-series expansion of Hilbert processes. 1.

Differentiation of Hilbert Processes

a. A random process C(t) for t e Z is said to be mean-square differentiable at a point t0 if the limit

c'(t0) = l.i.m. C(t0 + h) - C(t0) t0, t0 + h e `. h-.0

h

2.

HILBERT RANDOM FUNCTIONS

exists.

183

The random variable C'(t(,) is called the mean-square derivative

of the random process at the point to. It is easy to find necessary and sufficient conditions for meansquare differentiability of a random process. Since M C(to + h) - C(to)

C(to + h) - C(to)

h

hl

hIh {B(to + h, to + h) - B(to, to + hi) - B(to + h, to) + B(to, to)}

(4)

it follows on the basis of Lemma 1 that a necessary and sufficient condition for mean-square differentiability of the process C(t) at the point to is that the generalized mixed derivative a2B(t, t') t=t-=to atat'

lim B(to + h, to + h) - B(to, to + hi) - B(to + h, to) + B(to, to) hh,

h,hj_o

exists.

The mean-square differentiability of the process at the point

t and the inequality

MC '(t) -

< fM

b(t + h) -

,

h

(t) -

b(t + h) - b(t)

h

1

imply that dMC(t)/dt exists and that MC'(t)

= dt

MC(t)

(5)

If a process, C(t) is mean-square differentiable at every point of an interval , then the derivative C'(t) constitutes a Hilbert random process on . Theorem 1. Let C(t) for t e Z denote a Hilbert random process and suppose that the generalized derivative a2B(t, t') tl=t otat' I

exists at every t e Z.

Then the process C(t) is mean-square differentiable

on X and

t') =

a2B(t, t') atat'

aB(t, t')

t) = where

at

(6)

(7)

LINEAR TRANSFORMATION OF RANDOM PROCESSES

184

t') = MC'(t)C'(t') is the covariance of the process C'(t) and

t') = MC(t%'(t) is the mutual covariance of the processes C(t) and C(t). We need to prove formulas (6) and (7). We have t') = MC(t%'(t)

C(t + h) - C(t) h

``

= lim B(t + h, t') - B(t, t') h

consequently the derivative aB(t, t') exists and mutual covariance of the processes C'(t) and C(t) is given by formula (7). lim M C(t' + h') h'

C(t + h) - C(t) h

h,h'-w

lim

Furthermore,

B(t+h,t'+h')-B(t,t'+h')-B(t+h,t')+B(t,t') hh'

h,h'-O

From this it follows that the generalized second derivative a2B(t, t') atat'

exists (the hypothesis of the theorem assumes only that this derivative exists at t = t'). Formula (6) follows. If the process C(t) is stationary in the broad sense, then B(t, t') _

B(t - t'), and the results just obtained take the form of: Corollary 2. A necessary and sufficient condition for differentiability of a process C(t) for t e Z that is stationary in the broad sense is that the generalized second derivative of the covariance B(t) = MC(t)C(0)

exist at t = 0.

Then the generalized derivative d2B(t)/dt' exists and

to + t)

d2B(t) dt2

t, to) = Bas(t) =

dB(t) dt

Analogous results hold for mean-square derivatives of higher orders. 2.

Integration of Hilbert Processes

Let C(t) for t e [a, b],denote a measurable process and suppose that

2.

HILBERT RANDOM FUNCTIONS

185

bM I C(t) 11 dt < 0O . a

On the basis of Fubini's theorem, a

a

Consequently the integral I C(t)12 dt, and hence the integral bC(t)dt, a a exists with probability 1. If the functions f;(t) are square-integrable b

in the sense of Lebesgue over the interval [a, b], the integrals fj(t)C(t)dt

By again using Fubini's theorem, we obtain

also exist.

f2(t)C(t)dt = MIbJ f (t)f(v)C(t)Y(r)dtdz a

a

a

a

= 1J fi(t)B(t, z)f (z)dtdv .

(8)

J

REMARK 2.

The integral

(9)

6C(t)dt J

is defined for a measurable process as a quantity that exists with probability 1 for an arbitrary sample function of the process. But we can proceed in a somewhat different manner. In the first place, the integral (9) can be defined as the mean-square limit of integrals of simple functions that converge to C(t). It is easy to show that this definition coincides with the original one. To prove this, we may confine ourselves to nonnegative random variables.

By definition

the integral (9) is the limit as n --. 00 of Ja

where {C,(t)} is a nondecreasing sequence of random variables that assume finitely many values and such that {C (t)} converges with probability 1 to C(t). Since I C(t) - C(t) it follows from Lebesgue's theorem that M

J'C(t)dt a

E s MJ b I C(t) - C .(t) 12 dt -, 0 - J bbn(t)dt a a

as n --. = , so that bC(t)dt

= l.i.m. aC.(t)dt a

In the second place, the integral (9)

is

often defined as the

LINEAR TRANSFORMATION OF RANDOM PROCESSES

186

mean-square limit of the Riemann approximating sums n+ yy

LE b(tnk)Atnk, Otnk =

k=1

tnk - tn,k-1, a = t., < tn1 < ... < t,. = b

For the mean-square limit of these sums to exist it is necessary and sufficient, by virtue of Lemma 1, that the limit n+

n

m

Mk=1 L

yy

,n

yy

k-1

k=1,

B(tnk, 1

exist as n, m oo, that is, that the function B(t, z-) (for a < t, z- < b) be Riemann-integrable. As we can see, such a definition of the

integral is more restricted than the original definition but it is not connected with the concept of measurability of the process. One can easily show that when the second definition of the integral is used, it leads (mod p) to the same result as the original one. Specifically, 2

M f abC(t)dt - Ek=1 b(tnk)At,tk n

('tnk

(tnr L J J k=1 r=1 tn,k-1 n

tn,r-1

[B(t, Z) - B(t, t_) - B(tnk, r) n

n

L SZnkr --a 0 k=1 r=1

+ B(tnk, tnr)]dtdv < 2 E

where SZnkr is the oscillation of the function B(t, z) in the rectangle tn,k_1 < t < tnk, tn,r-1 < Z < tnr Example 1. Let C(t) (for a < t < b) denote a Hilbert random process and suppose that the derivatives aB(t, t')

and

at

a2B(t, t') atat'

exist for arbitrary t and t' and are finite and Lebesgue-integrable with -respect to their arguments. Let us look at the process C'(t) where C'(t) is the mean-square derivative of the process Z(t). Let us also assume that C'(t) is a measurable process. Then by virtue of equation (8) and Theorem 1 we obtain M

C(a) - C(t)

ar

2

rt

r) dzdr' - 2 Re - 3jfa a2B(Z, aVaZ' t

aB(Z, t) a

aZ

-

aB(r, a) aZ

/

/dr

+MIC(t)-C(a)I2 On the basis of the assumptions we have made, we can use the Newton-Leibnitz formula in evaluating the integrals on the right side of this equation. This leads to the equation

2.

HILBERT RANDOM FUNCTIONS

187

aC'(r)dr+C(a)-C(t)

M

0;

that is, for every t, C(t) = C(a) + JtC'(t)dt

(mod

P)

(10)

.

Thus the process C is with probability 1 representable as an integral of its mean-square derivative, and C(t) is with probability differentiable for almost all t e [a, b]. If we assume that C(t) is a separable process, equation (10) will with probability be simultaneously satisfied for all t; that is, C(t) is, with probability 1, differentiable. Example 2. Let C(t) (for - 00 < t < oo) denote a measurable Hilbert random process that satisfies condition (8) for arbitrary finite a and b. Let us examine the question of the existence of 1

1

l.i.m. 1 ra+TC(t)dt T-- TJa

(11)

.

For the mean-square limit (11) to exist it

is,

by virtue of

Lemma 1, necessary and sufficient that the limit a+T C(t)dt a ,(t)dt a+)a+T' lim

T,T'-.oo

exist.

M-

Ta

Ta a

= lim

T

a

B(t,'r)dtdzr

Furthermore, for the equation l.i.m.

IT (12)

Ja+T C(t)dt - T a+T MC(t)dt} = 0

to hold, it is necessary and sufficient that a+T a+T

1

lim

T2

a

R(t, z)dtdz = 0

(13)

a

where R(t, r) is the correlation function of the process. For a process that is stationary in the broad sense, R(t, r) = R(t - v) and Mg(t) = M. Since TJa+TR(t

TQ a

- r)dtdz- =

J 1

1z

T2

fT o

JT R(t - v)dtdv o

('T

LT

-

R(u)dudt = 1 R(u)(1 u ) du . t-T \ T T T -T We obtain the following result: If C(t) is a process that is stationary in the broad sense, for the equation E

1

t

Jo

l.i.m. T-.oo

1

('a+TC(t)dt

Ta

= MC(t)

(14)

LINEAR TRANSFORMATION OF RANDOM PROCESSES

188

to hold, it is necessary and sufficient that T

lim

T

_TR(u)(1

/du = 0 .

T

(15)

In particular, this condition will be /satisfied if the mean value of the correlation function is 0: T

lim - R(u)du = 0 1

T-- 2T -T

Expansion of a Random Process in a Series of Orthogonal Functions Let C(t), for t c= [a, b] denote a measurable mean-square continuous Hilbert process. Its covariance B(t1, t2) is a continuous nonnegativedefinite kernel in the square [a, b] x [a, b]. According to integralequation theory, the kernel B(t1, t1) can be expanded in a uniformly convergent series of its eigenfunctions an(t): 3.

B(t1, t2) _

n=1

where

bB(t, r)T.(r)d-r ,

Unm

and the eigenvalues X. are positive. We set J

This integral exists (cf. Part 2 of Section 2) and by virtue of (8), Menem = bJb

Xn3nm

mmB(t,

that is, the sequence of random variables {en}, for n = 1, 2, orthogonal. Furthermore, J

bB(t, r)q (r)dv = XncPn(t)

From this it follows that M

/

n

y/

b(t) -

k=1

kcDk(t)

/

n

B(t, t) - 2 k=1 B(t, t)

-

tt)MC(t)_

5k

Xk I Tk(t) 2 k=1

which approaches 0 as n --> -.

n

k + E,k=1)1k ,

I cPk(t) 2

, is

2.

HILBERT RANDOM FUNCTIONS

189

Theorem 2. A measurable mean-square continuous Hilbert process

((t), for t e [a, b], can be expanded in a series b(t) = Lk=1 ekcPk(t)

(16)

In this expansion, {$k} is an orthogonal sequence of random variables M I $k I2 = Xk where the Xk are the eigenvalues and the gk(t) are the eigenfunctions of the covariance

that converges in L2 for every t e [a, b].

B(t, r) of the process. REMARK 3.

If the process C(t) is a Gaussian process, its mean-

lt)C(t)dt are also

square derivative and all integrals of the form

Jm

Gaussian.

REMARK 4. If C(t) is a real Gaussian process and MC(t) = 0, then the coefficients ek in the series (16) are independent Gaussian variables and the series (16) converges with probability 1 for every t. To see this, note that independence of the variables ek implies

that they are orthogonal and Gaussian. For the series (16) to converge with probability 1 it is sufficient that the series E M($kgk(t))2 = Ek=1 X 'k I qk(t) I2 k=1

converge. B(t, t)).

But we have already seen that this series converges (to

Example 3. Let us look at the expansion of a process of Brownian motion C(t) (where C(0) = 0 and MC2(t) = t) on the interval [0, 1] in an orthogonal series. Here MC(t) = 0, B(t, S) = MC(t)C(s) = min (t, s). The eigenvalues and eigenfunctions of the kernel B(t, s) are easily found. From the equation Xnq)n(t) = J1 min (t, s)co (s)ds = tsgn(s)ds + J1tpn.(s)ds 0

t

o

we see first of all that q).(0) = 0. Differentiating with respect to t, we obtain X, p,' (t) = cp,,(s)ds from which we get p ,'(l) = 0. By t successive differentiation we arrive at the equation ) q;;(t) _ -an(t). The normalized solutions of this last equation that satisfy the boundary conditions are of the form

Xn1=(n+2)tr2, z

Pn(t)=1/2sin(n+ Thus

Dirt

n=0,1,...

LINEAR TRANSFORMATION OF RANDOM PROCESSES

190

e(t)=1/2 Len n=0

(17)

where {fin} is a sequence of independent Gaussian random variables with parameters 0 and 1. For fixed t this series converges with probability 1. Another expansion of a process of Brownian motion can be obtained as follows: Let us set e(t) = C(t) - tC(l). Then c(t) is a Gaussian process with convariance B1(t, s) =min (t, s) - is and Me(t)= 0. The eigenvalues and eigenfunctions of the kernel B1(t, s) are

found in the same way as in the preceding case. We again arrive an(t) with boundary conditions q),(0) at the equation X, (t) qn(1) = 0. The solutions of this equation with these boundary

conditions are of the form an(t) = 1/ 2 sin n7rt, A,n1 = n27r2, n = 1, 2,

.

Thus

VT

fi(t) = fi(t) -

n=1

sin n7rt

en

(18)

n7r

where {fin}, for n = 1, 2, , is again a normalized sequence of independent Gaussian random variables. Moreover,

n = 1/ 2 `le(t) sin nirtdt

.

0

Since MC(l) = 0, MC2(l) = 1, and

VT M(C(t) - tC(l))g(l) sinn7rtdt = 0, 0

if we set 0 = C(l) we obtain fi(t) =too + 1//-f

n-1

and the sequence {fin} for n = 0, 1, 2, as the sequence {fin} where n = 1, 2,

n sinn7rn7rt .

,

(19)

enjoys the same properties The series (19) converges

(for every t) with probability 1. 3. STOCHASTIC MEASURES AND INTEGRALS

In a number of processes an important role is played by integrals

of the form (1 ) Jf(t)<(t) , where f(t) is a given (nonrandom) function and C(t) is a random process. The sample functions of the process C(t) are, in general, functions of unbounded variation, and we cannot regard the integral

3.

STOCHASTIC MEASURES AND INTEGRALS

191

(1) as the Stieltjes or Lebesgue-Stieltjes integral that exists for almost

all sample functions C(t). Nonetheless, the integral (1) can be defined in such a way that it enjoys the properties associated with an ordinary integral. In this section we shall consider the definition and properties of the integral corresponding to integration with respect to a random Such integrals are called stochastic integrals.

measure.

Let { U, e, P} denote a probability space, let L2 denote L2{ U, C5, P},

let E denote some set, and let 31 denote a decomposable family of subsets of E. Suppose that to each A e T? is assigned a complexvalued random variable p(0) satisfying the following conditions: 1.

p(0) e L2, fi(o) = 0;

2.

[t(A1 U A) = lt(&) + )u(02)(mod P) if A, n A,

3.

M("(,N1)p(L2) = m(z1 n A2);

where m(A) is a set function defined on 3J2.

We shall refer to the family of random variables {p(A)} for as an elementary orthogonal stochastic measure and we shall refer to m(0) as its structural function. The orthogonality of a stochastic measure is expressed by condition (3) if A, n A. = q, the e

t

variables p(A) and t(A) and lc(A) are orthogonal. It follows from the definition of m(A) that m(A) is nonnegative:

m(0) = 0

m(A) = M I P(A) I2 > 0 ,

and additive: A1n02=o, 12

m(A1 n A2) = M I p(A) + = m(A1) + m(A2) + 2m(A1 n o2) = m(A1) + m(A2)

Thus m(A) is an elementary measure on P2 (cf. Definition Section 7, Chapter II). Let L(IJ2) denote the class of all simple functions f(x):

f(x) =

ki

ckxok(x) ,

Ok et

k = 1, 2, ..., n ,

,

5,

(2)

where n is an arbitrary natural number and XA(x) is the characteristic

function of the set A. Let us define the stochastic integral of a function f(x) e L{It} with respect to an elementary stochastic measure p by ftx)ft(dx) =

i

Ckf"(Ak)

(3 )

2 is a decomposable family of sets, every pair of functions in L{P2} can be represented as linear combinations of the characteristic functions of the same sets in R. Therefore, if f and Since

LINEAR TRANSFORMATION OF RANDOM PROCESSES

192

g belong to L{Jt} we assume that f(x) is given by formula (2) and that g(x) = Z dkxok(x)

where pk l pr = o for k

r.

The orthogonality of u implies that E ckdkm(Ok)

(4)

k=1

We assume that the elementary measure m is subadditive and hence can be extended as a complete measure {E, 2, m} (cf. Theorem

2, Section 7, Chapter II). Then L{WZ} is a linear subset of the Hilbert space L2{m} = L2{E, 2, m}, and L2{m} is the closure of L{J2}

in the topology generated by the scalar product

(f, g) = 5f(x)jm(dx)

(5)

(cf. Theorem 7, Section 7, Chapter II). Here equation (4) can be written in the form M(J f(x),u(dx)Jg(x) fc(dx))

= f(x)g(x)m(dx) J

(6)

for an arbitrary pair of functions f(x) and g(x) in L{1Z}. We now introduce the linear hull L{/t} of the family of random variables { fc(0)} defined for A e TI-that is, the set of random variables that can be represented in the form `(3)-and the space L2{1 }, which is the closure of L{,u} in the Hilbert space of random We note that equation (3) sets up an isometric correspondence rf = *(f) between L{ lT } and L{,u}. On the basis of Theorem 1 of Section 1, this correspondence can be extended as variables L2{U, e, pl.

If rf = 'Jr(f) for f e L2{ l}, then we use rf ='Jr(f) as the definition of an isometric correspondence between L2{m} and L2{Ft}.

ff(x)f-i(dx)

,

(7)

and we call the random variable rf the stochastic integral of the function f(x) with respect to the measure ,u.. From this definition we get: Theorem 1. a. For an arbitrary function (2) the value of the stochastic integral is given by formula (3); b. equation (6) holds for arbitrary f(x) and g(x) in L2{E, 2, m}; c. J

[af(x) + ,Gg(x)]p(dx) = a f f(x)ft (dx) + +(x)p(dx);

3.

STOCHASTIC MEASURES AND INTEGRALS

d.

193

for an arbitrary sequence of functions f(")(x) e L2{E, 2, m} such

that - f ("")(x)

I

12 m(dx) ---* 0 ,

n

(8)

we have

f(x)1t(dx) = l.i.m. f(n)(x) ft(dx) J

J

REMARK 1.

In particular, if the f(n)(x) are simple functions,

+ h A$ a = Lk=1c$ XAkn)(x), and (8) is satisfied, then /'(n)(x)

gy

l

,

mn

x ,/ d x

l.i.m.

c

n = 1, 2, ... i(i

It follows from Theorem 7, Section 7, Chapter II that an arbitrary function f(x) e L2{E, 2, m} can be approximated by some sequence of simple functions. Thus the stochastic integral can be regarded as the mean-square limit of suitable approximating sums. Let 20 denote the class of all sets A e 2 for which m(0) < co . We define a random set function ft(A): f-t(A) = XA(x) ft(dx) = A ft(dx) . (9) following properties: This function enjoys the a. ft(A) is defined on the class of sets 20 such that for arbitrary

A e 20 the class of sets {B (1 Al for B e 2 constitutes a a-algebra and M I f7(A) 2 = m(A) < °°.

If A. e 20 (for n = 0, 1, 2, ), Ao = U An, and Ak fl Ar = n=1 for k r# r (where k and r are both positive), then b.

fi(AO) =

c. d.

n=1

fi(An)(mod P) ;

Mft(A) ft(B) = m(A n B), A, B e 2o;

P(A) = ft(0) for O e fit. A random set function satisfying conditions a to c is called a stochastic orthogonal measure. Property (d) means that ft(A) is the extension of an elementary stochastic measure It. Thus we have: Corollary.

If the structural function of an elementary stochastic

measure ft is subadditive, then ft can be extended as a stochastic measure. REMARK 2.

Since L2{ f_t} = L2{ft}, we have

1 f(x) ft(dx) = f(x) ft(dx) .

LINEAR TRANSFORMATION OF RANDOM PROCESSES

194

In what follows we shall identify the stochastic integral with respect to an elementary orthogonal measure t(whose structural function is subadditive) with the stochastic integral with respect to a measure ft defined by equation (9). Let us make a few comments about the definition of a stochastic integral over an interval. Suppose that C(t) (for a < t < b) is a process with orthogonal increments; that is, M(S(t2) - S(t1))(b(t4) - WO) = 0

if t1 < t2 < t3 < t,, and suppose that C(t) is mean-square continuous from the left: M I C(t) - C(V) 11 ---> 0

as

rI

t.

We define F(t) = M I C(t) - C(a) 12

.

It follows from the orthogonality of the increments of the process C(t) that for t2 > t1, F(t2) = M I C(t2) -

yy b(t1) + yy b(t1)

-y

b(a) 12

= F(t1) + M I W2) - b(t) , from which we get F(t2) > F(t1) and F(t) = lim F(v). 12

Thus F(t) is a nondecreasing function that is continuous from the left. Let T1 denote the class of all left-closed right-open intervals A = [t1, t2) such that q, < t1 < t2 < b. Define =tt

b ([ t1, t2)) =

y Y(t2)

-y

Y (t1)

and m([t1, t2)) = F(t2) - F(t)

Then T1 is a decomposable class of sets MS(i1)bA) = m(i1 I A2) , and C is an elementary orthogonal stochastic measure whose structural function can be extended as a measure. Thus, we can define the Stieltjes stochastic integral a

5f(tg(dt)

,

d

where C(t) is a process with orthogonal increments. A stochastic integral over an open interval (a, b) or over the entire real line o, oo) is defined analogously. Let us prove some propositions regarding stochastic integrals. Let j_c denote an orthogonal stochastic measure with structural function m that is a complete measure on {E, 2} and let g(x) denote

3. STOCHASTIC MEASURES AND INTEGRALS

195

a member of L2{m}. We set X(A) = XA(x)g(x)ft(dx) ,

AeL.

Then MX(A)X(B) _

XA(x)XB(x)

I g(x) 12 m(dx) = LB I g(x)

12 m(dx)

,

On 2 we define a new measure 1(A) = A I g(x) 12 m(dx) .

We see that X(A) is an orthogonal stochastic measure with structural function 1(A) for A e 2. Lemma 1. If f(x) e L2{1}, then f(x)g(x) e L2{m} and

f(x)X(dx) = f(x)g(x) p(dx) . J

The assertion of the lemma is obvious for simple functions f(x) of the form f(x) = El CkXAk(x), where Ak e 2. FurtherProof.

more, if { fk(x)} is a fundamental sequence of simple functions in L2(1}, then 2

.f%(x)X(dx) - fn+m(x)x(dx)

I

n(x) - n+m(x)

12

1(dx)

I fa(x) - J n+m(x) IZ I g(x) 12 m(dx)

that is, { f, (x)g(x)} is a fundamental sequence in L2{m}. limit as n 00 in the equation

Taking the

fn(x)X(dx) = fm.(x)g(x)p(dx) ,

we obtain the assertion of the lemma in the general case. Lemma 2.

If X(A) = XA(x)O(x)f (dx)

for g e L2{m}, then for every A e 2,

p(A) = Proof. We note that g(x) = 0 on a set of 1-measure 0. 00 (mod 1). Furthermore,

1/g(x)

Thus

LINEAR TRANSFORMATION OF RANDOM PROCESSES

196

g(1

if A e 2.

XA(x)l(dx) I2

= JA g(x) 2

1 g(x) 12 m(dx) = m(A) <

Consequently we may use lemma 1: ZA(x)l(dx) = 1

p(A) J g(x) XA(x)g(x)fi(dx) =

This completes the proof of Lemma 2.

Let Z denote a finite or infinite interval, let 0 denote the aalgebra of Lebesgue-measurable subsets of Z, and let 1 denote Lebesgue measure.

Let us suppose that g(t, x) is (0 x 2)-measurable, that g(t, x) e LQ{1 x m}, and that g(t, x) e LQ{m} for arbitrary t e Z. Consider the stochastic integral

fi(t) = g(t, x) p(dx) .

(10)

For every t this integral is defined with probability 1. Lemma 3. The stochastic integral (10) can be defined as a function of t such that the process e(t) is measurable. Proof. If g(t, x) = E ckXBk(t)X Ak(x) ,

for Bk e O and Ak e 2, then

(1 1 )

function/

e(t) = L CkXBk(t)p(Ak)

is a (0 x e)-measurable function of the variables t and u for t e Z and u e U. In the general case we can construct a sequence of x)} of the form (11) such that simple functions I g(t, x) - g (t, x) 1E m(dx)dt , 0 as

n -- o

Let {e,a(t)} denote a sequence of processes constructed in accordance with formula (10) for g = g,,. Then there exists a process fi(t) such that

M I fi(t) - $.(t) I2 dt --p- 0

as

n

o,

and fi(t) is a (0 x Cam)-measurable function of t and u. hand, M I e(t) - e.(t) 12 dt =

J

I g(t, x) -

On the other

x) 1E m(dx)dt -, 0 ,

so that the processes e(t) and fi(t) coincide almost everywhere in

1 x U. We define

3.

STOCHASTIC MEASURES AND INTEGRALS

e(t),

{e(t),

197

if P{e(t) fi(t)} = 0 if W(t) = fi(t)} > 0 .

The process e'(t) is measurable (since c'(t) differs from the (0 x C25)measurable function e(t) on a set of measure 0), and it is stochastically equivalent to e(t). This completes the proof of the lemma. REMARK 3. Throughout what follows, when studying processes

defined by stochastic integrals of the form (10), we shall assume that the processes are measurable if they satisfy the conditions of Lemma 3.

Let g(t, z) and h(t) denote Borel functions such that

Lemma 4.

-,

a

J a6

h(t) 12 dt < 0 ,

and let It denote an orthogonal stochastic measure on B. bh(t) f

g(t, c) p(dr)dt = `W g,(z),u(dz)

(12)

Then (13)

,

J

where

gl(z) = bh(t)g(t, z)dt Proof. The mathematical expectation of the square of the integral on the left side of equation (13) is equal to

-

55h(ti)h(t2)5g(ti, z)g(t2, z)m(dz)dtldt2 = 5a

oe

So I h(t )12

dt5-Ja

bz)dt 5h(t)g(t,

2m(dr)

a

g(t, z) 12 dtm(dz) .

For t he square of the mathematical expectation of the integral on the right side of equation (13), we have the inequality shown in the last two rows of the expression displayed above.

Consequently

the right and left sides of equation (13) are continuous with respect

to passage to the limit on sequences of the form {g,(t, z)} that converge in L2{t}, where t is the direct product of Lebesgue measure and the measure m in the strip [a, b] x (- -, -). Furthermore, the set of functions g(t, z) for which equation (13) holds is linear and it contains all functions of the form E ckZAk(t)XBk(z).

Consequently it contains all functions belonging to L2{c}. Lemma 5. If the conditions of Lemma 4 are satisfied for every finite interval (a, b) and if the integral h(t)g(t, z)dt = lim 1 bh(t)g(t, r)dt a-.- J a

LINEAR TRANSFORMATION OF RANDOM PROCESSES

198

exists in the sense of convergence in L2{m}, then h(t)

g(t, v)ft(dv) _

j1(v)1t(dv) ,

(14)

where

h(t)g(t, v)dt . fl(r) = The proof follows directly from the fact that the left-hand

member of equation (14) is the mean-square limit of the left-hand member of equation (13), and from the possibility of taking the limit under the stochastic integral sign in the right-hand member of formula (13). Let us now consider a generalization of the preceding results to vector-valued stochastic measures. We shall confine ourselves to the simplest case of integration of scalar-valued functions-which

differs only slightly from integration with respect to numerical stochastic measures.

Let R. denote a complex vector space of dimension p. For simplicity we assume that some basis in this space is fixed. Let us suppose that to every 0 e EN is assigned a vector-valued random variable p(0) with values in RP: P(A) = {/1l(A), I12(A ... , Let I p(A) I denote the norm of the vector p(0):

AA) 12 =

k=1

I Ilk(o) I2

Let us assume that 2.

u(c) = 0, oo, p(Ol U Q = p(O&) + p(A2)(mod P), if of n A.

3.

Muk(Ol)1-'j(02) = mk9(Ol n 02), 0; e n, i = 1, 2; k, j = 1, ..., p,

1.

M I p(o) 12 <

We shall call a family of random vectors {p(0)}, for 0 e St, an elementary vector-valued stochastic (orthogonal) measure and we

shall call the matrix m(0) = (mk;(0)) = Mp(0)p*(0) its structural matrix. 02,

We note that the matrix m(pl n 02), as a function of Ol and enjoys properties 14 to 17 (cf. Chapter I, Section 2) of the

covariance matrix of a vector-valued random function.

Furthermore,

if Ol n p2 = cF, then m(& U 02) = m(A1) + m(A2) .

From this it follows that the diagonal elements of the matrix m(0) are elementary measures.

Furthermore it follows from the inequality

S.

STOCHASTIC MEASURES AND INTEGRALS

(15)

I mk.i(A) I G 1/mkk(A)m11(A)

that

+ IJ r

+

I mk.i(Ar) I < {L1 mkk('&r) r> r

m.iA(Ar)}1/E

,

(16)

so that the set functions mk;(0) (for k, j = 1, , p) will be of bounded variation on Do if m;;(D0) < - for j = 1, , p. We set mo(A) = tr m(A) _

P

k=1

mkk(A)

From (16) it also follows that if Z'=N1 mo(or) approaches 0 as N, 00 , EmN I mk;(Or) I also approaches 0 as N, 00. From this it follows that the functions m;k(A) can be extended as countably additive set

functions on e (in the sense established in Theorem 8, Section 7, Chapter II) if the function m0(A) is countably additive (or subadditive) on Z (cf. proof of the theorem just cited).

In what follows we shall say that the matrix functions obtained by such an extension from a structural function of an elementary orthogonal stochastic measure are positive-definite matrix measures. Above, 0 denoted the completion of s{JZ} with respect to the

extended elementary measure m0(A). We shall keep the original notations for the extensions of the functions mk; and mo and of the matrix m on Cam. In addition we shall assume that m(,(A) is subadditive on TZ. On L{1Z} we define a stochastic integral by the formula

'2 = f(x)p(dx) =

CkP(1k) ,

where f(x) = Y, ckxok(x) and Ok G TZ (for k = 1,

(17)

, n). The value

of this integral is a random vector (a column vector) with values in R. We let Lp{p} denote the set of all random vectors'2 of the form (17). If g(x) _ E dkxok(x) , then k=1

ckdkm(Ok) ,

which, can be written in the form g(x)p(dx))*) = f(x)g(x)m(dx)

(18)

From this it followsJJthat M .f(x)1u(dx)

I' =

J

I.f(x) IE mo(dx) .

(19)

LINEAR TRANSFORMATION OF RANDOM PROCESSES

ZOO

In L{fit}, we define a scalar product by

(f, g) = tr f(x)g(x)m(dx) = trace

fix)g(x)m(dx) ,

J

Formula (18) establishes an isometric mapping 7) = Jr(f) of the space L(I7t) onto LP{ ji} if the scalar product of two elements 72, and 72, of LP{ p} is defined as M7a V1 Let the closure-of the space of random variables LP{ p} be denoted by and let the completion of L{DZ} be denoted L2{7t}.

In a manner analogous to the derivation of inequality (16) we can derive (first for simple functions, and then by taking the limit, for arbitrary Cs-measurable functions) the inequality I fix) I I mkJ I (dx) < fJ I fix) I mkk(dx)

,

I fix) I

(20)

m>;(dx)}1I2

where I mk; I (A) is the absolute variation of the function mk;.

Inequality (20) implies the existence and continuity of the integral f(x)g(x)mk;(dx) as a functional of f and g in L2{mo}. J

4.

INTEGRAL REPRESENTATIONS OF RANDOM FUNCTIONS

By using the results of the preceding section, we can obtain various representations of random functions with the aid of stochastic integrals. Let us first suppose that a p-measurable vector-valued random

function e(O) for 0 e O can be represented in the form

e(°) =

Jg(0,

x)m(dx)

(1 )

,

where y is a stochastic measure with values in RP and with structural

matrix m(A), in the notation of the preceding section, and g(0, x) is a scalar-valued function such that for every 0 e O the function g(0, x) belongs to L2{mo}, where mo(A) is the trace of m(A). On the

basis of formula (18) of Section 3, the covariance matrix of the random function (0) is of the form R(0, 02) = ME(01)E*(02) = g(01, x)g(02, x)m(dx)

,

(2)

and it follows from equation (19) of Section 3 that Me*(o2)e(Bi) = g(01, x)g(02, x)mo(dx) .

(3)

We recall that L2{mo} is a Hilbert space of (25-measurable complexvalued mosquare-integrable functions. We let L2{g} denote the

4.

INTEGRAL REPRESENTATION OF RANDOM FUNCTIONS

201

closure in L2{mo} of the linear hull of the system of functions g(o, x) for B e 0. If L2{g} = L2{mo} the system of functions {g(0, x)} for 0 e 0 is said to be complete in L2{mo}.

Theorem 1. Suppose that the covariance matrix of a vector-valued random function e(o), for B e 0, has a representation of the form (2), where {m, 0 1 is an orthogonal positive-definite matrix measure, where g(o, x) e L2{mo}, and where mo(A) is the trace of m(A). Then there exists a stochastic measure p(A), for A e e, with structural matrix m(A) such that equation (1) holds with probability 1 for arbitrary 0 e 0. Proof. To every linear combination of the form

f(x)

= L akg(Bk, x) , k=1

Bk G (3 ,

(4)

let us assign the random vector 7) = *(f) defined by

(5)

'/ = k=1 L akggS(Bk)

Let L(f) and L{g} respectively denote the sets of all random vectors (5) and functions (4). In L{g} we define the scalar product by (f1, f2) = 5fi(x)fs(x)mo(dx) .

(6)

The relation 72 = 'i/r(f) is an isometric mapping of L{g} into L(f). Consequently (cf. Theorem I of Section 1), it can be extended as an isometric mapping of L2{g} onto L2{e}, where L2{e} is the closure of L{f} in the sense of mean-square convergence of random vectors

and L2{g} is the completion of the space L{g} with respect to the metric generated by the scalar product (6). Let us first suppose that the system of functions {g(0, x)} for 6e@ is complete in L2{mo}. If A e C25, then XA(x) e L2{mo} = L2{g}. We set p(A) _ 1/r(XA). Then p(A) is a vector-valued stochastic

measure and its structural function coincides with m: MP(A1)f2*(A2) = XA1(x)XA2(x)m(dx) = m(A1 fl A2)

Now let us define a random function r(o) as the stochastic integral

Vo) = g(o, x)m(dx)

.

Since

Me(o)p*(A) = g(o, x)XA(x)m(dx)

the isometry of the correspondence rJ = '/r(f) implies that Me(o)r(o) = g(o, x)g(o, x)m(dx) .

202

LINEAR TRANSFORMATION OF RANDOM PROCESSES

From this we obtain M I e(0) - x(0)12 = M([e(o) - r(0)]*[e(e) - C(B)])

= Me*(e) (o) - Mr(e)e(e) - M *(0)V0) + W(0)V0) = 0 which proves the theorem for the particular case that we are considering.

Let us now turn to the general case.

Let 01 denote an arbitrary

set having no points in common with O. Let us define a system of functions {u(0, x)}, for 0 E 01 and x e E, that is complete in L2{mo} (B L2{g}, and let us define a Gaussian vector-valued random process e1(0), for 0 e 01, that is independent of E(0) for 0 e (9, with

mean zero and with correlation matrix R1(01, 02) = u(01, x)u(02, x)m(dx) .

We set r(0) = (0) if 0 e O and r(0) = X1(0) if 0 e 01. We set f(0, x) = g(0, x) for 0 E O and f(0, x) = u(0, x) for 0 E 01.

Then the

covariance matrix of the process r(0) is equal to f(01, x)f(02, x)m(dx) (0a e 0 U 01, i = 1, 2)

and J f(6, x)} for 0 E (9 U 01, is a complete system of functions in L2{mo}.

According to what we have proved, there exists an orthogonal

stochastic measure p on C such that V0) = f f(0, x) p(dx)

for 0 e 0 U 01. Therefore, (0) =

Jg(0, x)

p(dx)

,

0e0.

This completes the proof) of the theorem. REMARK 1. If {g(0, x)}, for 0 e 0, is a complete system of

that is, the value of the stochastic measure p can be established from the values of the sample function of the process. In the general case, this is not true. If p(A) e L2{e} for arbitrary A E C25, we shall say that the stochastic measure p is subordinate to the process E. We now present a number of examples illustrating the use of Theorem 1. In what follows we shall consider processes e(t) that functions in L2{mo}, then p(A) E

are stationary in the broad sense and that satisfy the condition ME(t) = 0.

For the sake of brevity we shall use the word "stationary"

instead of the phrase "stationary in the broad sense." On the basis of Theorem 2, Section 5, Chapter I, the covariance

4.

INTEGRAL REPRESENTATION OF RANDOM FUNCTIONS

203

matrix of a stationary mean-square continuous process can be represented in the form R(t1, t2) = DR(t, - t2) = 5et1_t2)dF(o.))

(7)

where OF = F(u2) - F(u1) is a nonnegative-definite matrix interval function of bounded variation (the spectral matrix of the process). The function F((o) defines a positive-definite matrix measure {O, F}, where 0 is the or-algebra of subsets consisting of all Borel sets of the real line. (This a-algebra is complete with respect to tr F(A) = mo(A).) Equation (7) is the particular case of equation (2) in which the functions g(O, x) correspond to ei°t, 0 t, w x. Here, the function set {ei,t, - 0 < (o < oo} is complete in L2{mo} where mo is an arbitrary bounded measure on the real line. Thus Theorem 1 is applicable and we obtain the following result:

Theorem 2. A vector-valued mean-square continuous stationary random process fi(t) (for - 00 < t < oc) such that mg(t) = 0 has a representation of the form e(t) = 5evp(doi)

(8)

where p(A) is a vector-valued orthogonal stochastic measure on 0 subordinate to t(t). Between L2{e} and L2{mo} there exists an isometric correspondence under which a.

E(t) t--> ec.,t, p(A)

b.

if ')i

`-' n(w)

gi((o)(i = 1, 2), then/' 12i =

and

M?1?2 =

gi(w)p(dw) g#o)g#o)df((o)

Formula (8) is called the spectral decomposition of the stationary process and the measure p(A) is called the stochastic spectral measure of the process. It follows from Theorem 2 that MP(A1)P*(A2) = A1nA2dF((0) . REMARK 2.

For arbitrary rJ c L2{e} we have MY) = 0.

(9)

In par-

ticular, for arbitrary A e r8 we have Mp(A) = 0. REMARK 3.

If ME(t) = a # 0, Theorem 2 can be applied to

the process E(t) - a. On the other hand, the representation (8) can be preserved in the general case if we add to p(A) a measure concentrated at the point (o = 0 where its value is a.

LINEAR TRANSFORMATION OF RANDOM PROCESSES

204

As an example of the application of Theorem 2 we give the Kotel'nikov-Shannon formula for a random process whose spectral function is concentrated in a finite interval [-B, B]. Let us expand the function esmt on the interval [-B, B] in a Fourier series. We have esmt =

sin (Bt - 7cn) exp (i 7rn cvl

n Bt-7cn

\

B

The series on the right converges uniformly in every interval of the form [ - B', B'], where B' < B, and the sequence of its partial sums is bounded. Therefore it also converges in LZ{mo}. By virtue of the isomorphism between the spaces Lz{mo} and LZ{e}, we have (in the sense of mean-square convergence) sin (Bt - 7rn) E(7rn 1

(10)

n=_ Bt - 7rn `` B J Thus the value of the random function E(t) at an arbitrary instant t is uniquely determined from its values at equally spaced instants . of time 7cn/B, for n = 0, ± 1, ±2, REMARK 4. For stationary vector-valued sequences {E,,} for n = 0, ± 1 , ±2, , it is possible to formulate a theorem completely analogous to Theorem 2. The only difference is that the spectral measure of the sequence is concentrated on the half-closed half-open

interval [ -7c, 7c) and not on the entire real line as in the case of a process with continuous time (cf. Theorem 3, Section 8, Chapter I). We can treat processes with increments that are stationary in the broad sense in analogous fashion. Since the structural function

of such a process has (cf. Theorem 4, Section 5, Chapter I) a representation of the form D(tl, tz, ta, t4)

e itgm

-

eit1m

i(t)

-

e- it`. e- {tam dF(w) - iU)

where F((o) is the spectral matrix of the process, it follows, by virtue

of Theorem 1 (with '(B) = r(t2) - r(tl), where 0 = [tl, t2)), that

00 - Vtl) =

eitam - eitlm

iw

p(dw) ,

(1 1)

where p(A) is an orthogonal stochastic measure with structural matrix F(A).

Here, the system of functions (eitm - 1)/icv (for - oo < t < 00) This is easily seen if we represent the characteristic function of the interval A = (a, b), at whose endpoints the spectral matrix F(u) is continuous, in the form is complete in LQ{mo}.

4.

INTEGRAL REPRESENTATIONS OF RANDOM FUNCTIONS

205

(b sin A(t - w) dt

1

lim xo((o)=A-.ao7C

t-U)

that is, as the limit of linear combinations of the functions (eit.,

- 1)/iw. Theorem 3. A random mean-square continuous process

with

stationary increments r(t) has a representation of the form (11), where p(A) is an orthogonal stochastic measure, where M(p(A)p*(A)) = F(A) is the spectral matrix of the process, and where 1, two

K(w)d tr F((o) < -

K(w)=

,

1

,

12,1 c0 1 >1

The measure p(A) is subordinate to r(t). Let us make a few comments regarding processes with stationary increments. We confine ourselves to the one-dimensional case. For

a process C(t) to have a mean-square derivative at a point t, it is necessary and sufficient (cf. Theorem 1, Section 2) that the limit D(tl, t2, t3, t4)

lim t1.t2.t3.t4-.t

(t2 - t1)(t4 - t3)

eimt2 _ e-i.itl

lim t1.t2.t3.t4-.t j -°°

exist.

eimt4

-

ei.,t3

-uv(t4 - t3)

ioi(t2 - t1)

dF(w)

This condition will be satisfied if

dF(m)<-. In such a case the process is mean-square differentiable at every point and its mean-square derivative C'(t) has correlation function B1(tl, t2) = Iim

D(t1, tl + it1, t2, t2 + Ot2) At'At2

At1.At2'0

ei.,(t2-t1)dF(w) .3

On the other hand, we can take the limit under the integral sign in the relation ei(t+A)w - eit'

C(t + 0) -fi(t) = f °° J-o

O

as A, 0.

icv0

u(dw)

Thus eit.,p(dw)

V(t) =

is a process that is stationary in the broad sense and that has J)

spectral function F(w).

Consequently, for F(+ oo) < oo a process

with stationary increments can be regarded as the mean-square integral of the stationary process and, in the general case, as the

LINEAR TRANSFORMATION OF RANDOM PROCESSES

206

mean-square limit of such integrals. An interesting example of a process with stationary increments that is not an integral of a stationary process is a process for which dF(w) = dcw. Such a process can be regarded as the mean-square limit as A, + co of integrals of stationary processes having spectral density equal to 1

on the interval (-A, A), but equal to 0 outside that interval. Speaking crudely, we say that such a process is an integral of a fictitious stationary process having constant spectral density on the entire real line. This fictitious stationary process is called "white noise" and is often encountered in nonrigorous expositions of the theory of random processes and its applications. On the other hand,

the theory of such processes is part of the theory. of generalized random processes. The process r(t) with stationary increments, which was introduced above, may be interpreted as white noise integrated and will serve as a substitute for white noise itself in our discussion.

We note that this integrated white noise is a process with In fact, using Parseval's equality for the Fourier integral, and the fact that (e "°' - e '-)/io. is the Fourier orthogonal increments.

transform of the function Z(a,b)((o), where 1 is Lebesgue measure on

the real line, we have pt2) MM O

y b (t(t4) - WO)

27r

eit2w - eitlw

e-it4' - e-it3m

i()

-io)

J_ X(tl.t2)((O)X(t4.t3)((O)dw

do) = 21rl[(tl, t2)

'

(t4, t3)]

Let us now look at the matter of spectral decomposition of a mean-square continuous isotropic two-dimensional random field. On the basis of Theorem 5, Section 5, Chapter I the covariance function

of the field takes the form R(xl, x2) = R(p) =

Ju

Jo(op)dg(oi) ,

(12)

where xl and x2 are points in the plane and p is the distance between

If (ri, pi) are the polar coordinates of the point xi (for i =1, 2), then p = Vri + r2 - 2r1r2 cos (q - P2) them.

Using the addition formula for the Bessel function J, (see for example

Gray and Mathews, p. 39), namely

J,(a) p) =k=-co E

Jk(wrl)Jk(wr2)eak('i-W2)

5.

207

LINEAR TRANSFORMATIONS

we rewrite formula` (12) in the form R(p)

J.(wrz)e-'Lll2dg(w)s(dv)

= JT

where s(dv) is a measure concentrated at the points k = 0, ± 1, ±2, and s({k}) = 1. On the basis of Theorem 1, a plane isotropic homogeneous mean-square continuous field E(x) (such that Mt(x) = 0)

where x = rei' , has a representation of the/ form

E(x) = k eikc Jk(wr)pk(dw)

(13)

where {Pk} is a sequence of mutually orthogonal stochastic measures on the ray [0, ca ). 5.

LINEAR TRANSFORMATIONS

Consider a system E (a device or a procedure) designed to transform signals (functions) x(t) depending on the time t. The function to be transformed is called the input function of the system

and the transformed function is called the output function or the response to the input function. Mathematically, every system is defined by a class D of "admissible" input functions and a relation of the form

z(t)=T(x1t), where x = x(s) (for - ca < s < -) is the input function (belonging to D) and z(t) is the value of the output function at the instant t. A system Y, is said to be linear if (a) the class D of admissible functions is linear and (b) the operator T satisfies the superposition principle

T(axl +

8x,

I

t) = aT(x, I t) + RT(x2 I t)

Let us introduce the operation of displacement of time S, (for - 00 < z < -) with the aid of the equation xr(t) = S=(x I t) = x(t + v) This operation is defined on the set of all functions of the variable t (for - ac < t < co) and is a linear operation. The system E is said to be homogeneous with respect to time (or simply homogeneous) if T(x, I t) = T(x j

t + t)

or

T(Szx I t) = S=T(x I t)

;

that is, if the transformation T commutes with the time-displacement

S, (for -oa
A simple example of a linear transformation is one of the form

LINEAR TRANSFORMATION OF RANDOM PROCESSES

208

z(t) _

h(t, r)x(z)dz

(1 )

,

for which the class of admissible functions D depends on the properties of the function h(t, r). the form 0

Consider the set of functions of

for It - aI > -

,

2

fort - aI<E, 2

E

where s > 0. The function x,(t) has the nature of an impulse: it acts for a brief instant of time (of duration s) but its value is great,

x,(r)dr = I

since

.

It is convenient to regard the limit as s

0

of the set of impulse functions x,(t) as the generalized function va(t),

which is the instantaneous unit impulse at the instant a. This function is equal to 0 for t # a but it satisfies the condition a}h

-a-h

v(t, a)dt = 1, where h > 0. There are no functions, in the

usual sense of the word, with such properties. The theory of generalized functions including functions of the type described, is

developed in the books by L. Schwartz, and I. M. Gel'fand and G. E. Shilov. The function va(t) is called the delta function acting at the point a. Corresponding to the output function x,(t) of the system is the function

z;(t) = 1

a}EI2

s Ja-Ef2

h(t, r)dz

,

which approaches h(t, a) under certain hypotheses (for example, if h(t, z) is a continuous function of z at the point a). Thus we should interpret h(t, r) as the response of the system

to the delta function at the instant

z-.

In accordance with this,

h(t, z) is called the impulse transfer function of the system. If the system Z is homogeneous with respect to time, then formally,

h(t, a - c) = T(va-o I t) = T(S03a I t) = ScT(va I t) = h(t + c, a)

or by setting a - c and t -t - c, h(t - c, 0) = h(t, c)

.

The function h(t) = h(t + c, c) is called the impulse transfer function of the homogeneous system. Thus for a homogeneous system, equation (1) takes the form

5.

LINEAR TRANSFORMATIONS

z(t) =

209

h(t - z)x(v)dv .

(2)

The operation on the right side of equation (2) is called the convolution

of the functions h(t) and x(t).

If the input function of the system differs from the output function only by a scalar factor (the transformation T does not change the form of the signal), that is, if T(f I t) = xf(t) for (- ac < t < Do ), then f(t) is called an eigenfunction and A, an eigenvalue

of the transformation T. For systems that are homogeneous with respect to time and have an integrable impulse transfer function, I h(t) I dt < oo, functions of the form eitw (where (o is an arbitrary

real number) are eigenfunctions. This is true because all bounded measurable functions are admissible and

h(t - v)ei-dr =

h(u)eawct_w)du

= H(io))eiwt

where

H(i(o) =

h(u)e-awudu

(3)

the Fourier transform of the impulse transfer function, is an eigenvalue

of the transformation.

Thus the ratio of the response of the system to the simple harmonic input function eiwt to the latter function 10 H(im) = T (e'-ei

(3)

is independent of time. The function H(i(o) is called the frequency characteristic of the system or the transfer coefficient.

It is possible to interpret the frequency characteristic of the system (2) in a somewhat different way by considering another class of admissible functions. Suppose that x(t) is integrable. On the basis of Fubini's theorem,

J I z(t) I dt <

=

h(t - r)

x(v) I.drdt

LxvdvLht)dt< °°

;

that is, the function z(t) is also integrable. Now consider the Fourier transform of the function z(t). Applying Fubini's theorem, we obtain

LINEAR TRANSFORMATION OF RANDOM PROCESSES

210

z(w) =

e-iotz(t)dt

=

e-c-(e-i)h(t - z)e-ti°'zx(z)dzdt

= H(iw)2(w), z(m) =

Consequently the ratio of the Fourier transform of the output function to the Fourier transform of the input function is independent

of the input function of the system and is equal to its frequency characteristic:

H(i(o) = 2(w)

z(w)

In formula (1) the response of the system at the instant t depends on the values of the input function both at instants z < t and at instants z > t. In physical constructions, however, it is impossible to anticipate the future. Therefore, for physical constructions,

h(t, z) = 0 for t < z . (4) Equation (4) is called the condition of physical realizability of the system. For systems satisfying condition (4) formula (1) assumes the form t

z(t) =

h(t, z)x(z)dz

,

J

(5)

and if the system is homogeneous,

z(t) = L h(t - z)x(z)dz = o h(u)x(t - u)du .

(6)

If a function is introduced at the input of the system beginning at the instant of time 0 (that is, x(z) = 0 for z < 0), then z(t) = 0 th(t - z)x(z)dz

(7)

In studying such systems, it is convenient to use not the Fourier transform but the Laplace transform

2(p) = e-Ptz(t)dt

(8)

.

It follows from formula (7) that 2(p) = H(p)2(p) ,

x(p) =

f Jo

e-Ptx(t)dt

(9)

for Rep > a if the functions e-ath(t) and e-atx(t) are absolutely integrable.

Let us turn to the basic theme of the present section, namely, For the most part, we

linear transformations of random processes.

5.

LINEAR TRANSFORMATIONS

211

shall consider homogeneous (with respect to time) transformations of stationary processes. In regard to the more general case, we shall confine ourselves to some simple comments. Let C(t) (for - - < t < -) denote a measurable Hilbert process with covariance B(t, z).

Suppose that the function B(t, t) is integrable

with respect to t over every interval and that for every fixed z, the function h(z, t) is also integrable over every interval. Then with exists for arbitrary a and b. probability 1, the integral h(t, b

Let us define the improper integral from - - to - as the meansquare limit of the integral over finite intervals of integration as their endpoints approach - Co and + oo : h(t, z)(z)dz = l.i.m. h(t, z)e(z)dz . a-w Ja 6-, b

From part 2 of Section 2, we recall that for this limit to exist it is necessary and sufficient that the integral LL h(t, zl)B(v1, z2)h(t, z2)dzldz2

exist in the sense of a Cauchy improper integral over the plane. If it exists for t e Z, then z(t) for t e Z is a Hilbert random process with covariance h(t1, r1)B(z1, z2)h(t2, z2)dz1dz2 .

Bz(t1, t2) =

(10)

Let us suppose now that E(t) is a stationary process in the broad sense with spectral function F((o) and mg(t) = 0. This assumption will be retained until the end of the present section. The integral V(t)

=

h(t - z)e(z)dz

(11)

exists (in the sense mentioned above) if and only if the integral

JIB Ih(t - z1)R(z1 - z2)h(t - z2)dv1dz2 h(zl)R(z2 - Zl)h(z2)dzldz2

exists, where R(t) is the covariance function of the process. For this in turn it is sufficient that the function h(t) be absolutely

integrable over (- 0 ,

oo ).

In this case, by using the spectral

representation of the covariance function R(t) (cf. (1), Section 5, Chapter I), we obtain the following expression for the covariance function Rn(t1, t2) of the process ra(t):

LINEAR TRANSFORMATION OF RANDOM PROCESSES

212

R,,(t1, t2) =

j_ht1 -r1)R(r,

= r°°J°°J_h(tl e=ct1

Z2

- z2)h(t2 - r2)drldr2

- z )ei-(r1-=2)h(t2 - r2)dzldz2dF((0)

H(i(o) IZ dF((o) = Rn(t1 - t2)

Thus the process Y)(t) is also stationary in the broad sense.

For a process fi(t), a transformation T is called an admissible filter (or, more briefly, a filter) if it is defined by formula (11) where h(t) is an absolutely integrable function, or if it is the mean-square limit of a sequence of such transformations Definition 1.

(in L2(e}).

A condition for convergence of a sequence {)),(t)} = {Tje I t)} of transformations of the form (11), with impulse transfer functions h,(t) and frequency characteristics Hn(i(o), consists in the following: M )In(t) - /n+m(t) I2

I Hn(i(o) - Hn+m(i(o) 11 dF(m) - 0 ,

(12)

that is, in the requirement that the sequence {HH(iw)} be a fundamental sequence in L2{F}. But then the limit H(im) = l.i.m. H%(i(o) exists in L2{F}. This limit is called the frequency characteristic of the limiting filter. If c(t) rP= l.i.m. r)n(t), then R, (t) =

eta, I H(i(o) I2 dF((O) .

(13)

Conversely, every function H(i(o) e L2{F} can be approximated in the sense of convergence in L2{F} by functions that are the Fourier transforms of absolutely integrable functions. Thus it is convenient to define filters by their frequency characteristics. Theorem 1. For a function H(i(o) to be the frequency characteristic of an admissible filter, it is necessary and sufficient that H(im) belong to L2{F}. The covariance function of the process at the output of the filter with frequency characteristic H(i(o) is given by formula (13).

If we recall the energy interpretation of the spectral function, it follows from formula (13) that I H(i(o) I2 shows by how much the energy of simple harmonic components of a process with frequencies

in the interval (w, w + d(o) is multiplied by passage through the filter.

Theorem 2. If a process e(t) at the input of a filter with frequency characteristic H(im) has the spectral representation

S.

LINEAR TRANSFORMATIONS

213

e(t) = Letp(dw),

(14)

then the process 72(t) at the output of the filter is of the form y)(t) =

Proof.

.

(15)

If the filter has an absolutely integrable impulse transfer

function, then on the basis of Lemma 5 of Section 3, Y(t) =

ea"H(iw)p(dw) .

h(t - v)E(z)dv =

Proof in the general case is obtained by taking the limit with respect to sequences {H,1(iw)} that converge in L2{F} to H(iw). Let lyh(t) denote a process at the output of a filter with frequency characteristic Hh(iw) (for k = 1, 2). Let us find the mutual covariance function of the processes 7),(t) and iy2(t). It follows immediately from the isomorphism of the spaces L2{fit} and L2{F} that R12(t) = M?1(t +

e"H1(iW)H2(iW)dF(W) .

(16)

Let us give some examples of filters and their frequency characteristics.

A band filter admits (without modification) only harmonic components of a process with frequencies in a given interval (a, b). The frequency characteristic of the filter is equal to H(iw) = and the filter is admissible for an arbitrary process. The impulse transfer function is found from Fourier's formula 1.

h(t)

2.

f e"-dw =

e%bt _ eM1at

2'rit

A low-pass or a high-pass filter admits (without modification)

only harmonic oscillations with frequencies not exceeding or not less than some value b. Such a filter is admissible for an arbitrary process. Its frequency characteristic is equal to H(ZCV) = X(-oo,b)()) (h(b,+oo)(0)))

and the impulse transfer function does not exist. 3. Consider the operation of mean-square differentiation of a process that is stationary in the broad sense. A sufficient condition for a process e(t) to have a mean-square derivative is the existence of R"(0) (cf. Corollary 2, Section 2). This condition is equivalent to the requirement (cf. Theorem 2, Section 5, Chapter I) that LW2dF(w) < 00

.

(17)

LINEAR TRANSFORMATION OF RANDOM PROCESSES

214

On the other hand if this condition is satisfied, then

-1

ei"h

(in Lz{F})

--. ico

h

and in the relation (t + h) - e(t) =

we may take the limit as h

e""

f

1 ,(dw)

ean-

0 under the stochastic integral sign.

Consequently, (18)

eaa°,iw fi(des)

V(t) =

Thus, corresponding to the operation of differentiation is a filter with frequency characteristic ia), which is admissible for all stationary processes satisfying condition (17). The impulse transfer function

does not exist, but the filter can be regarded as limiting (ass for filters with impulse transfer functions of the form

for ItI> s

0

h,(t)_

sgn t 62

for

0)

,

I

t

I <s,

to which the frequency characteristics 4 sine (As 2

correspond. 4. The time-displacement operation.

Since

ei°,eeiwr p(d(o)

z) _

corresponding to the time-displacement operation ST defined by t) = (t + z) we have the frequency characteristic H(ico) = e"°T. There is no impulse transfer function. Sr(e I

5. Differential equations. Consider a filter defined by a linear differential equation with constant coefficients

L77 =Me,

(19)

where dn

L = a0 dtn +

d"-1 a1

M = b0 d- + dtm

+ dm 1 + ... + bm

dtn-1 b1

dtm_1

Equation (19) is meaningful only when the process e(t) is m times

S.

LINEAR TRANSFORMATIONS

215

mean-square differentiable. Then we seek an n-times mean-square differentiable stationary process Y)(t) satisfying equation (19). Let us suppose that (19) has a stationary solution. Then 7)(t) can be represented in the form y)(t) _

eiwtH(ic))p(dw)

.

If we apply the operations M and L to the processes e(t) and ra(t) respectively, we obtain e'1tL(ic))H(iw) p(dco) _ e`wtM(i(o)p(dco)

,

so that if L(iw) does not have real roots,

H(ie)) = M

(20)

L(ico)

Conversely, if the process e(t) is m times mean-square differentiable, if M(iw) e L2{F}, and if L(iw) =/- 0 (for - < (o < oo ), then the process 17(t) =

eb.t L(M

ia))

p(dw)

is n times mean-square differentiable and satisfies condition (19). Thus under the conditions M(ic)) e L2{F} and L(iw) # 0, there exists a unique filter satisfying the differential equation (19). We note,

however, that it is possible to determine the solution of equation (19) under more general conditions. Let us suppose that the polynomial L(iev) has no real roots. A filter with frequency characteristic M(iw))/L(ier)) exists, and without requiring that M(i(o) belong to L2{F}, it is sufficient to require that M(i(o)/L(i(o) belong to L2{F}. This

will always be the case when the degree n of the polynomial L is not less than m. Thus for n > m, the filter with frequency characteristic

(20) whose denominator does not vanish for real w is admissible for an arbitrary input process, and we identify the output process of the filter with the stationary solution of equation (19). Let us confine ourselves, as before, to differential equations for which the polynomial L(x) does not have purely imaginary roots. Let P(x) denote the polynomial portion of the rational function M(x)/L(x) (which will be nonzero if m > n) and let us decompose the remainer into partial fractions. Then n tk ) ckII C" (w) M(iw)

where

P(Zer)) +

k=1

(I0)

Pk)8

+

(uo

p'k )8

LINEAR TRANSFORMATION OF RANDOM PROCESSES

216

P(i(o) =

ak(iw)k k=0

for m > n and P(iw) = 0 for m 0, where pk and p'k are the roots of the polynomial L(x) = 0. Since

_

1

am

P)

8

d

1

°°epte-m.t,tdt

dPeb

TS _-l !

=

0

t

o

ePte-i.,tdt

(s 1 )

(Rep < 0) and 1 = fo (1w - p)8 - J-em

-

t8

e,-te-cwtdt

(Rep > 0)

(s - 1)t

the output process i)(t) of the filter can be represented in the form

)7(t) = k=0 E

a&kV) +

v)G1(z)dt + e(t +r)Gs(-r)dv , 0

0

where ekete ){exppt (t > 0)} G,(t) =k=1I (E 8=1 (S - 1)!

,

11

tk

G2(t)

(E

s

(scketl)i

){exppk't (t < 0)}

We note that if the polynomial L(x) has roots with positive real part, the corresponding filter is physically unrealizable. 6.

PHYSICALLY REALIZABLE FILTERS

In the present section we shall consider the question: What spectral functions can be obtained at the output of a physically realizable filter? At the input of the filter we shall consider the random process that is simplest in a certain sense. The processes considered in the present section are invariably assumed to be homogeneous and stationary in the broad sense. Therefore, the word "stationary" will sometimes be omitted, and the words "in the broad sense" will always be omitted. We shall begin by considering stationary sequences. We shall

not carry over to sequences all the definitions and heuristic considerations that were given for processes with continuous time although we shall use the corresponding terminology. Consider a

6.

PHYSICALLY REALIZABLE FILTERS

217

system whose states at the input and output are regulated only at integral-valued instants of time t = 0, ± 1, ± 2, .. Suppose that a unit impulse takes place at the input of the system at the instant of time 0. We let at denote the response of the system to that impulse at the instant t.

If the system does not anticipate the future,

then a t = 0 for t < 0. If the system is homogeneous with respect to time, the response of the system to the unit impulse applied to the system at the instant v is equal to at_z. The response of a linear, homogeneous, physically realizable system at the instant t to the

sequence of impulses E(n) (for - co < n < a.) will be )7(t) _ n=-. at_ E(n)

n)

n=0

.

(1 )

In a certain sense, the simplest hypothesis is that e(n) is a stationary sequence with mean value 0 and uncorrelated values Me(n) = 0, M(e(n)S(m)) _ a...(- - < n, m < co) . We shall call such a sequence uncorrelated. Its covariance function has the spectral representation n-n

R(n) =

ei'""dw

2ir

Consequently the spectral density of this sequence is constant.

For the series (1) to converge in mean-square in the case of an uncorrelated sequence e(n), it is necessary and sufficient that

(2)

E I an lZ < oo

n=0

If this condition is satisfied, the process 'J(t) the broad sense and M)7(t) = 0 ,

also stationary in

is

Ra(t) = M(-I(t + v))7(v)) _

n=0

an+tan

(3)

What sequences can be obtained in this manner? Lemma 1. For a stationary sequence 7; (n) to be the response of a physically realizable filter to an uncorrelated sequence, it is necessary and sufficient that the spectrum of 7J(n) be absolutely continuous and that its spectral density f(co) have a representation of the form

f(w) _

I2

g(eil) = E bneinm

,

n=0

E I bn Iz < n=0

(4)

Proof of the Necessity. Suppose that the sequence has a representation of the form (1). We set g(eiw) = .,

V

/1

i anein

2ir n=0

(5) \

LINEAR TRANSFORMATION OF RANDOM PROCESSES

218

Then from Parseval's formula, g(eiu) I2dw

an+tan = rz eitm

Rn(t) =

n=0

I

-rz

that is, the sequence i7(n) has an absolutely continuous spectrum with density f(m) = I g(eil) 12, where is defined by formula (5). Proof of the Sufficiency. Suppose that )7(n) is a sequence with covariance function eit) f(w)do '

Ra(t) = rz rz

and suppose that this sequence admits a spectral representation rI(t) = J`rz eit. p(dw)

.

rz

Suppose that f(tv) = I g(eiw) 12, where g(eil) is defined by relations (4).

On the o-algebra of Borel subsets of the interval (-7r, 7r), let us construct the stochastic measure rz

(A) =

1

-- V 27C g(eiw)

XA((o)p (d(o)

Then, 1

XA((0)Xs(0j) rz

_

1

27C .)Ana

27c

g(eim)

12

f([e)dco

dw,

that is, (A) is an orthogonal measure with structural function l(A fl B), where d denotes Lebesgue measure. py using Lemmas 2

and 1 of Section 3, we obtain )7(t) = Jrz

eit0fI(dw)

L V n=0

J-s

where an = V27Cb(n),

=

Vrzrz

cit'V27cg(ei-)e(d(o)

ei(t-n) 0 (dw) _

n=0

a,

(t - n)

rz

ei'%'e(dw) _rz

,

and

rz

M((n)(m)) =

-rzei(n-m)md(v

= 6nm

27C

Thus the sequence e(n) is uncorrelated. This lemma gives a simple answer to the question asked. But this answer is not sufficient for us in the general case since we still

do not know when the spectral density can be represented by formula (4).

6.

PHYSICALLY REALIZABLE FILTERS

219

Let us find conditions under which f(w) admits a representation of the form (4). Let H2 denote the set of all functions f(z) that are analytic in the disk D = {z, I z I < 11 and that satisfy the relation anzn , then = Jim Ir I fire M) 12 d8 < oo . If f(z) I f(z) 112

rT1

I

ff(Ye") = E anrneine n=0

that is, the anrn are the Fourier coefficients of the function f're'e). On the basis of Parseval's equality,

rzIfrei")I2d8

I an 12 r2'

= 21c

From this it is clear that f(z) e H2 if and only if

an I2 < co n=o

Consequently for every function f(z) e H2 it is possible to define a series f(e'B) = Z aneine that converges in L2(1), where I is Lebesgue n=0 measure on (- 7c, 7r). The function f(z) (for I z I < 1) is determined by the function f(e2B) in accordance with Poisson's formula f(reto) =

' t- f(e;°')P(r, 0, w)dw

2_

(6)

,

where

P r 8 w)

('

)

r2

1-2rcos(0-w)+r2

=

rinieance-.

Proof of this assertion follows immediately from Parseval's equality. It is shown in the theory of functions (see Privalov, or Hoffman) that if the function f(e'w) in formula (6) is Lebesgue-integrable, the limit lim f(re'B) = f eil) exists for almost all 0. The function f(e") *tl

is called the limiting value of the function f(z) (for I z I < 1). Theorem 1. Let f(w) denote a nonnegative function that is Lebesgue-integrable on the interval [ - tr, tc]. For the existence of a function g(z) e H2 such that

f(w) = I

g(ea.,)

12

,

(7)

it is necessary and sufficient that

rzIInf(w)Idw
(8)

Suppose that

g(z) _

n=0

anzn e H2

and that (7) holds. We may assume that g(O) # 0 (because if

LINEAR TRANSFORMATION OF RANDOM PROCESSES

220

g(O) = 0 we can consider not g(z) but z-mg(z), where m is the order

of the zero of g(z) at z = 0) and in fact we may assume g(0) = 1. Let r denote a number in the open interval (0, 1) and define A = {w; I g(re4l) I < 1 } and B = {w; I g(re°w) I > 11.

Then d(j)

= JB In I g(re'W) I dw - JA In I g(rez,) I = 2f B In I g(rei") I dw - ,-r In g(re"") I dw

rz I In I g(Yez,) II dw rz

From Jensen's inequality it follows that rz

I 27r

In If(rei-)Idw=In I r >>0, k=1 I Zk

-rz

where the zk are the zeros of the function f(z) inside the disk I z I < r. Consequently, Jrz

I In I g(reiw) II do) < rz

JB

In I g(re"-) I dw < B I g(re'-) I2 dw 2dw=27rjIa.12

Applying Fatou's lemma, we obtain lim rzrz I In I g(Yez') II dw < 7 lim I In I Jrz

J

II dw

InIg(ei-)Ildw < 27r Ian 2, n=0

a

which proves the necessity of condition (8). Proof of the Sufficiency. The function u(r, 0)

Suppose that condition (8) is satisfied.

27r JR rz

In f(w)P(r, 0, w)dw

is harmonic in the disk D: I z I < 1. Jensen's inequality that

We note that it follows from 1

rz

u(r, 0) <- In

27t

- f(w)P(r, 0, d(o) }

Let T(z) denote an analytic function defined in D with real part u(r, 0) that assumes a positive value at z = 0. We set g(z) = exp

g(re'B) I2 = e'

'

2

= eucr'° < 27r

R f(w)P(r, 0, w)dw

6.

PHYSICALLY REALIZABLE FILTERS

221

and

I g(rei°) 11 do < J rz

J

rz

rz .f(co)dw < rz

Thus g(z) e H2 and

\

lim I g(rei') I2 = exp (limu(r, 0) = f(O) )

almost everywhere.

This completes the proof of the theorem.

As we can see from the proof of the theorem, the function g(z) can be chosen so that it has a positive value at z = 0 and has no zeros in D. Combining Lemma 1 and Theorem 1, we obtain Theorem 2. For the sequence ra(t) to have a representation of REMARK 1.

the form

11 an 12<-, n-o

72(t)=Ea, n=o

(1')

where e(n) is an uncorrelated sequence, it is necessary and sufficient that the spectrum of ra(t) be absolutely continuous and that the spectral density f(m) satisfy the requirement rz

In f(m) > - o.

.

rz

REMARK 2. The function 99(z) is an analytic function in D, and its real part has limiting values In f(co). Consequently, ,P(z)

In f(c))

= 27c

rz

+

eim

z do)

.

(9)

Expanding the function g(z) = exp (I /2q(z)) in a power series g(z) E;,=o bnzn, we obtain the following values for the coefficients an in

formula (1'): an = 1/Tbn. On the other hand the expression for g(z) can be transformed as follows:

Since eim

+z

+

2ze-i.

+

I - zeiw

ei0 - Z

kik. k=1

we have

g(z) = exp

27r

'-, lnf(w)do +

where dk - rz eika In f(c))dco rz

27r 1 k=1 j dkzk}

LINEAR TRANSFORMATION OF RANDOM PROCESSES

222

Setting

P = exp ( 47C

1a

In f(w)dw

,

exp C 1 j dkzk) = j Ckzk / k=o \ 22V k=1

we obtain g(Z) = P E CkZk . k=o

Thus,

a = 1/27rPC . REMARK 3. The function g(z), whose existence was established by Theorem 1, is not uniquely defined. However, if g(z) satisfies the two conditions

a. g(z)#O for z e D, b. g(0) > 0,

it is unique and hence coincides with the function that we have To see this, let g1(z) and g2(z) denote two such functions. Then 'r(z) = g1(z)/g2(z) is analytic in D, it does not vanish in D, and its absolute value is equal to 1 on the boundary of D. The function In 1r(z) is analytic in D and its real part vanishes on the boundary of D. Therefore In *(z) = ik, where k is real. Since In p(0) is real, we have In /r(z) = 0. Let t1(e) and C2(O), for 0 e O, denote two Hilbert random functions. Let denote the closed linear hull of the system of found.

random variables {C;,(0), 0 e O} in L2.

Definition 1. If HH1 c Hc2,then the random function C1(0) is said to be subordinate to C2(0). On the other hand, if Hsl = Hc2, then C1(0) and C2(0) are said to be equivalent. REMARK 4.

As one can see from the proof of Lemma 1, the

sequences E(n) and fi(n) are equivalent.

Let us turn to processes with continuous time. Consider a process 77(t) that has a representation of the form 77(t) =

Jo

a(v)de(t - v)

,

(10)

where e(t) is an integrated white noise, that is, a process with orthogonal increments, such that M Ae(t) = 0, M I A4(t) I2 = At. Corresponding to the process e(t) is an orthogonal stochastic measure on the a-algebra of Lebesgue-measurable sets (cf. Section 3). The

integral (10) exists if and only if

6.

PHYSICALLY REALIZABLE FILTERS

223

Ia(t)12dt<-.

(11)

Jo

The process ra(t) can be regarded as a process at the output of a physically realizable filter at the input of which there is a white Then formula (10) applies to all such admissible filters, since by definition every admissible physically realizable filter is either of noise.

the form (10) or is the limiting value of a sequence of filters of such a form. A condition for mean-square convergence of filters of the form (10) with impulse transfer functions an(t) is as

n, n' -- - .

But if this condition is satisfied, the limit l.i.m. an(t) = a(t) exists (with respect to Lebesgue measure on (0, oo)() and an(r)de(t - v) = Jo a(v)dd(t - z)

U.M. Yen(t) = l.i.m.

.

JO

Thus passage to the limit in filters of the form (10) does not broaden the class of filters. Formula (10) can be rewritten ra(t) _

a(t - r)de(z)

a(t) = 0 for t < 0 .

,

Consequently the covariance function of the process ra(t) is equal to

a(t + v - u)a(z - u)du

R(t) = or

R(t) =

a(t + u)a(u)du .

(12)

J0

Lemma 2. For a process ra(t) that is stationary in the broad sense to be the response of a physically realizable filter to a white noise subordinate to the process, it is necessary and sufficient that the process ra(t) have an absolutely continuous spectrum and that the spectral density of the process f (o) have a representation of the form f(w) = I h(icj) 12

(13)

,

where

h(i(o) = o

b(r)e-'.rdr

I b(v) 12 dr <

,

(14)

JO

Suppose that the process representation of the form (10). Define Proof of the Necessity.

h(ic)) =

a(z)e-t-dr

1

1/270

0

.

i7(t)

has a

LINEAR TRANSFORMATION OF RANDOM PROCESSES

224

On the basis of Parseval's equality,

R(t) = 5a(t + u)a(u)du = 5

I h(ic)) J2 dcv

that is, the spectrum of the process is absolutely continuous and the spectral density is of the form (13)-(14). Proof of the Sufficiency. Suppose that the conditions of the lemma are satisfied. Consider a spectral representation of the process 72(t) _

and a stochastic measure 5-_

(A) _

(15)

h(im)

The stochastic integral (15) is meaningful for an arbitrary bounded Borel set A since XA((o)lh(i(o) e L2{F}, where F is the spectral function

of the process and F(A) = f

I h(im) 12 dw . A

We can easily see that C(A) is anLB orthogonal measure and that M(aS(A)S(B))

dw

=

.

We define the integrated white noise e(t) by m

1

e(tJ = V27r

--

e-iwt2

-

eti0,tl

im

C(dw) .

(16)

On the basis of Lemma 1 of Section 3 and formula (15) we obtain rJ(t) = 5

ea' th(im)C(d(o)

(17)

.

We note now that for an arbitrary function h(im) = 27r

where I I a(z) I2 dz < -, we have h(i(o)C(dm) = 5

a(r)e(dz)

.

(18)

This is true since the spaces L2{Q and L2{e} are isomorphic to the L2{1 }, where 1 is Lebesgue measure on the real line (- -, -) and the scalar product in L2{1 } is invariant under the Fourier transform; hence, it will be sufficient to prove formula (18) for simple functions. Suppose that a(t) = E CkX k(t) where Ok is the interval (open or

6.

PHYSICALLY REALIZABLE FILTERS

225

closed on the left, open on the right) (ak, bk). Then esmbk

ck

ck$(Ok)

-

e-i° 'k

1/2tclm

k

1(dw)

which is a special case of (18). Thus formula (18) is established. It follows from (18) that r

e'wth(i(w)C(d(o) = 5

a(v)d$(t - z)

,

(19)

5

since, on the basis of formula (16), multiplication of the measure C by e:o,t leads to displacement of the argument of the function $ by an amount t. (Here in formula (19) the sign of the argument of the function is changed, which causes a change in the sign of the stochastic interval.) Applying formula (19) to the expression (17) for ' J(t), we obtain

)7(t) - f a(z)de(t - z), where a(t)

h(ic))e-'0'tdw

o

This completes the proof of the lemma.

Suppose that we are given the spectral density f(w) of a process, J(t).

The following questions arise:

When does the spectral

density admit a representation of the form (13)-(14)? (When does it have a factorization?) How can we find the function h(i(o) (and hence the function a(t)) from the function f(cv)? The answers

to these function-theoretic questions can be obtained by reducing them to the questions that we have already answered for the case of functions on a circle. We introduce the space H2 of functions h(w), where w = x + icv,

that are analytic in the right half-plane and that satisfy the relation I h(x + 1w) 12 dw < c for all x > 0. The functions in H2 are just the functions that are necessary for realization of the factorization (13) and (14). This fact follows from the Paley-Wiener theorem: For h(w) to belong to HI, it is necessary and sufficient that it have ,a representation of the form h(w) _ 5:e_wtb(t)dt,

where

I b(t) I2 dt < cc

(20)

5o

(cf. Wiener and Paley, and Hoffman). The sufficiency of the condition of the theorem follows immediately from Parseval's equality. If h(x + ico) has a representation

of the form (20), then h(x + ico) 12 dw = 27c 5:_2xt 1 b(t) 12 dt

LINEAR TRANSFORMATION OF RANDOM PROCESSES

226

We introduce the transformation w = (1 + z)/(1 - z), which maps the disk D in the z-plane into the right half of the w-plane. On the boundary of the corresponding regions (w = i(o, z = e'°), this transformation takes the form co = cot (0/2). Let us suppose that f(c)) admits a factorization of the form (13)-(14). We set g(z) = h(w) = h(1 + z / '

The function f(9)

is

f(0) = f (cot 2) _ .f(w)

integrable over the circle and it admits a

factorization f(O) = I g(e'B) 12, where g(z) is analytic in D and

Il((J) IZ do) that is, g(z) e H2.

g(ei') IEd0

<0

But then on the basis of Theorem 1,

-- IIn f( w) dw =

In f(0)dO > - 0

.

(21)

n

The converse follows from analytic considerations: If (21) is satisfied, the factorization (13)-(14) exists. With regard to this

factorization, we might make remarks similar to those regarding the factorization of functions on the circle. The expression for h(w) is obtained from formula (9) by replacing z with w and making the corresponding change of variables under the integral sign: h(w) = exp

1

r°°

In f(co) i

cow dwl

2tc J-- 1 + (01 a) + iw

J

(22)

Theorem 3. For a nonnegative integrable function f(co) (for

- 0 < w < oc) to admit a representation of the form (13)-(14), it is necessary and sufficient that

lnf(w) dco>

(23)

Under the additional conditions that h(w) 0 (for Re w > 0) and h(1) > 0, the function h(w) is unique and is given by formula (21). Theorem 4. For a stationary process ra(t) (for - oo < t < o o) to have representation of the form (10), it is necessary and sufficient that it have an absolutely continuous spectrum and that its spectral density satisfy condition (23). 7.

PREDICTION AND FILTERING OF STATIONARY PROCESSES

An important problem in the theory of random processes, one that has numerous practical applications, is that of finding as close as

7.

PREDICTION AND FILTERING OF STATIONARY PROCESSES

227

possible an estimate of the value of a random variable C in terms of the values of the variables ea(for a e ?X). It is a matter of finding a function f(ea I a e 1X) depending on Ea for a e ?I with least possible

error, and satisfying the approximate equation y

b fyb =flea Iae%) .

( I )

An example of such a problem is the prediction (extrapolation) of a random process. In this case, the problem is that of estimating

the value of the random process at the instant t* from its values on some set of instants of time Z preceding t*. Another example is the problem of the filtering of a random process. This problem consists in the following: A process e(t) = ra(t) + C(t), representing the sum of the "useful" signal C(t) and the "noise" Y)(t), is observed at instants t' a V. The problem is to separate the noise from the signal; that is, for some t* e Z : ', we need to find the best approximation of C(t) of the form

W *) - C= f e(t') t' e ') The statement of the problem is not yet complete, since it has not been shown what is meant by "best approximation." Of course the criterion of optimality depends on the practical nature of the problem in question. With regard to the mathematical theory, the methods of solving this problem are primarily based on the meansquare deviation as a measure of accuracy of the approximate I

equation (1). The quantity a = {M[ - f(ea I a e I)]2}1/2 is called the mean-square error of the approximate formula (1).

(2) The

problem consists in determining the function f so that (2) is minimized. In the case in which S?t is a finite set, we mean by f(ea I a e'? I) a measurable Borel function of the arguments a for a e W. On the other hand, if ?I is infinite this symbol denotes a

random variable that is measurable with respect to the a-algebra 2`7S = Q(ea, a e %).

In what follows we shall assume that both C and f( a a e s?I) have second-order moments. We define

(3) (cf. Section 8, Chapter III). Then 2 = M{C - f(EEa I a e ?X)}2

= M(C - 7)2 + 2M(C - 7)(7 - flea I a e sVI))

+ M(7 -

I a e L[))2 .

LINEAR TRANSFORMATION OF RANDOM PROCESSES

228

Since y - f(ea I a e s?I) is 3-measurable, we have

M( - 7)(7 - f(ea I a e s?1)) = MM{(C - 1)(7 - f($a I a e W)) = M(7 - .f($a I a (z at)) M{(e - 7) I

}

}=0

Thus

2 = M(C

- 7)a + M(7 - f(a I a e))Z

from which we get Suppose that a random variable C has a finite secondorder moment. An approximation of C with minimum mean-square error obtained with the aid of a o'{ea, a e %}-measurable random variable is

Theorem 1.

unique (mod P) and is given by the formula REMARK 1.

7=M{CIA}. The estimate = y of a random variable C

is

unbiased; that is,

M7 = MM{C} = MC and the variables C - y and are, for arbitrary a c t, uncorrelated: M(C - 7)ea = MM{(C - 7)ea IM } = Me M{(C - 7)

}=0.

Unfortunately, the use of Theorem 1 to obtain actual approximation formulas is extremely difficult in practice. In the case of Gaussian random variables, however, we can proceed further. We note first of all that the simplest statement of the problem leading in a number of cases to a final and analytically attainable solution is the problem of finding the best approximation & not in the class of all measurable functions of given random variables but in the narrower class of linear functions. More precisely this means: Let {U, Cam, P} denote a basic probability space. Let us suppose that the variables ea and C have finite second-order moments. We introduce

the Hilbert space H{ea, a e I}, which is the closed linear hull of the variables Ea, for a e fit, together with all constants. The subspace H{ea, a e S?I} can be regarded as the set of all linear (nonhomogeneous) functions of Sa with finite variances. The best linear approximation

& to the random variable C is that element H{ea, a G W} that lies nearest C:

3E=MIA-CI2 for arbitrary C' e H{ea, a e S?t}.

M

C' - CIZ

We know from the theory of Hilbert

spaces (cf. Section 1 of this chapter) that the problem of finding the element

in the subspace H. that lies nearest the given element

7.

PREDICTION AND FILTERING OF STATIONARY PROCESSES

229

is the projection C always has a unique solution. Specifically, of C onto Ho. The element can always be determined (uniquely) from the system of equations (C - , C') = 0 for arbitrary C' G Ho, where (x, y) denotes the scalar product of x and y. In the present case this system of equations can be written in the form

(4) M(M) Since (1) belongs to H{ea, a e 5?X}, we have M = MC, so that the best linear estimates of are necessarily unbiased. Furthermore we may assume that W. = 0 for arbitrary a. Therefore in what follows we shall confine ourselves to a study of the subspace of random variables in L2{ U, Cam, P} with mathematical expection 0.

Of course, we do not always have grounds for assuming that a linear estimate of the quantity C is acceptable. For example, if e(n) = e'(°'"+w),

where v is uniformly distributed on (-ir, ir), then

m and the best linear approximation of the variable gy(m) from the values of all E(n) for n # m is of the form M(e(n)e(m)) = 0 for n

Vi(m) = 0; that is, it does not use values of the variables E(n), whereas any pair of observations (k) and E(k + 1) are sufficient to determine the entire sequence E(n) precisely, si(n) _

(e(k+l))n-k (k)

Let us suppose now that all finite-dimensional distributions of the system {C, a, a e 51x} are normal and M. = MC = 0. In this case it followss from the uncorrelatedness of the variables C - & and a that they are independent. Therefore C - C are independent of the a-algebra

and

M{CI }=M{C-&+&I`}=M{CTheorem 2. For a system of Gaussian random variables (C, ea, a e 51X), the best approximation (from a standpoint of mean-square deviation) of the variable C with the aid of a e 51X} of a measurable function coincides with the best linear approximation in H{ea., a G 5I}.

We now consider a number of particular problems on the construction of best linear approximations. 1.

The number of random variables a. (for a = 1, 2,

, n)

is finite. This problem has a simple solution as we know from linear algebra. Assuming that the a are linearly independent, we can construct the projection & of the variable C onto the finite-dimensional

space H. generated by the quantities a (for a = 1, of the formula

, n) by means

LINEAR TRANSFORMATION OF RANDOM PROCESSES

230

(e1, e1) (S n, e1) e1

en) 0

,fin) is the Gram determinant of the system

where P = P( e2, of vectors 1, e2,

,

n, (e1, e1) .

g P(S1, 2, .. . ,n) _

. .

. ... . . ... .g.... . e1) .. . (S n, en)

r)) = M(e9) ). The mean-square error 3 of the approxiand where mate equation C & is equal to the length of the perpendicular

onto the space H. and is given by the

dropped from the vector formula a2 = {l (e1, e2,

, n)}

, en, S)/ry (e1, e2,

Consider the problem of approximating a random variable from results of observation of a mean-square continuous random process c(t) on a finite interval of time T = [a, b]. Let R(t, z-) denote the covariance function of the process e(t). On the basis of Theorem 2 of Section 2, the process e(t) can be expanded in a series, 2.

L V TkPIWt k=1

k

where {Pk(t)} is an orthonormal sequence of eigenfunctions, where the Xk are the eigenvalues of the correlation function on (a, b): XkPk(t) =

5Rt, m b

and where ek is the normalized uncorrelated sequence Meker = 3kr Obviously, {ek}, for k = 1 , 2, , constitutes a basis in H{e(t); t e (a, b)}.

n = 1, 2,

Therefore ,

_ E7=1

(' b

where

for

Ja

and

cn = MUn = :&(t),pn(t)dt a

&(t) =

The mean-square error a of the approximation can be found from the formula y 32=MI2-MI

6RsF(t)pn(t)dt SI2=MI C2 I -E n=o

2

a

In practice, the application of this method is made more difficult by the complication of calculating the eigenfunctions and eigenvalues

of the kernel R(t, z). 3.

The integral equation for filtration and prediction.

Wiener's

7.

PREDICTION AND FILTERING OF STATIONARY PROCESSES

231

Let E(t) and C(t) for t e Z denote two Hilbert random

method.

Let us suppose that the process e(t) is observed over some set T of values of the argument t. Consider the problem functions.

of determining the best approximation of the value of C(t0) for to e Z from the observed values of E(t) where t c T. If we assume that

the desired approximation is of the form

(5)

(to) _ Tc(B)E(O)m(dO)

where m is some measure on T, and if the conditions under which the above integral is meaningful are satisfied, then equation (4) takes the form Tc(B)Rff(B, t)m(d6) = R3'e(to, t)

(6)

tcT,

,

where RU is the covariance function of e(t) and RAF is the mutual correlation function of C(t) and e(t). Equation (6) is a Fredholm integral equation of the first kind with symmetric (Hermitian) kernel. By no means does it always have a solution. However, if T

M I e(t) 12 m(dt) < _ ,

the integral equation (6) has a solution c(O) e L2{m} if and only if the best linear approximation of C(t0) of the quantity C(t) is of the form (5). Suppose that Z is the real axis and that T = (a, b), that the processes and C are stationary and stationarily connected (in the broad sense), and that the measure m is Lebesgue measure. Then equation (6) takes the form bc(B)R,# - t)dO = &(to - t)

,

(7)

t e (a, b)

If C(t) = e(t) (for - oc < t < -) and tQ > b, that is, if the problem consists in finding an approximation for the quantity E(to) from the

values of c(t) in the past, we - shall call the problem one of pure prediction.

Let us look in greater detail at the problem of prediction of the variable C(t + T) from the results of observation of the process E(O) from the instant t, where 0 < t. Let us treat the predicting variable fi(t) as a function of t for fixed T. We can easily see that C(t) as defined by equation (5) is a stationary process. To see this, note that equation (7) is of the form Jt

ct(O)RF£(B - u)dO = &(t + T - u)

,

uGt

.

The change of variables t - is = v, t - 0 = z transforms the preceding

LINEAR TRANSFORMATION OF RANDOM PROCESSES

232

equation into the following:

ct(t - r)Ree(v - r)dr = Rse(T + v)

V>0.

,

(8)

From this we see that the function ct(t - z-) is independent of Let us set c(r) = ct(t - r). Equation (8) can now be written

Oc(r)Ree(v-r)dr=&e(T+v),

v>0,

t.

(9)

and formula (5) for the predicting function takes the form

c(t - r)e(v)dr =

&(t)

c(r)c(t - r)dr .

(10)

Jo

Thus the process C(t) = CT,(t) is stationary. It follows from formula (10) that c(t) is the impulse transfer function of a physically realizable filter that transforms the observed process into the best approximation

of the quantity C(t + T). It is easy to exhibit an expression for the mean-square error 8 of the predicting function fi(t). Since 82 is the square of the length of the perpendicular dropped from the vector C(t + T) onto H2{e(r), -r:!5; t}, we have

32=MI&(t+ T)I2-MI&(t)I2 = 5i5R25(t - r)c(r)dtdr Setting &(0) = a and shifting to the spectral representation of the covariance function RFF(t), we obtain 32

=v

(12)

I C(iw) 12 dFFF(m) ,

where FFF((o) is the spectral function of the process e(t) and C(ic)) =

Jc(t)e-i0tdt 0

We shall explain briefly a method proposed by N. Wiener for solving equation (9). Suppose that the spectrum of the process e(t) is absolutely continuous and that the spectral density ffF(w) admits a factorization (cf. Theorem 3, Section 6): .fee((o) = I h(i(o) 12, h(z) _

'

a(t)e_ztdt,

Vfi to

Re z > 0

It follows from Parseval's equality for the Fourier transform that Ree(t) =

e-at0, I h(i(w) I2 dw

=

Jo

a(t + u)a(u)du

Let us suppose also that the mutual spectral function of the processes

7.

PREDICTION AND FILTERING OF STATIONARY PROCESSES

233

Z(t) and e(t) is absolutely continuous and that its density ffe((o) satisfies the condition ffe((O) = k(i(o) E LZ .

(13)

h(i(o)

Then eit0ffe((o)dw

Rse(t) =

=

Jo

=

b(t + u)a(u)du

etitwk(i(o)h(i(o)dm

,

where b(t) 1

'

With the aid of the expressions obtained we can rewrite (9) as follows:

[b(T + v + u) - o c(v)a(v - v + u)dr]d(u)du = 0

,

v > 0 . (14)

A sufficient condition for equation (14) to hold is that c(t) satisfy the equation

b(T + x) =

c(v)a(x - v)dv

x>0.

,

Jo

(15)

Equation (15) is of the same type as equation (9) except for the important fact that the function a(t) vanishes for negative values of t. If we write equation (15) in the form

b(T + x) = Jc(v)a(x - v)dr

x>0,

,

(16)

we can immediately solve it with the aid of the Laplace transform. Multiplying equation (16) by e-11 and integrating from 0 to o, we obtain BT(z) = C(z)h(z), where BT(z) = (1/1/fi)Sb(T + x)e-22dx, 0

C(z)

1/27c Jo

c(t)e-ztdt

Thus

C(z) = BT(z) h(z)

c(t) =

,

1

eio,t

-c

BT(im) dm h(i(o)

,

(17)

where the expression for BT(z), for Re z > 0, can be written in the form BT(z)

_

1

27r

e%T°, J (m)

dm

h(im) z - iw

(18)

LINEAR TRANSFORMATION OF RANDOM PROCESSES

234

Formulation of the assumptions under which formulas (17) and (18) are valid is extremely laborious. In solving specific problems, it is simpler to verify directly the validity of the proposed transfor-

mations that lead to the solution of the problem. 4. Yaglom's method. With this method, in contrast to that of Wiener, we seek not the impulse transfer function of the optimum filter, which may not even exist, but instead the frequency characteristic. We shall not give general formulas for solving the problem but shall present only a method of choosing the desired function by starting with those requiremements that it must satisfy. In many important cases, this choice is easy to make. Suppose that a two-dimensional stationary process (e(t), ((t)) has

a spectral representation of the form fi(t) =

e'.t p (du)

eq.t

E(du)

with spectral density matrix Jee(w)' Jk(w) .fsf(m)' .fss(co)l

As before, let us consider the problem of the best approximation of the quantity ((t + T) from the values of the process e(-r), for v < t. The predicting process &(t) is subordinate to e(t). Therefore ecwtc(iw)1_t1(dw)

fi(t) =

,

_m I c(iw)

12

ff(w)dw < °

(19)

The equation defining the process fi(t), namely MC(t + T)e(T) = MC(t)e(v)

Z. < t

,

takes the form fSe(m) - c(io))fff(o))}dw = 0

,

V>0.

(20)

In addition to conditions (19) and (20), we also have the requirement that c(im) be the frequency characteristic of a physically realizable filter. These conditions will be satisfied if a. b.

the function ff(w) is bounded, c(iw) is the limiting value of the function c(z) E HZ (cf.

Section 6), c. gr(im) = e"'Tfff(w) - c(i(o)fff((o) is the limiting value of the

function *(z) E Ha, where Ha, is defined analogously to H; except that Ha consists of functions that are analytic in the left half-plane.

To see the validity of this assertion, note that condition (b) implies that

c(iw) 11 dw < oa, and this, together with condition

7.

PREDICTION AND FILTERING OF STATIONARY PROCESSES

235

a, ensures satisfaction of condition (19). Also, condition (b) implies that c(ic)) is the frequency characteristic of a physically realizable filter. It follows from condition (c) that e'lT ff,((o) - c(io)f££(oi) is the Fourier transform of a function that, by virtue of equation (20), vanishes for positive values of the argument. If we confine ourselves to condition b we rule out filters whose frequency characteristics can increase at infinity. Such frequency characteristics correspond to operations connected with differentiation of the process e(t) and are often encountered in the construction of optimum filters. Therefore it is desirable to replace condition b

with a less restrictive one. Let us suppose that c(z) is a function that is analytic in the right half-plane and that c(z) I approaches - no faster than the rth power for some r as I z We define the functions c (z) by cn(z) =

r+ l nI

c(z)

C1+

e H2

we have

Since I c(z)

lim

I c(uo) - t(ie)) I2f

(o))d(o = 0

if condition (19) is satisfied. Thus c(i(y) is the limit in L2{FF} of frequency characteristics of admissible physically realizable filters. Therefore c(ity) is also the frequency characteristic of such a filter. Thus we have proved: Theorem 3. If the spectral density fFF(w) of a process E(t) is bounded, then the three conditions

c(i() )d(o < b. c(ity) is the limiting value of a function c(z) that is analytic a.

in the right half-plane and that increases no faster than some power of

zI as zI,o-;

c. ' 1r(iw) = e24TffF(m) - c(im)fEF((o) function 1/r(z) e HZ ;

is the limiting value of a

determine uniquely the frequency characteristic c(ic)) of the optimum filter approximating the quantity e(t + T).

The mean-square error 3 of the best approximation is equal to

6={MIC(t+ t(i(d) 2fee((o)de)}2

Example 1.

(21)

Consider the problem of a simple prediction of a

LINEAR TRANSFORMATION OF RANDOM PROCESSES

236

process e(t) (where e(t) = C(t)) with covariance function R(t) _ 62e-a 1 t I

for a > 0. The spectral density is easily found; 62a

Jf((0)

7r

1

w2+a2

The analytic continuation of the function 'r(i(o) is of the form = c(z) - eIT Y'(Z) (z + a)(z - a) This function *(z) has only one pole in the left half-plane: z = - a. To neutralize it with the aid of a function c(z) that is analytic in the right half-plane, it suffices to 'set c(z) = const = e-aT. Here condition (a) of Theorem 3 is satisfied. Thus ezte-aT p(do))

e-aT

c(i(w) =

that is, the best formula for the prediction of the quantity (t + T) is (t + T) P,-s which depends only on the value of e(v) at the last observed instant of time. The mean-square error of the extrapolation is equal to 3 = 6V1 Example 2. Let us consider the problem of pure prediction of a process e(t), that is, of finding an estimate for (t + T) from observed values of e(v) for z < t. If the spectrum of the process fi(t) is absolutely continuous and if condition (23) of Section 6 is e-1,T.

satisfied, then the spectral density of the process admits a factorization fff(tn) = I h(i(o)12, where h(z) belongs to H2 and has no zeros in the right half-plane. Let us consider the case which is very important for practical applications and in which h(z) is a rational function h(z) = P(z)/Q(z), where P(z) is a polynomial of degree m and Q(z) is a polynomial of degree n > m. Let us suppose also that the spectral density f;(co)

is bounded and that it does not vanish. Then the zeros of the polynomials P(z) and Q(z) lie in the left half-plane. Let P(z) and Q(z) be represented in the forms 9

P(z) = A H (z - zj)aj

,

Q(z) = B R (z - 2j)lj

j=1

j=1

where

Eaj=m,

j=1

j=1

,6j=n.

Let us define

P1(z) = (- l )-A fJ (Z + 2j)ai 7=1

,

Q1(z) _ (- 1)" Bj=1II (z + 2jyj

7. PREDICTION AND FILTERING OF STATIONARY PROCESSES

237

The analytic continuation of the function 1/r(iw) is of the form #(z) = (e.=T - c(z)) P(z) Pa(z) Q(z) QAz)

The function c(z) must be analytic in the right half-plane and 1/r(z) must be analytic in the left half-plane. Therefore c(z) must be analytic in the entire complex plane and it may have poles at the zeros of the polynomial P(z). The order of such a pole cannot

exceed the order of the corresponding zero of P(z).

Therefore,

c(z) = M(z)/P(z), where M(z) is an analytic function in the z-plane that has no singularities for finite z. Since c(z) has no more than a power order of growth, M(z) is a polynomial. In view of the square-integrability of the absolute value of the function c(iw)

P(iw) = M(i(e) Q(iw) Q(iw)

the degree m, of the polynomial M(iw) does not exceed n - 1, that

is, ml < n - 1. On the other hand, the indicated choice of the function c(z) ensures satisfaction of conditions (a) and (b) of Theorem 3. We must choose the polynomial M(z) in such a way that the function '/'(z) = (eZTP(z).- M(z)) Pa(z) Q(z)

Q'(z)

or, what amounts to the// same thing, the function *AZ) =

ezT P(z) - M(Z) Q(z)

has no poles in the left half-plane. For this it is necessary and sufficient that

=

d'M(z) dzi

1z=7,

di(e$TP(z))

dzi

z=zk

i = 01 11 ... , 16k - 1

k=1,...,q.

(21')

The problem of constructing a polynomial M(z) satisfying conditions (21') is the usual problem in the theory of interpolation and it always has a unique solution in the class of polynomials of degree n - 1. In finding the polynomial M(z), we automatically find the

frequency characteristic of the optimum predicting filter c(iw) _ M(iw)/P(aw).

We may also use the following procedure for determining the function c(z). Let us decompose the functions P(z)Q-'(z) and M(z)Q-1(z) into partial fractions:

LINEAR TRANSFORMATION OF RANDOM PROCESSES

238

P(z) = Q(Z)

q

rykj

M(z)

Ck'

k=1 j=1 (Z - Zk)j '

k=i j=1 (Z - Zk)j

Q(Z)

For the function '(r1(z) not to have poles at the points 2k, for k = 1, , q, it is necessary and sufficient that

_ dzi

0, 1, ..., (3k - 1

(Z - 2k)PkY'1(Z) L-zj, = 0, Ek=1

where *1(z)

k1 (CkjezT - ,Ykj)I(Z

,

- 20j. Simple calculation

shows that rykj =

[kJ

T2

+ TCk.j+1 +

TPk-j

Ck.j+2 + ... +

(Rk - j)!

2!

CkPkI eakT

k=1,...,q,

Knowing the coefficients yk j, we can write the expression for c(iw): q

C(lw)

-

1

h(am) k=1 j=1 (Z

7kq2k)9

=

Pk

k-1 j=1 qqE

Pk

k=1

ryk j

(Z - 2Jj Ckj

(22)

(Z - Zk)q

Suppose that we are observing a process (z) (for z < t) but that the results of our measurements of C(r) are distorted by various errors, so that the observed values yield a function (v), for z < t, different from ((z). Let us suppose that the magnitude of the error (or, as we say, the noise) ra(t) = fi(t) - C(t) is a stationary process with mean value 0. Suppose that we wish to find an estimate of the value of C(t + T) from the results of observation of the process (v) = C(v) + Y)(z) for z < t. Such problems are called filtering or smoothing problems (we say that we need to filter the noise )a(t) out of the process fi(t) or that the process fi(t) must be "smoothed," that is, the irregular noise needs to be subtracted from it). Here for T > 0, we have a Example 3.

problem of filtering with prediction and, for T < 0, we have a problem of filtering with lag. Let us suppose that the noise ra(t) and the process C(t) are uncorrelated and have spectral densities{ fe(w) and ff,((o). Then Ree(t) = Rno(t) +

and J,(w) = .f,,((o) + there exists a mutual spectral density of the

Since RMt) = processes ((t) and e(t) and we have f;,((o) =

f"(w) = w2 + a2 Then

and f,,(w) =

Suppose that ( 1 )2+6 2

7.

PREDICTION AND FILTERING OF STATIONARY PROCESSES

££((0) -

((0 2

C3(0 + 72) //

C3 = C, + C2 ,

,

+ 32) + For the function *r(z) we obtain the expression

239

72 = c1a2 + C2 92 C1 + C2

a2)(w2

- C1ezT(Z2 - 32) +

C3C(Z)(Z2

- 72)

Y'(Z) _

(Z2 - a2)(Z2 - N2)

Suppose that T > 0. The function 'i/r(z) must be analytic in the left half-plane and it must belong to H,1. For this it is necessary

that the numerator vanish at the points z = - a and z This leads us to the equations C(- = 0 a) = C(R)

e "T(a2 C1 C3

2R2)

a - Y

(23)

Furthermore c(z) must be analytic in the left half-plane (and also in the right half-plane by virtue of condition b) except at the single point z = -7 where it may have a simple pole. Thus c(z) = q(z)/(z + 7), where T(z) is an entire function. From the condition of finiteness of the integral

I C(i(o) I2 f j(O)dw,

it follows that p(z) is a linear function, p(z) = Az + B. From (23) we obtain

c, 8+ae_.T c3 7+a

c(z)=AZ+(3

Z+7

Therefore the formula for optimum smoothing with prediction is of the form

T(t)=A

e24't10)+Rp1(dco).

ico + 7

Remembering that (i(o + 7)-1 is the frequency characteristic of a physically realizable filter with impulse transfer function e-71, we obtain CT(t) = C1 ,Q + a e_.T J e(t) l\

- (R - ry) f t

e

C37+a

(24)

For T < 0, formula (24) is not valid. Formally, this is connected with the fact that the function *(z) is not bounded in the left halfplane in this case. For T < 0 the function #(z) may be determined from the following considerations. Suppose that '1/r1(z) = - c1eQT(Z2

- R2) + C3C(Z)(Z2 - 72)

Then c(z) must be analytic in the left half-plane except at the point

z = - 7 and we have *,(-a) = 1/! l(- /3) = 0. Since C(Z) = #1(z) + c1ezT(Z2 - R2)

C3(Z2 - 72)

LINEAR TRANSFORMATION OF RANDOM PROCESSES

240

and c(z) is analytic in the right half-plane, it follows that *1(z) is an entire function and we have ''lr1(7) = - c1erT (72 - 9Z)

(25)

.

If we set *1(z) = A(z)(z + a)(z + 6), the function A(z) must be an entire function. It follows from condition (8) of Theorem 3 that A(z) = const = A. The value of A is determined from equation (25): A=C1eTT7+IS

a-7

From this we get c(icv) = cl (a - 7)(W2 + 62)ei.T - e. T (,y + R)(i(o + a)(ico + R) (26)

(a-7)(w2+72)

c3

For the prediction and filtering of stationary sequences, we apply methods analogous to those that were discussed for processes with continuous time. The general solution of the problem of prediction of stationary sequences is given in the next section. Here we shall confine ourselves to a single example. Example 4. A process of autogression is defined as a stationary sequence e(t) satisfying the finite-difference equation

aoe(t) + a1e(t - 1) + ... +

p) = YI(t)

(27)

and subordinate to r)(t), where Y)(t) is an uncorrelated sequence such that Mr9(t) = 0 and Dra(t) = 62. Suppose that rz

eitwdd(w)

)7(t) = f rz

is the spectral representation of the sequence ra(t) and that C((o) is a process with independent increments and with structural function (1/27r)l(A (1 B), where l denotes Lebesgue measure. Then the spectral representation of the sequence e(t) must have the form e(t) =

e{t°q?(c))dC((o) J rz

,

where

I cp((o) 12 do) < - . J rz

rz

(28)

rz

Substituting (28) into (27), we obtain rrz eito,p(ei')co(w)dd(w) J

n

where P(z) = Yk=o akzk

= rz

ett",dC(w) rz

From this it follows that

'P (0)) = P(ei0))

(mod 1)

.

Let us suppose that the function P(z) has no zeros in the closed

disk I z I< 1.

Then 1/P(z) e H2.

If

8.

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

P(Z)

=j

bkZk(b0

=

Qo)

241

,

then

e(t) _

n=0

byj(t - n)

and we have obtained a representation of the sequence E(t) in the form of the response of a physically realizable filter to the uncorrelated sequence Y)(t). 1

(t)

ao

Since

[a,e(t - 1) + ... + a9E(t - p) + )7t]

(29)

the optimum prediction e(t) from given (t - n), where n = 1, 2, is of the form fi(t)

,

1 (a,$(t - 1) + a2E(t - 2) + ... + a,e(t - p)) ao

The minimum mean-square error of the prognosis is equal to {M_(t)

I211/2

I aoI2

a

=

1ao

Successive use of formula (29) enables us to obtain the optimum prediction several steps in advance. 8.

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

In this section we shall consider certain general theorems on the prediction of stationary sequences and processes with respect to the infinite past. By a stationary process we mean a process that is stationary in the broad sense and that has mathematical expectation 0. 1.

Prediction of stationary sequences

Let e(t), for t = 0, ±1, ±2, , denote a stationary sequence. Let HF denote the closed linear hull generated in L2 by all the quantities e(t), and let HF(t) denote the closed linear hull generated by the quantities e(n) for n < t. Obviously HF(t) C Hf(t + 1) and HH is the closure of u-,HF(t). Consider in HF the operation S representing time displacement. For elements of H£ of the form Y7 = E c (tk) this operation is defined by S)J = E Ckl(tk + 1)

LINEAR TRANSFORMATION OF RANDOM PROCESSES

242

The operation S has an inverse S-1:

S-1' = E ckWk - 1) and it preserves the scalar product, M(S(E Cke(ik))S(L dkee(rk))) - E Er ckdrM(e(tk + 1)ee(rr + 1)) k _ E E ckdrM(e(tk)ee(zr)) = M(E cke(tk)

dre(rr))1

.

Therefore S can be extended as a continuous operator to H. It then becomes a unitary operator in H. We introduce the spectral representation of the sequence e(t) (cf. Remark 4, Section 4), e(t) =

e'.,t p(da)

where p is a spectral stochastic measure with structural function F.

In what follows we shall not distinguish the measure F(A) from

the spectral function (generated by it) of the sequence F(w) _ F[-7r, w). We recall that a random variable 7) belongs to HH if and only p(D)/.t(dw), where p e LZ{F}. Consider the sequence ra(t) if = J n of random variables 7)(t) = Str) (t = 0, ±1, ±2, ). .1 rz

Lemma 1. The sequence r)(t) is stationary and subordinate to e(t) and it has the spectral representation

YAO = I eit.,p(w)1 (dw) . J ,

(1 )

That Y)(t) is subordinate to the process e(t) is obvious. That it is stationary follows from the unitariness of S: MO7(t + r)ri(r)) = 01(t + r), r7(r)) _ (St+ry), S "Y)) = (St)7, rI) = M(7)(t)21(0))

Finally, the spectral representation (1) is easily verified for elements

i of the form 17 = Z

(where p(o)) = E akeiltk) and it

is

obtained for arbitrary Y) by passage to the limit. We also note the following obvious properties of the operator S: a. SH£(t) = HF(t + 1); b. if e(P)(t) is the projection of e(t) onto HF(t - p), then Sea)(t) = eM(t + 1) , q) Since

M I (P)(t + q) I2 = M I S9e(P)(t)

IE

= M I e(P)(t)

the quantity M I e(P)(t) 12 is independent of t.

12

,

Therefore the quantity

8.

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

32(P) = MIfi(t) - e(t)I2= MIfi(t)I2-MI

243

I(t)i2,

which is equal to the square of the minimum mean-square error of the prediction of the variable e(t) with the aid of the quantity e(n),

for n < t - p, is also independent of t. Obviously, a2(1) < 32(2) < ... < 62 = M I fi(t) 12

The equation 32 (n) = 62 means that e(t) is, for arbitrary t, uncor-

related with all the variables ek, for k < t - n, so that the value of these terms yields nothing for the prediction of the variable E(t). If 3(1) = 0, then e(t) e HF(t - 1), so that HF(t - 1) = HF(t). Let us set HE = nt HF(t). In the present case, HF = He. This means

that if we know the sequence of values of the process e(k), for k < to, then all the subsequent terms of the sequence have with probability 1 a precise linear expression in terms of the observed values. In a certain sense we have a contradiction in the case in which Hs = 0, where 0 denotes the trivial subspace of HF consisting only of the element 0. Here, knowing the terms of the sequence (k) (for k n) yields little for the prediction of the variable (n + t) for large values of t since lime . M I I e"'(n) I I = 0 and limt__ 52(t) = Q2. Deflnltlon 1. If H£ = HH, the process e(t) is called a singular (or determined) process. If 8(l) > 0, the process e(t) is called an undetermined process.

If HF = 0, the process is called a regular (or

completely undetermined) process.

Theorem 1. the form

An arbitrary sequence has a unique representation of

fi(t) = es(t) + I(t) ,

(2)

where e,s(t) and 77(t) are mutually uncorrelated sequences subordinate to is singular, and where ra(t) is regular.

Proof. Obviously, SHF = H. Since S is a unitary operator, it follows that HF reduces S; that is, S is a one-to-one mapping of the subspace H1, = HF e Hs onto itself (cf. Lemma 2, Section 1). Let es(0) denote the projection of e(0) onto Hs and let 17(0) denote the projection of (O) onto H. Let es(t) = and 7)(t) = S°7)(0) for t = 0, ±1, ±2, Since (0) = es(O) + r(0), we ra(t). Here the sequences 77(t) and e(t) have e(t) = are stationary, mutually uncorrelated, and subordinate to e(t). Furthermore, since, in the equation fi(t) = e8(t) + ra(t) it is true

.

that es(t) e Hs and ra(t) e HF, we have HF(t) n Hs c HFS(t).

Therefore

HF c Hs. On the other hand, the inclusion relation

e8(t) a HE

implies that HHS(t) c Hs.

Thus for arbitrary t, we have H s(t) _

LINEAR TRANSFORMATION OF RANDOM PROCESSES

244

HF = HAS; that is, the sequences E2(t) is singular. Furthermore, the equation ' J(t) = fi(t) - e,(t) implies that 7)(t) e HF(t). Therefore

H, = n H,,(t) c Hs. On the other hand, by definition H,(t)

is

Thus Hs = 0; that is, the process 7)(t) is regular. The uniqueness of the representation (2) follows from the fact that under the hypotheses of the theorem the projection of ra(t) onto HF is equal to 0, Hs = H 8 = HFS, and consequently $,(t) is the orthogonal to H,8.

projection of 72(t) onto Hs. This completes the proof of the theorem.

The sequences ?7(t) and $s(t) are called respectively the regular and singular components of the process e(t).

Theorem 2. The regular component ra(t) of a stationary sequence can be represented in the form

ra(t) _ E a(n)g(t - n) , ri=o

(3)

where C(t) (for t = 0, ± 1, .) is an uncorrelated sequence, where Hi(t) = H,(t), and where En=o I a(n) 11 < -. Proof.

We introduce the subspace G(t) = H,2(t) e H,(t - 1).

This subspace is one-dimensional (if it were the zero space we would have 32(1) , 0 and 7)(t) would be a singular sequence). Let us choose in G(0) the unit vector i;(0). Then the sequence C(t) = SIC(O) is orthonormal (C(t) e Hi(t) (D HH(t - 1), therefore C(t) is orthogonal

to Hn(t - 1); also C(k) e H,(t - 1) for k < t), Hi(t) c H,(t), and n Hi(t) c n H,(t) = 0. This means that the sequence C(t) constit

'tutes a basis in HE. Expanding 7)(0) in elements of this basis, we obtain

72(0) = E a(n)i;(-n) , k=0

a(n) I2 = I 117(0) I I2 = M I)7(0) I2 <

Applying to r7(0) the operator St, we obtain

1)(t) = Z a(n)C(t - n) ri=0

The inclusion relation H,(t) c Hi(t) follows immediately from (3), and the opposite inclusion relation follows from the definition of C(t).

This completes the proof of the theorem. REMARK 1.

We may assume without loss of generality that

a(0) is positive. Lemma 2. Suppose that the spectral function F(6o) of a stationary process e(t) is equal to F1((o) + F2(w), where the F;(w) are nonnegative

8.

245

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

nondecreasing functions and the measures F;(A) corresponding to the functions F1(c)) are singular. Then there exists a decomposition e(t) = E1(t) + e2(t), where the processes $,(t) are subordinate to e(t), are

orthogonal, and have spectral functions F;(w) for i = 1, 2.

To prove this, we represent the interval [-7r, ir] as the union We set

of two disjoint sets P1 and P//2 such that F2(P1) = F1(P2)//= 0.

1(t) =

,$2(t) _ rz etitmXP2(w),1(dw) , /l.

Jrz

rz

rz

is a stochastic spectral measure of the process e(t) and where x1Jco) is the characteristic function of the set Pi. Then e1(t) + e2(t) _ (rz e;tlu(dw) = fi(t) M(e1(t1)S 2(t2)) =

Ve'('1-t2)'X,,(oj)X,,(co)dF(o)) = 0

,

rz

M(e (t1)e,(t2)) =Jrz e'(tl-t2)'XPJ(oJ)dF(co) = .ei( t J-t2).dF;((o)

,

which completes the proof of the lemma. Theorem 3. For a sequence e(t) to be undetermined, it is necessary and sufficient that rz

In f(w)dcv > - oo

(4)

,

rz

where f(u) is the derivative of an absolutely continuous component of the measure F(A) (with respect to Lebesgue measure). Proof of the Necessity.

Let F,((o) and Fs((o) denote the spectral

functions of the sequences i)(t) and eS(t). It follows from the uncorrelatedness of ii(t) and $,(t) that F(w) = F,((o) + FS(w)

It follows from Theorem 2, Lemma 1, Section 6 and Theorem 1, Section 6 that F,((o) is absolutely continuous and that the condition Jrz

In ff,(o))da) > - o0 rz

is satisfied for f,((o) = F,;,(w). Decomposing the measures F(A) and FS(A) into absolutely continuous and singular components with respect

to Lebesgue measure (cf. Theorem 1, Section 6, Chapter II), we obtain

F(A) _ Af(w)dw + F* (A) ,

FS(A) _

fs(w)dw + FS (A) ,

from which it follows that f(m) = f,((o) + fs(w) and

LINEAR TRANSFORMATION OF RANDOM PROCESSES

246

In f,((o)dw > - 00

In f(w)dw

.

Thus if the process is undetermined, (4) is satisfied. Proof of the sufficiency. Let us assume the opposite. Then the decomposition

F(A) = Fs(A) = f fs(w)dw + Fs (A) corresponds to the decomposition of e(t) into uncorrelated components

E1(t) and e2(t) that are subordinate to e(t) (cf. Lemma 2). If it were true that In f(w)dw =

In fs(w)dw > - 00

,

then on the basis of Theorem 2 of section 6,

ei(t) = R=o E a'(n)C'(t - n) where C'(t) is an uncorrelated sequence. Since

and n HF1(t) = 0 we have fl HF(t) c n HHE(t) c H 2, which does not coincide with He(t) c HE1(t) ® HHE(t)

He = He, Q) He2 so that the process e(t) cannot be singular.

In f(w)dw = - 00

Therefore

.

J rz

This completes the proof of the theorem. Let us consider the problem of prediction of undetermined processes. Using Theorems 1 and 2, let us write e(t) = es(t) + 7I(t)

7I(t) =%=o Z a,C(t - n)

,

Since es(t) is exactly determined by the past, it will be sufficient to consider the prediction of the regular component fi(t) of the process e(t). It follows from Theorem 2 that the projection i7(t) onto H,(t - q) coincides with the projection onto HH(t -q). Consequently,

72c9>(t) = q an:(t - n) *=a

.

(5)

The value of the mean-square error is determined from the equation q--+1

u2(q) = Ln=oI a. 12

.

(6)

We shall now obtain a formula for the best prediction that does not contain the sequence C(n). Since C(0) c H, we have

8.

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

247

I qp(w) l2 dF(w) < _ ,

C(0) = Jrzn p(w)v(dw) ,

Jrz

rz

where v is a spectral stochastic measure of the process 77(t), F(w) = E;,=o dne'' (cf. Lemma 1, Section 6).

I g(eiw) I' dw, and g(e") =

Consequently (cf. Lemma 1), fi(t) = SV0) = Jrz e:t' cp(w)v(d(v)

.

rz

To find the function p (o)), let us use formula (3). We have i7(t) = n=0 E anb(t - n) = f

rz

J

e:t' q (w) n=0

rz

Comparing this with the equation 7)(t) =

(w) _

dv(dw), we obtain

I

J rrz

2tr g(iw))-' f = (V

ane-an.l -1 n=o

We now have

y2so rz

ane-tinml

/1q(w)e.t.,v(dw)

n=qy

that

r J

eit.,r1 rz

L

-

gy(ea.)

v(dw) (7)

where y-1

gy(e:.,) =

Z ane,n.,

(8)

We shall now demonstrate a method of determining the function g(z) = Z =o bnzn, where bn = (l/1/2tr)an. In doing so, we shall obtain

both the general solution of the problem of the prediction of a stationary sequence and a formula for calculating the mean-square

error of the prediction. The function g(z) e Hz, g(O) = ao/V 27r is real (cf. Remark 1). With the aid of this function, the spectral density of the sequence r'(t), f,(w) = I 12 can be factored. On the basis of Remark 3 of Section 6, if g(z) has no zeros in the disk I z I < 1, it can be determined uniquely from f,(w). Therefore if the function g(z), constructed in accordance with Theorem 1, does not vanish for I z I < 1, it is identical to the function g(z) obtained in the proof of Theorem 1 of Section 6. Lemma 3.. For I z < 1, we have g(z) # 0. Proof.

We note first of all that if fe(w) _ I

12, where

LINEAR TRANSFORMATION OF RANDOM PROCESSES

248

h(z) _

c 12 < -, then 82(1) > 27r I co 12.

This is true because

N

M I7(0) - k=1 E N C1 - E dke-ik')(

l-a

Cke-ik.

dco > 27r co 2 .

k=0

k=1

Since this is true for arbitrary dk and N, we have

(9)

32(1)>27rIcolE.

Let us suppose now that g(zo) = 0 for I zo I < 1. The function p

gJz) =

E a"z"

Therefore g1(z) = (z - zo)=o b z", where

vanishes at the point zo. Then,

bo = -a0/(V27c 20). g(e;.,)

1

1/27C

= n=O

1/27c

a"e-u"0,

1 - e-i°zo

e-°' - 20 = I o brie-a". =

e -2lU I

- z0

o

b", e

".,

n=0

brie:".

where bo' = bo = -ao/(1/27r'2,). It follows from (9) that 31(1) > 21rIb0'12 =

a

°

za

which by virtue of (6) is impossible for I zo I < 1. This completes the proof of the lemma. Corollary. In formula (7) for the best prediction, the function g(z) e H2 is uniquely determined (under the hypothesis that g(0) is positive)

and it coincides with the function obtained in Theorem 1 of Section 6.

We have solved the problem of prediction for the regular part Now we need to clarify the questions: How can we express the spectral density of a sequence rJ(t) in terms of the spectral function of the process e(t)? What is the form of the formula for prediction for the sequence E(t) expressed in terms of the characteristic function of $(t)? of an undetermined sequence.

Lemma 4. Suppose that an undetermined process e(t) is represented in the form e(t) = 7)(t) + e,(t), where ra(t) and e,(t) are uncorrelated, $,(t) is a singular process, i7(t) is regular, and F(w), F,.((O), and F,((O) are the spectral functions of the sequences $(t),r)(t), and gs(t). Then

8.

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

249

the equation

F(w) = F,(w) + Fs((o)

(10)

is the decomposition of the function F(co) into an absolutely continuous component F,(co) and a singular component FF(oi) with respect to Lebesgue measure.

Proof. Formula (10) follows from the uncorrelatedness of the sequences ra(t) and es(t). We introduce the spectral represen-

tation of the uncorrelated sequence c(t) that appears in the representation (3): tC(dw) ,

(11) W) = f I r the stochastic measure with structural function rz

where C(A) is (1/2ic)l(A) (I denotes Lebesgue measure). we obtain

Substituting (11) into (3),

72(t) = Jrz e'1w1/27cg(e'w)C(dcn) J

rz

Suppose that

es(t) = J1 z

(12)

rz

is the spectral representation of the sequence Es(t). Then e(t) = 5ett0e(dw) =

g(e.w)C(dw)

+ ei (dw)) .

(13)

+ ps(dw))

(14)

It follows from equation (13) that 7rg(eaw)C(dw)

Jrz p(w)p(dw) = Jrz P(w)(1 rz

rz

for an arbitrary function p (w) e LZ{F}.

We can write yet another spectral representation for es(t). Since es(0) e He, we have

s(0) _ rz ps(w)p(dw) rz

so that

e'.tgs((o)II(dco)

es(t) = Stss(0) = Jrz

.

rz

Remembering (13) and (14), we obtain

s(t) =

Jrz

ecwt1ps((0)[1/27cg(e'IC(dw) + us(dw)] rz

Substituting the expression for es(t) given by equation (12) into this equation and transposing the second term in the above integral to

LINEAR TRANSFORMATION OF RANDOM PROCESSES

250

the left, we obtain e;wt(1

-

rz

The two sides of this equation contain elements of subspaces that

are orthogonal to each other.

Therefore they are both zero.

Consequently,

.ps(w) = 1 rps(w)g(e"w) = 0

(mod Fs) , (mod 1)

.

Since g(e°l) can be equal to zero only on a set of Lebesgue measure 0, it follows that Ts(w) is equal to 0 almost everywhere. Let S denote the set on which ps(w) = 1. Then 1(S) = 0. Thus FS(A)

I

= JA

FF(A) =

cps(w) 12 dF(w) = F(A (1 S) , g(e;.,) I2 dw .

27c 1 A

This completes the proof of the lemma. Lemma 5. three integrals

Suppose that g1(co), q2(w), and q)3(w) are such that the

q,1(w)ft(dw) J

,

s

Jrz q 2(w)v(dw) ,

JRn q,3(w)l(dw)

rz

are the projections of the quantities e(t+q), ri(t + q), and s(t + q) onto HE(t), H,(t), and Hes(t) respectively. T1(w) = q)2((O) = TAW) = e.t((1

Proof.

Then

- g(e:l)

(mod F) .

I

In view of formula (7), it will be /sufficient to prove

that PI(w) = 'P2((0) = 'P3(0). It follows from the equation S(t + q) - Jrz g1(w),u(dw) =

[(t + q) - Jrz g1(w)v(dw)] + [e2(t + q) -

gi(w),us(dw)] (15)

and the orthogonality of the terms in the bracketed expressions on the right that 2

81(q) = M rI(t + q) -

g1(w)v(d(o) 2

+M

I es(t

+ q) -

J

rz

with equality possible only when rpl(w) = p2((o)

(mod Fr) ,

8.

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

es(t + q) = J rz q 1((o)ps(dw) ,

q(w) _ q (w)

rz

(mod Fs)

251

.

On the other hand, 32(q) = 32(q) by virtue of the definition of e,s(t). This completes the proof of the lemma. The results obtained can be formulated as:

Theorem 4. Let e(t) denote an undetermined stationary sequence. Then the optimal prediction e(°)(t) of the quantity (t + q) from the results of observation of e(r) for v < t is given by the formula e(9)(t)

rz

-

1-

eitro

09(eiw)

f"(dw)

g(ei.,)

S-rz

where p is the spectral stochastic measure of the sequence e(t), 4-1

g(z) _

bnZn n=0

g4(z) =

,

n=0

bnZn

the function g(z) e H2 has no zeros in the disk I z I < 1, g(O) is positive, and I g(eil) 12 = f(c)), where f((o) is the derivative of the absolutely continuous component of the spectral function of the sequence e(t). The square of the mean-square error of the prediction is equal to 32(q) = 27c exp

tat

rz

In f((o)dw) E I C.

12

where the cn are determined from the equation exp {_i_

Inf(w)dll = 1

rz

2ir n=1

Zn -rz

n=0

In particular, rz

32(1) = 27r exp (

\ 27r

In f(co)dco) -rz

The theorem follows immediately from Lemmas 4 and 5 and formula (7) of the present section and from Theorem 1 and Remark 2 of Section 6. 2.

Prediction of stationary processes with continuous time

Let e(t) (for - oo < t < rco) denote a stationary process e(t) =

ei.t jt(dw)

,

where p is an orthogonal stochastic measure on the real line

(-- < (v< o.) and

LINEAR TRANSFORMATION OF RANDOM PROCESSES

252

R(t) = M(e(t + u) (u)) =

Me(t) = 0 ,

e;e,dF(w)

F(+oo)= a'. We introduce the Hilbert space HF = H{e(t), - 00 < t < oa } and < z < t}. In HF we define its subspaces H2(t) = H{e(t), the group of operators representing time displacement Sr (for - ca < v < ca) by setting ST1

Cke(tk))

= E Ck6(tk + v)

and extending the definition of Sr as a continuous operator to the entire space H2. The Sr constitute a group of unitary transformations of H6. This group has the same properties, with obvious modifi-

cations, as the group of transformations S% in the case of discrete time. The problem of optimal linear prediction for a process e(t) consists in finding a random variable eT(t) e HF(t) such that 12

for an arbitrary element 7) of HF(t). This problem has a unique solution: the variable eT(t) is the projection of (t + T) onto HF(t). We set

The quantity 8(T), the mean-square error of the prediction, is a nonincreasing function of T and 0 < $(T) < a. If lim 8(T) = a, T-,the process is said to be regular (completely undetermined). If 3(T0) = 0 for some To, then HF(t) C HH(t - To) for arbitrary t. Consequently,

He(t) C n Hf(t - kTo) k=1

for arbitrary t and 8(T) = 0 for all T > 0. In this case the process is said to be singular (determined). We shall call nonsingular processes undetermined processes.

The proof of Theorem 1 can be carried over directly to processes

with continuous time: an arbitrary stationary process admits a decomposition of the form e(t) = YI(t) + eS(t)

where ri(t) is a regular and es(t) is a singular process and the two are uncorrelated and subordinate to e(t). The analogue of Theorem 2 is: Theorem S. A regular stationary process r1(t) can be represented

8.

GENERAL THEOREMS ON THE PREDICTION OF STATIONARY PROCESSES

253

in the form

a(t

ra(t)

(16)

where C(v) is an integrated white noise, Hi(t) = Hi(t), and Ia(t)12 dt <

Here Hi(t) denotes the closure in L2 of the set of random variables of the form E ck[oyS(tk)

-

yy b(tk-)],

tk < t.

Proof of this theorem can be obtained from the preceding results for stationary sequences by means of a passage to the limit applied to the sequence )7,(k) = r)(k/n), k = 0, ± 1, ±2, . The results obtained for the prediction of stationary sequences can now be carried over to processes with continuous time with certain modifications in the wording and the proofs. Here, we need to use the spectral representation of stationary processes with continuous time and to refer to the results of Lemma 2 and Theorem 3 of Section 6. Thus we obtain the following theorem and formulas: Theorem 6. For a process e(t) to be undetermined, it is necessary and sufficient that In f(w)d(o _c I + 0)2

>

where f(w) is the derivative of the absolutely continuous component of the spectral measure F of the process e(t).

If E(t) =)7(t) + es(t) is the decomposition of the process e(t) into regular and singular components and if in accordance with Theorem 5, 72(t) =

a(z)<(t J0

- z)

then eT(t) _ Ta(z)dC(t - z) + es(t + T) T

and the optimum mean-square error of the prediction is determined from the relation 32(T)

- T a(z) I2 dz 0

Another expression for the optimum prediction is eT(t) _

r- hT(icv) e"[1

h(i(o) Ju(dw)

254

LINEAR TRANSFORMATION OF RANDOM PROCESSES

where pt is the stochastic spectral measure of the process e(t), h(ic)) =

1

1/27r

-a(r)e-i-d-c o

and

hT(ic)) =

1

1/270

T

a(r)e-°-dr

0

The function h(ic)) is determined from the spectral density of f(c)) in accordance with formula (22) of Section 6.

VI PROCESSES WITH INDEPENDENT INCREMENTS

A process e(t) defined on Z into R(- is called a process with independent increments if for all t0 < t1 < t, in Z, the quantities , $(tk) - E(tk_1) are independent. The simplest familiar examples of random processes with independent increments are Wiener and Poisson processes. Every stochastically continuous process with independent increments can be represented as a sum of independent Poisson and Wiener processes. (In the general case, e(to), e(t1) - e(to),

this sum contains a continuum of Poisson "addends," so that we should speak of the sum of a Wiener process and an integral of Poisson processes.)

Such a representation of the process is the basic

topic of the present chapter, and by means of it we shall study certain important properties of the sample functions. 1.

MEASURES CONSTRUCTED FROM THE JUMPS OF A PROCESS

Let E(t) denote a separable stochastically continuous process with independent increments that is defined for t e [0, T] with range in some finite-dimensional Euclidean space X. On the basis of Corollary 2, Section 4, Chapter IV, e(t) fails with probability 1 to have discontinuities of the second kind. For every s > 0, there will be with probability 1 only finitely many points t for which I (t + 0) - e(t - 0) 1 > s. Let X, denote the set of all x for which I x I > s and let 0, denote the a-algebra of Borel sets contained in X.. It follows that for every A e 0E, the number of points t in [0, T] for which e(t + 0) - e(t - 0) e A is finite with probability 1. Let v(t, A) denote the number of points s e [0, t) for which E(s + 0) - e(s - 0) e A. The process v(t, A) has independent increments, since v(t2, A) - v(t1, A), where t, < t2, is completely determined by the increments c(s) - e(t1) for s e [t1, t2]. Hence the increments of v(t, A) on the disjoint intervals are expressed 255

256

PROCESSES WITH INDEPENDENT INCREMENTS

in terms of increments of e(t) on disjoint intervals. Furthermore v(t, A) is a stochastically continuous process. (If v(t', A) - v(t, A) does not approach 0 in probability as t' - t approaches 0, we have P{I c(t') - e(t) I > e} -i-> 0, and this is impossible by virtue of the stochastic continuity of e(t).) Theorem 1. For every A e 0. and every r > 0, we have Mv(t, A)r < 0 . Proof. Suppose that 0 = ton < t1,, < ... < tnn = t and lim max (tk+l, n n-.o

k

-

tkn) = 0

If *1(t) = 1 for t > 1 and 11r1(t) = 0 for t < 1, the monotonicity of v(t, A) with respect to t and the fact that all the jumps of v(t, A) are equal to 1 imply that with probability 1, v(t, A) = lim

n-m k=0

Y 1ly(tk+1, n,

A) - v(tkn, A)).

Let us set r?nk = 1rl(v(tkn, A) - v(tk_1, n, A)).

I k=1 1 yy7.k > 1} = 1J

I 17.j

kP- 1

1

i k=1

{j=1 F-/nj

= 0, j=1 E

Then

=1}p{ y-k+1

-1

PI k=1 '/nk >

k

1 - 11 Pl k=1

0, Lj=1'/nj = 11

j=1

P11"=1 17nj>0}PlE17-i> j=1

ijnj>1-1}

1- 1}

Therefore

P{I1 /nk > 1} < (PI "

'/nj > 0})

1+1

Taking the limit as n --> o, we see that P{v(t, A) > 1} 5 (P{v(t, A) > 0})1+1 .

Observe that .p{v(t, A) > 0} < 1. would obtain

(1 )

If this were not true, then we

P{v(t, A) = 0} = jj p{v(tk, A) - v(tk_1, A) = 0} = 0 k=1

for 0= t0 < t1 < ... to = t and hence there would exist t' and t" arbitrarily close to each other such that p{v(t", A) - v(t', A) > 1} = 1,

which would contradict the stochastic continuity of v(t, A). existence of Mv(t, A)r for r > 0 follows from the inequality Mv(t, A)r < E p{v(t, A) > k}(k + 1)r k=0

E (P{v(t, A) > 0})k(k + 1)r < 0 . k=0

The

1.

MEASURES CONSTRUCTED FROM THE JUMPS OF A PROCESS

(The series Ek-o (k + 1)'xk converges for arbitrary I x I < 1.) completes the proof of the theorem. Corollary. Define II(t, A) = Mv(t, A). 11(t, A) is, for fixed t, a measure on 58E.

257

This

Then the set function

Proof. If A = U 7-1 Ak where the Ak are pairwise disjoint, then v(t, A) = E7=1 v(t, Ak), and consequently Mv(t, A) = Ek-, Mv(t, Ak),

in view of the fact that 0 < Y,k=1 v(t, Ak) < v(t, A). To study the properties of the quantity v(t, A), we find it useful to consider the process (t, A) defined by e(t, A) = E [e(S + 0) - e(S - 0)]XA(e(S + 0) - e(S - 0)) , s
(2)

where XA(x) is the characteristic function of the set A. In other words (t, A) is the sum of the jumps (of the process e(t)) that occur up to the instant t and that fall in the set A. If A e iE, then the number of such jumps is with probability 1 finite, so that (t, A) is meaningful. The stochastic continuity of v(t, A) implies the stochastic continuity of (t, A). Furthermore e(t, A) is a process with independent increments. The most important property of processes of the type g(t, A), namely the independence of the processes e(t, A), , (t, Ak) for pairwise disjoint sets A1, A2, , A, follows from:

Theorem 2. For every A e 0E, the processes (t, A) and g(t) - g(t, A) are independent processes with independent increments.

Proof. That e(t) - g(t, A) is a stochastically continuous process follows from the stochastic continuity of the processes e(t) and (t, A).

The considerations that we took into account in connection with v(t, A) enable us to assert that the process [e(t, A); g(t) - (t, A)] in X x X is also a process with independent increments. To prove the independence of the processes (t, A) and e(t) - $(t, A), it will therefore be sufficient to show that Meti(z1,£(t,A)-f(s,A))}i(z2,1(t)-£(t,A)-£(s)-I-£(a,A))

= (3) for z1 and z2 in X and s < t. Specifically, it follows from (3) and the independence of the increments of the process A); e(t) - e(t, A)]

that for arbitrary 0 < t, < ... < t = T and zij), z2k) e X we have

PROCESSES WITH INDEPENDENT INCREMENTS

258

M exp I l k , L(Zik), e(tk, A) + (Z2k), e(tk)

A))

- e(tk, A) -

e(tk-1) + e(tk-1, A))]

= M exp i E (Zlk), e(tk, A) k=1 x M exp

1

i

A))

(Z2k), e(tk) - e(tk, A) - e(tk-) + e(tk-1, A))

and and this equation means that the processes e(t, A) and e(t) - g(t, A) are independent. Let us prove equation (3) first for the case

H(t, PA) = 0 ,

where PA is the boundary of the set A. We note that in this case the process e(t) fails with probability 1 to have jumps on PA. Suppose that s = t, 0 < tnl < ... < tnn = t and lim max (t, , k+1 - tnk)- = 0 n-- k

Then if XA(x) is the characteristic function of the set A, the relationship

e(t, A) - e(s, A) llm

i X A(e(tnk) n-. o k=1

holds with probability 1.

k-1)J

(4)

We set E

EE

yS

- (tn,

nk = XA(e(tnk) - S(tn, k-1))Le(tnk) - W., k-1)]

I

/ nk = e(tnk) - W.' k-0 - enk

To prove (3) it will be sufficient, in view of equation (4), to show that n

n

n

n

i E (zi,fnk) i E (zi,fnk)+i E (z2,7nk) i E (z2,7nk) k=1 _ Me k=1 Me k=1 lim Me k1

= 0.

(5)

Using the independence of the pairs (ink, v .J and the inequality n

H ak k=1

ak - bk

bk k=1

k=1

for I ak I < 1 and I bk I < 1, we see that E (zl, £nk)+ E (z2,7nk) 1 i E (z1,fnk) Meik=1 E (zp,7nk) k=1 / - Me k=1 Meitk=1

=

7"1

Mei(z1,fnk)Mei(z2,r?nk) k=1

Ikk=1

Mei(z1,enk)+i(z2,'7nk) k=1

-

Mei(z1,enk)Mei(z2,7nk)

1.

MEASURES CONSTRUCTED FROM THE JUMPS OF A PROCESS

259

Since (z1, enk)(Z2, )7 k) = 0 (at least one of these must be zero), it follows that e'(z1,£nk)+'(z2,'Ink) = 1 +

=1

=1+ =

(t(Z1, enk) + I(z2, ./n) k m!

+

Enk))m m-1 E(i(z',M!

ei(z1,£nk) +

ei(z21n7 k)

(t(Z2, 7)nk))m

mi

-1

J

.

Therefore,

I Mei(zl,£nk)+"2''7nk) - Mei( z1,£nk)Mei(z2.rlnk) I Mei(z1.£nk)

=I

-1

I

I Mei(x2''ink) - 1 1.

But Mei(.1.£nk)

-1

I < M I ei(z1.£nk) - 1 I < 2P{enk > 0}

= 2P{XA(e(tnk) - e(tn, k-1)) > 0}

and for every p > 0, Mei(Z2,',n)

-1

p7 < Sup I 1 - ei(z2'x) I+ 2P{I ? k I > p}

x1--P

Consequently,

/

hm IMey (/

`zl. E

kz1

/

/

\

£nk\ I+il z2. E ]-kJ k=1

- M

eil z1.

\

k-1

£nkJ

/

\

E 'Ink) M eil z21k1

< lim (sup I ei(z21x) - 1 I + 2 sup P{I /nk I > p}) n-- xISP k yy x llm 2 E P{XA(e(tnk) - S E(tn, k-1)) > 0}

n-.oo

k=1

= 2 lim (sup I ei(z2.x) - 1 I+ 2 sup P{I Ix15P

k

I> p})

/

x lira ME XA(E(tnk) - E(tn, k-1)) n-oo k=1

Since P{I )7nk I > p} < P{Ie(tnk) - W(tn, k-1) I > p}

for p < s (where

A e 0E), the uniform stochastic continuity of e(t) implies that

>p}=0. Let us show that _llm n-.oo

M k=1 E X A(c(tnk) - E(tn, k-1)) < C <

(6)

Just as in the proof of Theorem 1, we can show that PIE XA((tnk) - S(tn, k-1)) > 1} k=1 n g

g

y (PIE XA(S (tnk) - S (tn, k-1)) > 0})'

k=1

(7)

PROCESSES WITH INDEPENDENT INCREMENTS

260

Furthermore it is easy to see that in the present case, lim

- k=1 XA(e(tnk) - Vtn, k-1)) = v(t, A) - v((s, A) with probability 1. H n--,m

Therefore

P1k%=1 I XA(e(tnk)

- (tn, k-1)) > 0 < 1 I

(8)

since P(v(t, A) - v(s, A) > 0} < 1. (This inequality was established in the proof of Theorem 1.) From inequalities (7) and (8) we get (6).

Thus, til zl, E £n,k)td z21 E pyyk)

lim Me \

\

k=1

k=1

k=oe

eti(az,:)

2C sup

-1

-

Me41 z1 kEl f,yk 1Me21

(9)

.

LISP

Taking the limit as p --+ 0, we obtain (5), which completes the proof

of the theorem for the case H(t, PA) = 0. Turning to the general case, let us note as a preliminary that the collection 21 of sets for which the theorem is valid constitutes a monotonic class (cf. Section 1, Chapter II), since for an arbitrary sequence of sets A. e ,, the relations E(t, U An) = llm n-,oo It

(t, k=1 -4k)

E(t, fl An) = llm E(t, n

n-+ao

Ak) k=1

are valid with probability 1, and the operation of passing to the limit does not destroy the independence of the random variables. One

can easily show that in the case in which a is a positive number such that II(t, PX) =0, the sets A in0, for which II(t, PA) = 0 constitute an algebra of sets. But every monotonic algebra is a aalgebra (cf. Theorem 3, Section 1, Chapter II), so that s?t is a aalgebra. Finally we note that at each point there are spheres of arbitrarily small radius that belong to fit. (Since the boundaries of the spheres SP(x) with the same center x but different radii p have no common points, the inequality II(t, PSP( )) > 0 holds for only a countable set of values of p.) Therefore I contains all open sets belonging to 8,, so that A contains ,. This completes the proof of the theorem. , Ak are pairwise disjoint sets belongCorollary 1. I f All A2, , E(t, Ak) ing to, for some s > 0, then the processes e(t, A), g(t, A2), and E(t) - Ej=1 e(t, A1) are mutually independent.

1.

MEASURES CONSTRUCTED FROM THE JUMPS OF A PROCESS

261

c(t, U j=1 A;) is Proof. The process e(t) - Ej=1 e(t, A;) independent of the process g(t, U j=1 A;). The processes c(t, A;) are completely determined by the process g(t, U;=1A;). Therefore they are all independent of the process e(t) - Ej=1 e(t, A;). Analogously,

the collection of processes (t, A;), for j z i, are completely determined by the process e(t) - e(t, Ai), which is independent of e(t, Ai),

so that this collection is also independent of the process e(t, Ai). Thus each of the processes

A,), j = 1, 2, ..., k; e(t)

k

j=1

A5)

is independent of all the others. Our assertion now follows. Corollary 2. For pairwise disjoint sets A1, A2, , A, in 1BE, the processes v(t, A),

, v(t, Ak) are mutually independent.

This follows from the preceding assertion since the process v(t, A) is completely determined by the process e(t, A). Let us now study the distribution of the quantity v(t, A).

Theorem 3.

The process v(t, A) is a Poisson process, that is, a

process with independent increments with Poisson distributions: For

s
P{v(t, A) - v(s, A) = 1} _ [II(t, A) - II(s, A)1 exp {- [II(t, A) - II(s, A)]} 1!

Proof.

Suppose that s = tno < t,1 <

< tnn = t and

lim max (tnk - t,, k-1) = 0 n-.m k

Then n-1

P{v(t, A) - v(s, A) = l} = lim E P{v(tn;, A) - v(s, A) = 0} n-

i=0

x P{v(tn, i+1, A) - v(tn;, A) = 1 }P{v(t, A) - v(tn, i+1, A) = 1 - 1 }

We note that P{v(tn,:+1, A) - v(tn;, A) = l} i=1

i=1

M[v(tn, i+1, A) - v(tni, A)] = M[v(t, A) - v(s, A)] < o

Furthermore it follows from the stochastic continuity of the process v(t, A) that lim max p{v(tn, i+1, A) - v(tni, A) > 1 } = 0 .

It follows from inequality (1) that

PROCESSES WITH INDEPENDENT INCREMENTS

262

- v(tai, A) > 1} (P{v(tn,A) - v(tni, A) > 0})E

P{v(tn,i+1, A)

Therefore

L (p{v(tn, ill, A) - v(tni, A) > 0} - P{v(tn, ill, A) - v(tni, A) = 1}) i=O

_

n-1

i=1

P{v(tn, i+1, A) - v(tni, A) > 1 }

< max p{v(tn, ill, A) - v(tni, A) > 0} i

n-1

x L p{v(tn, i+1, A) - L(tni, A) > 0} a=0

and the last expression approaches 0 as n

00 .

This means that

p{v(t, A) - v(s, A) = 1}

= lim

n-1

P{v(tni, A) - v(s, A) = 0}p{v(tn, i+1, A) - v(tni, A) > 0}

x P{v(t, A) - v(tn,i+1, A) = 1 - l} n-1

= lim Z P{v(t, A) - v(tn, ill, A) = 1 - 1 } n-- i=0

x [P{v(tn, i+1, A) - v(s, A) > 0} - P{v(t i, A) - v(s, A) > 0}]

Since p{v(t, A) - v(v, A) = 1 - 11 is a continuous function of r, by virtue of the stochastic continuity of v(r, A), it follows that p{v(t, A) - v(s, A) = 1}

_ P{v(t, A) - v(r, A) = 1 - 1}drP{v(r, A) - v(s, A) > 0} .

(10)

For r1 < v, we have p{v(r, A) = 0}

= P{v(r1, A) = 0}P{v(r, A) - v(r1, A) = 0} .

(11)

As shown in the proof of Theorem 1, p{v(r, A) = 0} > 0, and we may set p{v(r, A) = 0} = e'l(r). It follows from (11) that ez(r)-2(rl) . P{v(r, A) - (rl, A) = 0} = The function X(v) is nondecreasing and continuous, since the process v(v, A) is stochastically continuous. Thus,

p{v(t, A) - v(s, A) = 1} = te2(t)-2(r)d(1 - ea(r)-1(8)) e

= ea(t)-a(s)

t

e-a(r)d(- eacr>)

= eact)-2(8) Jt(- dX(z-)) = [X (s) -

a(t)]eace-aes

1.

MEASURES CONSTRUCTED FROM THE JUMPS OF A PROCESS

263

Substituting the value obtained for p{v(t, A) - v(s, A) = 1} into formula (10), we obtain P{v(t, A) - v(s, A) = 2} -

X(t)]2

[X(S)

2

By induction, it is easy to verify that for all 1 > 0, P{v(t, A) - v(s, A) = l} - [7(s)

l

M(t)]l

(12)

Thus v(t, A) is a Poisson process whose finite-dimensional distributions are determined by formula (12) and the relation v(0, A) = 0. In particular, Mv(t, A) = -X(t). On the other hand, Mv(t, A) = II(t, A) by definition. If we replace X (s) - %(t) with the expression II(t, A) - II(s, A) in equation (12), the proof is complete. REMARK 1. The characteristic function of the process v(t, A) is given by the formula Me+zvca,A> = exp {(ei2 - 1)II(t, A)} . (13) REMARK 2. Let us consider the set [0, T) x X,. If B* is a Borel set belonging to this Cartesian product, we can consider the random variable v*(B*) that is equal to the number of points t for which the pair

(t; c(t + 0) - s(t - 0)) belongs to B*. One can easily show that v* is a random measure on the v-algebra 0e of all Borel subsets of [0, T] x X. Let us set

II*(B*) = My*(B*) .

The quantity H*(B*) is a finite measure W.. The connection between the measures v(t, A) and the measure v* is obvious: v*([t1, t2] x A) = v(t2, A) - v(t1, A) .

(14)

Analogously,

II*([tl, t2] x A) = II(t2, A) - II(tl, A) . (15) Theorem 4. The measure v* is a Poisson random measure with The characteristic function of the quantity v*(B*) is given by the formula independent values.

Mei2v*(B*)

= exp {eiz - 1)II*(B*)} .

(16)

Proof. Let ?So denote the algebra of sets that is generated by sets of the form [tl, t2] x A, where [t t2] c [0, T] and A e 0E . If

PROCESSES WITH INDEPENDENT INCREMENTS

264 A*,

, At are disjoint sets belonging to Ko, there exist disjoint tai)] x Ai for i = 1, , N such that the At are the

sets A*

sums At. Here the sets i

can be chosen so that for distinct i both the intervals [tii), tai)] and the sets A; are either disjoint or coincident.

In this case independence of the v*(A) is a consequence of the independence of the v(t, Ai) for distinct A; and of the independence of the increments of v(t, Ai). The independence of the v*(i ) implies independence of the v*(A; ). These variables have a Poisson distribution since they are the sums of independent Poisson variables. To complete the proof, we need only note that a(s?t0) = 0,*. 2.

CONTINUOUS COMPONENTS OF A PROCESS WITH INDEPENDENT INCREMENTS

Let c(t) denote a separable stochastically continuous random process with independent increments and range X. Take x1, xE, , x,

to be an orthonormal basis for X. Denote by F(t) the process obtained from e(t) after discarding the jumps exceeding s in absolute value: c0(t) = e(t) - (t, XE), where X. is the set of all x such that I x I > a. The process c0(t) is stochastically continuous, has inde-

pendent increments, and possesses no jumps exceeding s. It is natural to expect that e0(t) converges to some continuous process with independent increments as a -+ 0. This turns out to be the case if we subtract from E,(t) specially chosen continuous nonrandom functions. To prove this fact, we need: Lemma. For every s > 0, M e0(t)12 < oo. Proof. Suppose that 0 = tno < ... < tnn = t and llm max (tnk - tn, k-1) = 0 . n-+m It

Let us set nk = '1A E(tnk) where *,(x) = x for I x < a and *a(x) = 0 for I x I > a. It is easy to see that with probability 1

E(t) = lim E nk n- oo k=1

(1 )

We note that no term in this sum exceeds 2a in absolute value. If the sum Ek=, D(enk, xi) were unbounded for some i, it would be possible to choose a sequence of values of n such that k=1

D(Enk, xi)

In such a case, the quantities

2.

265

CONTINUOUS COMPONENTS OF A PROCESS

yy77

,ink

(enk - Monk, xi)

=_

xi)

would satisfy the conditions of the limit Theorem 5, Section 6, Chapter I since P

=o, M"!nk=0, Y, DY7nk= 1 y7

72.k I

.

k=0

4E1 DS .k

Thus the sum Ek=1 rink would have a normal limiting distribution,

so that for arbitrary a, ll

+(Monk, xi)} lim (e-k, xi) > a JLk=1D(Enk, xi) + L n-- P{Li k=1 k=1 _

1

e_n2/2du

V 27

,

n

lim P{E (enk, xti) < -aY Ek=1D(enk, xi) + k -

L (MSFnk, xd)} k=1

e-n212du

1

2-7r

--

But these last relations contradict the boundedness nk=1 nk, xti), which follows from relation (1). Thus Ek=1 x;) is bounded for all i. We also note that by Chebyshev's inequality, n

E

L<

nn+

(S nk, xti) - L M(enk, xti) k=1 k=1

DE k=1

xi) L2

From this and the boundedness in probability of the quantity k=1

xti) ,

it follows that M Zk-1 (enk, xi) is bounded. n

M Z enk k=1

r In

2

M I $.(t) 12 < limM

ll2

= M i Z enk, xi) i=1

i=1

the quantity m

Since

k=1

LDk=1 L (enk, xti) + (M k=1 (enk, xi) 12 J ,

k=1 enk 12

is also bounded, as is M I eE(t) I2, since

k=1 nk I2

This completes the proof of the

lemma.

Let {sn} denote a sequence that decreases to 0. We let Ok denote the set of all x such that sk < I x 1 < sk-1 for k = 2, 3. ,

PROCESSES WITH INDEPENDENT INCREMENTS

266

and we let 01 denote the set of all x such that I x I > sl. that

We note

m

E e(t, Ak) + k=2 The terms on the right are independent by virtue of Corollary 1 to Theorem 2 of Section 1. Therefore, for arbitrary x, ,

k=2

D(e(t, Ok), X) < D(eE1(t), x)

,

and hence for arbitrary x the series Ek=2 D(e(t, Ok), X) converges. Let us choose a sequence {nk} (where n1 = 2) such that

Z D(e(T, 0;), x;) < .7=nk

1k6 for i= 1, 2, ' , r .

Then the sequence nk

(2)

[e(t, 0;) - Me(t, o;)] will, with probability 1, converge uniformly to some limit as k

oo.

To see this, note that

p sup

05t5T

nk+1

nk

E [e(t,4) - Me(t, 0;)] - Zj=2[e(t, 0;) - Me(t, 0;)] >k2} ;=nk+1 nk+1

pjsupT {=1

E (e(t, A j) - Mee(t, 0;), xa)

025t 5

nk+1

lim p{ sup

{=1 m-.o

=

15mT

1

}1(e(m,

Me((1

m'

/

x)

-

1

}

k2/-r

nk+1

lim rk°M t=1 m.-.o

E (W(T, A;) - Me(T, 0;), xi)

=nk+1

(Here we used Kolmogorov's inequality: Theorem 1, Section 4, Chapter III.) Since the series Ek=, r2/k2 converges, it follows from the BorelCantelli theorem (Theorem 2, Section 3, Chapter III) that the terms of the series nk+1

E (e(t, 0;) - Me(t, 0;)) E k=1,7=nk+1 .

are eventually majorized, with probability 1, by the terms of the convergent series Ek 11/k2. Hence, the sequence (2) converges uniformly with probability 1. Therefore, there exists a process e0(t) that is the uniform limit of the sequence nk+

eE1(t) -

L [e(t, 0;) - Me(t, 0;)] 3=2

2.

CONTINUOUS COMPONENTS OF A PROCESS

267

Since E(t, A,) is a stochastically continuous process and sup M I e (t, Oj) 12 < 05LST

it follows from Theorem 6, Section 5, Chapter II, that lim Me(t, Oj) = Me(s, A;) E-+8

Consequently, the process e,,(t) - J;k2 [e(t, A;) - Mf(t, p;)] fails with probability 1 to have jumps exceeding e,, in absolute value, and the process e0(t) (the uniform limit of such processes) is continuous with probability 1. We note that in accordance with Kolmogorov's theorem (Theorem 4, Section 4, Chapter III), the series j=2

[e(t, Oj) - Me(t, Oj)]

converges by virtue of the convergence of the series,

D(e(t, Oj), x)

for every x. For every t, the sum of the series ET, [e(t, 0 j) - Me(t, /.j)]

coincides (mod P) with the sum of the series

i

k+1

k=1j-nk+1

[e(t, Aj) - Me(t, Oj)]

Thus we have Theorem 1. For every separable stochastically continuous process with independent increments, there exists a continuous process E0(t) such that E(t) = e0(t) + e(t, 01) +j=2 E [E(t, Oj) - Meet, 0;)] REMARK 1.

The process e((t), being the limit of the sequence

of processes 9=2

is independent of each of the processes e(t, A,), where j = 1, 2,

.

, m.

Since

e0(t) + E (e(t, A,) - MSF(t, A,)) = es1(t) j=2

and m I eE1(t) I2 <

and since the terms on the left are independ-

ent, it follows that M I e0(t) I2 < - . The following theorem completely characterizes the process e0(t).

Theorem 2. The process E0(t) has independent Gaussian increments; that is, for every z e X,

PROCESSES WITH INDEPENDENT INCREMENTS

268

M exp i(z, e0(t2) - e0(tl))

= exp i(z, M[e0(t2) - S0(t)]) - 2 D(z, c0(t2) - e0(t1))

(3)

Proof. go(t) is a process with independent increments since it is the limit of processes with independent increments. It follows from Theorem 3, Section 5, Chapter IV that if

tl - tn0 < tn1 < . . . < tnn. = t2 and limn_ maxk (tnk = 0, then for every p > 0, lim E P{I go(tnk) - g0(t,, k-1) I > p} = 0 n-.oo k=1

(4)

On the basis of formula (4) we may assert that there exists a sequence {pn} that converges to 0 as n oc and such that lim E P{I e0(tnk) - g0(tn, k-i) I > pn} = 0 n-.oo k=1

Set gnk = (z, kPn(g0(tnk) - g0(tn, k-i))), where *P(x) = 0 for I x I > p

and *P(x) = x for I x I < p. numbers I z I pn.

i

Pt

The quantities gnk are bounded by the

Since enk # (z, e0(t2) - e0(tl))1

k-1

+ (

L Plgnk z (z, g0(tnk) - g0(tn, k-1))} k=1

n

< Ek=1 P{I g0(tnk) - g0(tn, k-1) I > pn}

,

the sequence {Ek=1 gnk} converges in probability to the quantity g0(t2) - g0(t1). Let us suppose that D(z, go(t2) - g0(t)) > 0. Then n

lim E De%k > D(z, g0(t2) - g0(t1)) > 0 n-.oo k=1

Therefore Theorem 5, Section 3, Chapter I is applicable to the quantities gnk - Mgnk k=1

Dgnk

and hence,

lim M exp J i? I

Furthermore,

L1=1 gnk - k=1 L Me,k) L Dgnk k=1

= e-2

2

(5)

2.

CONTINUOUS COMPONENTS OF A PROCESS

lim M exp {i?. %--

e%k}

269

= M exp {iX(z, c0(tz) - eo(t1))}

k=1

From these two relations one easily shows that (z, co(t2) - eo(t1)) has a normal distribution. On the other hand if D(z, co(t2) - eo(t1)) = 0, then (z, eo(tz) - eo(t)) = M(z, eo(tz) - e0(t1)), and formula (3) is obviously valid. This completes the proof of the theorem. REMARK 1.

The expressions

M(z, e0(t) - e0(0))

are continuous functions of t. proved that P{1 (z, SM(tz))

and

D(z, eo(t) -bo(o))

It follows from the theorem just

- (z, eo(t1)) I > s} =

2

e-uz1zdu

1

,

U

(6)

where U = {u: I u + M(z, eo(tz)

- o(t1)) >

eo(tz) - ea(t1)))-1} .

If there were a t such that D(z, eo(t2) - e0(t1)) -%- 0 as t, - t and t2 - t,

the right-hand member of (6) would also not approach 0 and this would contradict the stochastic continuity of the process e0(t). For t1
D(z, o(t1) -X0(0)) + D(z, o(tz) - o(t1)) D(z, e0(tz) by virtue of the independence of the increments e0(t). Consequently D(z, eo(t) - e0(0)) is a continuous function. Furthermore, if lim I M(z, e0(tz) t1-.t

- eo(t1)) > b

t2-.t

it follows from (6) that lim P{I (z, e0(tz) - eo(t1)) I > E} >

I

.

tz-.t

This inequality ensures the continuity of M(z, e0(t) - e0(0)). REMARK 2. If we let a(t) denote the quantity M(e0(t) - e0(0)) and if we let A(t) denote a nonnegative symmetric linear operator

in X such that D(z, e0(t) - e0(O)) = (A(t)z, z) ,

then the distribution of the process e0(t) is determined by the characteristic function M exp {i(z, e0(t))}

=

Mei ,e0(O)) exp {i(a(t), z)

- 2 (A(t)z, z)}

(7)

PROCESSES WITH INDEPENDENT INCREMENTS

270 3.

REPRESENTATION OF STOCHASTICALLY CONTINUOUS PROCESSES WITH INDEPENDENT INCREMENTS

Let us consider stochastic integrals with respect to the measure v(t, A). As we mentioned in Section 1, the measure v(t, A) is a countably additive nonnegative function of the set A e 0E Suppose that a measurable function fi(x) is bounded on every compact subset

of the space X and is equal to 0 for I x I < s (where s is some positive number).

Then the integral T(x)v(t, dx) can be defined

in the usual way. This follows from the finiteness of the measure v(t, A) on 0. and also from the fact that v(t, X,o) = 0, where Xp is the set of all x such that I x I > p and

p=maxIe(s+0)-e(s-0)I 025sst Thus we are actually considering the integral only over the set Is < I x I < p} on which the function g(x) is bounded. that

Let us show

e(t, A) _ ` xv(t, dx) . J

(1 )

A

If A = U 7-1 Bk where the Bk are pairwise disjoint sets with diameters not exceeding 8 and if x, e Bk, then e(t, Bk) - xkv(t, Bk)

e(t, A) - E xkv(t, Bk)

k


(The process (t, Bk) represents the sum of the jumps v(t, Bk) belonging to Bk and therefore differing from Xk by no more than 8.) Equation (1) follows from this. Since

M (t, A) -

xkv(t, Bk) I

k

SMv(t, A)

M e(t, A) - E kxkv(t, Bk) 12 5 32M[v(t, A)j2

,

we have xII(t, dx)

Me(t, A) = lira M E xkv(t, Bk) = 8-0

JA

k

D(e(t, A), z) _

(z, x)211(t, dx) A

for every bounded set A lying at a positive distance from the zero point of the space X. Let us now consider the random set function

3.

REPRESENTATION OF STOCHASTICALLY CONTINUOUS PROCESSES

271

v(t, A) = v(t, A) - 11(t, A) which has the following properties: Mv(t, A) = 0 , 2) M(v(t, A)v(t, B)) = Mv(t, A fl B) = H(t, A fl B) (The last follows from the fact that v(t, ) is independent for disjoint sets and is Poisson distributed.) Equations (2) enable us to use the general construction of a stochastic integral with respect to an or-

f(x)v(t, dx) for all

thogonal measure to construct the integral measurable functions f(x) such that f(x)12II(t, dx) < 00

.

In the preceding section it was shown that D(c(t, &J, z) < 0 .

i

k=2

Since

D(E(t, Ok), z) =

(x, z#2II(t, dx) ek

it follows that for arbitrary z e X, (x, z)2II(t, dx) <

lim e-. 0

LIXI.,581

Consequently we have lim e-.0

.<x5e1

Therefore the integral

x 12II(t, dx) < 0 .

We note also that

xv(t, dx) exists. 0
xv(t, dx)

E (c(t,Ok) - Me(t, Ok)) k=2

and hence the series Ek=2 (M(t, AJ - Me(t, Ak)) converges in probxv(t, dx). Remembering that ability to 0
e(t, AJ =

xv(t, dx) , IxI>El

we obtain from Theorem I and Remark I of Section 2 the following result (for definiteness we set s, = 1). Theorem 1. Let e(t) denote a separable stochastic continuous process with independent increments. Then there exists an independent increment Gaussian process ea(t) such that ea(t) is independent of all the processes v(t, A) and has the representation e(t) = e0(t) +

xv(t, dx) + Ixl>i

xv(t, dx) . IxI

1

3)

PROCESSES WITH INDEPENDENT INCREMENTS

272

Let us find the characteristic function of the process e(t). Since the terms on the right side of (3) are mutually independent, we only need find the characteristic function of each term. The characteristic function of the process e0(t) is given by formula (3) or formula (7) of Section 2.

Let A denote a bounded set belonging to 0E. i z,Exkv(t, Bk)

Mei(s,e(t,A)) = Jim Me (

k

Then

(4)

)

2- 0

where A = U k-, Bk and the Bk are pairwise disjoint sets with diameters not exceeding X, and where each point xk belongs to Bk. The quantities v(t, Bk) are mutually independent, so that by using formula (13) of Section 1 we obtain i z,Exky(t,Bk)

Me (

k

n

n

) = TT Mei(z,xk)v(t,Bk) = JI exp kk=1 k=i

exp

= Taking the limit as ?

{(ei(a,xk)

- 1)H(t, Bk)J

{(eik) - 1)H(t, Bk)} 0 in this last equation, we see that

Mei(z,e(t,A)) = exp {5(ei' x) - 1)II(t, dx)}

( 5) .

Formula (5) is also valid for unbounded sets belonging to 0, since such sets can be represented as sums of increasing sequences of bounded sets A. for which (5) is valid and we can take the limit as n . -> -. It follows from (5) that M exp i(z, e(tt,A) - Me(t, A))

that is,

{5(e' x) - 1 = exp

ll

i(z, x))ll(t, dx) ( ,

(6)

M exp i(z,5A xv(t, dx)) \\

= exp

//

{5(ei(8,x) _ 1 - i(z, x))n(t, dx)}

(7)

By taking the limit with respect to A, we see that formula (7)

is

valid for all sets A for which the right-hand member of this equation

is meaningful. Now by taking formula (7) of Section 2 and formulas (5) and (7) of the present section, we can write the characteristic function of the process g(t): Mei($,e(t))

=

Mei(z,e(o)) exp {i(a(t), z)

+

(ei(z,x)

-- 2 (A(t)z, z)

- 1)II(t, dx)

5 Ixl>1

+

( J e
(ei(8,x)

- 1 - i(z, x))II(t, dx)}

(8)

3.

REPRESENTATION OF STOCHASTICALLY CONTINUOUS PROCESSES

273

With the aid of this formula we can determine the distribution e(t2) - c(t1) and hence all finite-dimensional distributions of the process e(t).

Since stochastically equivalent processes have identical finitedimensional distributions and since for every process there is a stochastically equivalent separable process (by Theorem 2, Section 2, Chapter IV), we have: Theorem 2. The characteristic function of a stochastically continuous process with independent increments is of the form (8), where a. the function II(t, A) is continuous, is nondecreasing with respect

to t for every A e U Z., and satisfies the condition E

lim e-.0

a
I x I2II(t, dx) < c ;

b. a(t) is a continuous function with range in X; c. the function A(t) is continuous and its range consists of nonnegative

symmetric linear operators in X; also, A(t2) - A(t1) is nonnegative for t1 < t2.

Sometimes instead of formula (8) we use the formula Me+(x.ec )) = Metc:'"° exp {i(ai(t), z)

- 2 (A(t)z,

Z)

12

i(z' x) )1 + I x G(t, dx)} 1 + xI2 IxI2

+

(9)

Here,

a1(t) = a(t) -

XIXI IxIs1 1 + I X 2

II(t, dx) +

x

IxI>1 1+ I x 2

II(t, A)

,

and the measure G(t, dx) is defined by the relation X 12 II(t, G(t, A) =

dx)

I

JA 1 +

Ix12

Formula (9) is convenient because the measure G(t, A) is finite and is defined on the a-algebra of all Borel subsets of the space X. A random process e(t) with independent increments is said to be homogeneous if it is defined for t > 0, $(0) = 0, and the distribu-

tion of the quantity (t + h) - e(t) depends only on h. Theorem 3. For a stochastically continuous homogeneous process e(t) with independent increments, there exist a vector a e X, a symmetric

linear nonnegative operator A, and a function II(A) that is for every positives a finite measure on 0. and satisfies the inequality lim1 e-.0 .1 .
IxI2II(dx)
274

PROCESSES WITH INDEPENDENT INCREMENTS

and such that the characteristic function of e(t) is given by the formula Me,c$,ect»

= exp 1t[i(z, a) - 2 (z, Az) +

ixi>l(e.c$'

J

+

r

- 1) II(dx)

(ei(',z) - 1 - i(z, x))II(dx)]} .

Proof.

(10)

Since (0) = 0 in the present case, it follows from (9)

that the finite quantity 'P(t) = In Me'c=,ect" exists for small t. The function q(t) is continuous by virtue of the stochastic continuity of e(t). Since for t, > 0 and tQ > 0, Me.(=,e(tl+t2)) = Me,(=,e(tl))+'(Z,e(tl+t2)-e(tl)) Me'(z,cctl>)Mei(z,e(tl+t2)-c(tl)) = Meac=,ect1>>Me,c=,ect2 = we have p(tl + t2) = q(tl) + p(t2). Therefore q(t) = tq(1). Formula (10) follows from (8) if we set

a=a(1), A=A(1), H(A)=H(1,A). This completes the proof of the theorem. 4.

PROPERTIES OF THE SAMPLE FUNCTIONS OF A STOCHASTICALLY CONTINUOUS PROCESS WITH INDEPENDENT INCREMENTS

We noted in Section 1 that a separable stochastically continuous process with independent increments fails with probability 1 to have discontinuities of the second kind. In this section we shall establish certain relationships between the properties of the sample functions

of a separable process e(t) and the properties of the functions a(t), A(t), and fl(t, A) which determine the finite-dimensional distributions

of the process c(t).

Definition. A process e(t) is said to be piecewise-constant on [0, T] if there exist points 0 = to < t, < . . < t = T such that e(t) is constant on each of the intervals (ti, tj+l). The points t;, will, of course, depend on the individual paths. The following theorem gives conditions under which a process will be piecewise-constant with probability 1. For a separable stochastically continuous process c(t) with independent increments to be piecewise-constant, it is necessary and sufficient that the characteristic function of e(t) be given by the formula

Theorem 1.

Mei(',ect>) = Me'(',e(° exp

l- 1]II(t dx)}

(1 )

4.

PROPERTIES OF THE SAMPLE FUNCTIONS

275

where H(t, A) is for every t e T a finite measure defined on the o'-algebra

of all Borel subsets of the space X, and for each Borel set A is a continuous nondecreasing function of t.

Proof of the Necessity. If XX = {x; I x I > s}, then for the step process e(t), the relation e(t) - (O) = limE-0 (t, XE) will with probability 1 be satisfied (since E(t) - (O) is the sum of its jumps). The characteristic function of the process e(t, XE) is given by formula (5) of Section 3. Therefore Mezcz,6ce>> = Me.cz.cco>> lim

[e:cz.x>

E-.O

- 1]II(t, dx)}

expdX1>8

a nd hence formula (1) will be proved once we show that the extension

of II(t, A) onto the a-algebra of all Borel sets is finite. To do this, it will be sufficient to show that the finite limit limE,0II(t, XE) ex-

But the limit limE.0 v(t, XE) exists for a step process with probability I and is equal to the number of jumps of the process. Being the limit of random variables with Poisson distribution, ists.

lim P(t, XE) E-,0

will also have a Poisson distribution. It follows from the convergence of the Poisson distributions that their parameters also converge;

that is, M lim V(t, XE) = lim II(t, XE) < E-0

C)O

E-.0

This completes the proof of the necessity of the hypotheses of the theorem. (The properties of H(t, A) as a function of t follow from Theorem 2 of Section 3.) Proof of the sufficiency.

Let us set

X(t) = lim II(t, XE) E-0

and

G(t1, t2, A)

= [II(t2, A) - 11(tl, A)][X(t2) - x(t1)]-1

Then G(t1, t2, A) is a probability measure on X.

Let Gk(tl, t2, A)

denote the k-fold convolution of the distribution G(t1, t2, A) with itself, so that ei(z,x)Gk(tl,

t2, dx) =

[ee.G(ti, t2, dx)] k

.

Then from formula (1) the characteristic function of the variable c(t2) - c(t1) can be written in the form

PROCESSES WITH INDEPENDENT INCREMENTS

276

e-[1(t2)-1(t1)] [X(t2) k=O k!

X(tl)]k

eti(Z,a)Gk(tl, t2, dx)

(2)

,

where G0(t1, t2, dx) is the distribution of a quantity that with probability 1 is equal to 0. Thus the distribution of the variable C(t2) - C(t1) coincides with the distribution of the variable S., where are independ, fin, S. = 0 and S. = 1 + + cn. Here i, ent random vectors with distribution G(t1, t2, A), and v is a random , fin, . and variable that is independent of the sequence C1, Indeed, has a Poisson distribution with parameter ? (t2) - X(t).

Mei(=,s,) = E M(ei(E,s,) I v = k)p{v k=0

which coincides with (2). p{e(t2)

= k} = k=0 E Mei(=.sk)p{v = k}

Thus

- e(t1) = 0} = p{SY = 0} > p{v = 0} = e-U(t2)-2(t1)]

Consequently p{ sup

I C(S)

t15a 5t2

- e(t1) I = 0}

= limp sup

n-o {15k5n

(tl +

= lim j{

- (t2 - tlJ)) _ e(t1` + k - 1 k

k (t2 n

n-.o k=1

n

n

- tl)) -

> lim lI exp { - [x(t2 +

kn

k=1

-

1(tl

e(t1 + k - 1 (t2 n

) =

t1) J

01

- t1)) = 0}

(t2 - ti)) k

+

(t2 -

n 1 (t2

- t1))]} =

e-[2(t2)-A(tl)]

, 2%. Let us find a Define hn = T/2" and tk = khn for k = 0, bound for the probability of the event that among the intervals

[tk, tk+1] there will be no fewer than 1 intervals for which sup

tk5tstk+l

I e(t) - C(tk) I > 0 .

Let this probability be denoted by pl. Then we may write sup IC(S) - W,,) I > 0} P1 <_ E Hj=1p{ ti.s8stij+1

< i1<...
<1 [E (1 - e-[d(tk+l)-1(tk)])]` 1! k=0 J < j) 1 1

k-0

[X(tk+1) - X(tk)]/ <

X(T)' 1(

4.

PROPERTIES OF THE SAMPLE FUNCTIONS

277

(In this chain of inequalities we use the relation 1 - e x < x for

x>0.) Let v. denote the number of intervals [tk, tk+1] for which sup tk5s5tk+1

I e(s) - e(tk) I > 0 .

Since v,+1 > vn it follows that the limit v = limn-- v, exists with probability 1. For the quantity v we also have P{v >- l} <

1 X(T)' .

- l!

The event {v = l} means that e(t) is constant on no more than l + 1 adjacent intervals. Since P{v > l} --- 0 as l , -, the process e(t) is

constant on v + 1 adjacent intervals and the quantity v is finite with probability 1. This completes the proof of the theorem. Suppose now that g(t) is a process with numerical values, that is, that X is the real line. Let us investigate the conditions under which the sample functions of the process E(t) are, with probability 1, monotonic functions. Theorem 2. For the sample functions of a numerical separable stochastically continuous process e(t) with independent increments to be nondecreasing with probability 1, it is necessary and sufficient that the characteristic function of the variadle e(t) be given by the formula Mei2ect> = Mei2e(o) exp {ix-l(t) +

1)II(t, dx)

(3)

0

where the measure II satisfies the condition \xfl(t, dx) < oo and 7(t) 0

is a nondecreasing function.

Proof of the Necessity. If e(t) is a nondecreasing function, the process e(t) has only positive jumps; hence II(t, A) = 0 for every set A lying on the negative half-line. We note also that in the present case the process E(t) - E(t, XJ is also nondecreasing (since removal of the jumps does not destroy monotonicity). Similarly, the process

e(t, XE) - e(t, X1)

for 0 < s < 1

,

is also a monotonic process; also 0 < W, XX) - e(t, X1) < e(t) - E(0) - E(t, X1)

On the basis of the lemma of Section 2, M[e(t) - E(0) - E(t, Xl)] < Therefore

PROCESSES WITH INDEPENDENT INCREMENTS

278

M[e(t, XE) - e(t, XM =

J

xH(t, dx) < M[e(t) - e(O) - c(t, X1)]

Taking the limit ass 0, we see that \xH(t, dx) has a finite value. We note also that the quantity e(t) - E(t, XE) decreases as s , 0 0

(since more positive jumps are discarded with decreasing s). Consequently the limit lim, [s(t) - g(t, XE)] = E0(t) exists with probability 1 and the process e0(t) is, with probability 1, continuous. As shown in Section 2, the increments of the process e0(t) will have Gaussian distributions. But the process e0(t), being the limit of nondecreasing processes, will itself be nondecreasing so that P{e0(t)

- 0(0) > 0} = 1

.

It follows from this relation that D(e0(t) - e0(0)) = 0 (since a normally

distributed variable a can be nonnegative with probability 1 only when D = 0). Thus where

spo(t) = e0(O) + 7(t),

7(t) = M[e0(t) - e0(0)]

and hence does not decrease. Formula (3) can be obtained from the relation Metiae(t) = lim Mei2eo(t)Met2 t.xd EGO

if we keep in mind the form of the process e0(t) and formula (5) of Section 3. This completes the proof of the necessity. Proof of the Sufficiency. Let us show that P {W(t2) - W(t1) > 0} = 1.

To do this, it will be sufficient to show that with probability 1 a nondecreasing random variable e whose characteristic function has the form

- 1)dG(x)}

Mebxe = exp {.:ei2

,

(4 )

where G(x) is a monotonic bounded function, (g(t2) - e(t1)) is the

sum of 7(t2) - 7(t) and the limit of a quantity with characteristic function of the form (4)). Let us set F(x) = c[G(+ oo) - G(x)], c = [G(+ oo) - G(+ 0)]-1 Then Me:ae =

c k=o k!

e-E(1r ei2ldF(x)

lk

J o

so that the characteristic function of the variable

coincides with

the characteristic function of the variable Sy, where So =0 and +,,. Here 1,2, is a sequence of independent Sn = 1 + identically distributed nonnegative variables with distribution function

4.

PROPERTIES OF THE SAMPLE FUNCTIONS

279

F(x), and v is a Poisson random variable independent of Consequently c is nonnegative. Thus P{c(t2) > c(t1)} = 1

for t1 < t2

.

It follows from the last relation that the event that the inequality e(t1) < e(t2) will be satisfied for all pairs of rational t1 and t2, where t1 < t2, has probability 1. Using the fact that g(t) is separable and has no discontinuities of the second kind, we conclude that P(W(t1) < Vt2), t, < t2) = 1

This completes the proof of the theorem. Let us investigate the conditions under which the sample functions of the process e(t) are with probability 1 of bounded variation. We recall that the variation of a function x(t) given on [a, b] with range in X is defined as n-1

var x(t) = sup E I X(ti) -

X(till)

i=0

[a,b]

with the supremum being taken over all possible partitions of the interval [a, b]: a = to < t1 < . . . < to = b. Theorem 3. For the sample functions of a separable stochastically continuous process e(t) with independent increments and defined on an interval [0, T] to be of bounded variation on that interval with probability 1, it is necessary and sufficient that the characteristic function of the variable e(t) be given by formula (8) of Section 3, with

var a(t) < o [0,T]

A(t) = 0, and the measure II(t, A) such that OQxI51

I x1II(t,dx) < - .

Proof of the Sufficiency.

Since the process defined by xv(t, dx)

e(t, X1) = IxI>1

is with probability 1 piecewise-constant, its variation, which coincides with the sum of the absolute values of the jumps, is finite. The function a(t) is, by hypothesis, of bounded variation. Therefore to prove the boundedness of the variation of e(t), it will be sufficient to prove the boundedness of the variation of the integral xv(t, dx) 0
PROCESSES WITH INDEPENDENT INCREMENTS

280

as a function of t (cf. formula (3) of Section 3). Consider the process

xv(t, dx) .

VE](t) = 5 e
It is easy to show that the variation of the process e(,)(t) on the interval [0, T] is equal to var (,)(t) = var [0,T]

xf(t, dx)

xv(t, dx) + var 5

[0,T]

[0,T]

LIx1:51

Therefore

[0,T]

I x II(T, dx)

I x I v(T, dx) +

var e(E)(t) <

LIZI!<1

LIX1251


J 0
IxjII(T,dx)

The existence and finiteness of the limit lim, 5

follows from the fact that the quantity

.

I x I v(T, dx) e
x I v(T, dx)

`

is a

monotonic function of s and M` 6<x1;51

IxIII(T,dx)<=.

1xIv(T,dx)<J 0
Since we can choose a sequence {sn} that converges to zero as n such that the sequence of the processes E(,-)(t) converges uniformly

with probability 1 to the process with probability 1, xv(t, dx) < 5

vary [0,T]

O
xv(t, dx), it follows that 5

I x I v(T, dx) + 5 0
(It is easy to see that if x, (t)

.

x(t) as n --+ 1o for every t, then /

m-1

mn-+1

I x III(T, dx) 0
// lim var xn(t) E I xn(tk+1) - xn(tk) I <_ n-.ao L I x(tk+l) - x(tk) I = lim n-- k=0 [0,T] k=0

and hence var[o,T] x(t) < HE,- var[O,T] xn(t). From this it follows

that the variation of 5

xv(t, dx) is bounded. 0
Suppose that the variation of c(t) on the interval [0, T] is bounded. Then it will also be bounded on every interval contained in [0, T]. Consider the process Proof of the Necessity.

C(t) = var e(s) . [0,t]

This process is also a process with independent increments since the increment C(t2) - C(t), where t1 < t2, is the variation of a process on

4. PROPERTIES OF THE SAMPLE FUNCTIONS

281

the interval [t1, t2] and hence depends only on the increments of the process e(t) on the interval [t1, t2]. Hence it is independent of the values of C(t) for t < t1. The process C(t) is a stochastically continuous process since the variation var[o,t] x(s) of the function x(s)

can have discontinuities only at points of discontinuity of the function x(s) itself and the process e(t) has no fixed discontinuities (because of its stochastic continuity). Finally it is obvious that C(t) is a nondecreasing function of t. From what was said above it follows that

(t+0)-c(t-0)=Ie(t+0)-e(t-0)I. Therefore, if CW(t) is the sum of all jumps of C(t) exceeding s and occurring up to the instant t inclusive, then I x I v(t, dx) .

VE)(t)

xl>E

Since C(0) = 0, it follows from the monotonicity of the process C(t) that

fi(t) = 7(t) + lim

C(E)(t)

= 7(t) + lim

E--.0

E -.0

I x Iv(t, dx) Izi>z

where 7(t) is the variation of the continuous component of the process e(t). Furthermore, 5

(t)] < c°

I x I II(t, dx) =

IxIs1

.

Finally we note that if eo(t) is the continuous component of the process e(t), then I

0(t) -

I < var Se0(s) = 7(t) [0,T]

and hence

I (e0(t) - e0(°), Z) I < 7(t) I z

But a normally distributed variable with positive variance cannot be bounded with probability 1 by any constant. Consequently for arbitrary z, D(z, 0(t) - e0(0)) = 0

that is, A(t) = 0. Thus 7(t) is the variation of the function

a(t) - f

o
xH(t, dx) .

Since the variation var

xf(t, dx) < LIXI51 I x I H(T, dx)

[0,T] JO
PROCESSES WITH INDEPENDENT INCREMENTS

282

is finite, the variation var[o,T] a(t) is also finite. This completes the

proof of the theorem. REMARK. It follows from the proof given of the necessity that if a process e(t) is a bounded variation, then we can write

e(t) = g(0) + al(t) +

xv(t, dx) .

.,1 xl>0

Also, we have seen that fi(t) = var E(s) = var al(s) + [o,t]

[o,t]

Jxj>o

I x I v(t, ds)

and the characteristic function of the variable C(t) is given by Meiz;[t) = exp {iX var a (s) + [o,t]

5.

lei' 1- 1 - l]II(t dx)1( IxI>o

1

'

J

PROCESSES OF BROWNIAN MOTION

A process w(t) of Brownian motion is defined as a separable homogeneous Gaussian process with independent increments such that Mw(t) = 0 and D(z, w(t)) = ct(z, z). In this section we shall consider a one-dimensional process of Brownian motion (a process with numerical values) and for brevity in writing we shall assume that c = 1 (this can always be obtained by changing the units of the time scale). For such a process, the variable w(t + h) - w(t) has a normal distribution with mean zero and variance h, so that the distribution density p,a(x) of the variable w(t + h) - w(t) is given by the formula p"(x) =

1

e-(x2/2h)

1/ 27rh

On the basis of Theorem 6, Section 5, Chapter IV a process of Brownian motion is continuous. REMARK 1.

The name "process of Brownian motion" can be

explained by the following description of the corresponding physical phenomenon. Consider a sufficiently small particle suspended in a liquid and consider its motion under the influence of collisions with

the molecules of the liquid, which are in chaotic thermal motion. In physics this phenomenon is known as "Brownian motion." In the probabilistic study of this phenomenon it is natural to consider the velocities of the molecules with which the particle collides as random, and in the case of a homogeneous liquid we

S.

PROCESSES OF BROWNIAN MOTION

283

assume that the distribution of the velocity is independent of the position of a molecule. (It depends on the temperature, which is everywhere uniform.) If we also assume that the velocities of the different molecules are independent of each other and if we neglect the mass of the particle, then the displacement of the particle from an arbitrary position after a certain interval of time will not be dependent on the position of the particle or its previous motion. Consequently, a process C(t) in R(3) that describes the position of a particle at an instant t is a process with independent increments. Furthermore it is obvious from physical considerations that it will be continuous and homogeneous with respect to time if the physical state of the liquid does not change with time. But every continuous homogeneous process with independent increments is a Gaussian process.

We shall assume that at the initial instant of time the

position of the particle coincides with the coordinate origin: t(0) = 0. Suppose that Me(t) = to and D(E(t), z) = t(Az, z), where a is a vector

in RM and A is a symmetric operator in R(3). In the case of a homogeneous liquid where there are no currents, the process must be isotropic (since the distribution of the projections of the velocity of the molecule of the liquid in an arbitrary direction is independent of that direction); that is, (a, z) and (Az, z) are independent of z for z I = 1. This is possible when (a, z) = 0 and (Az, z) = c(z, z). Thus from the most general considerations of the physical phenomenon of Brownian motion, we have obtained our definition of a process of Brownian motion.

Let a denote some number other than 0. Let za denote an instant of time at which w(t)/a < I for t S za and for arbitrary

8>0, sup

w(t)

>1

.

ra5t5rat8 a

If w(t)la for all t, we take za = + 0' . We shall call ca the instant of first crossing of the level a by the process w(t). Now suppose that z'a is that instant of time at which w(t)/a < 1 We shall call za the instant of first attainment of the level a by the process w(t). Obviously, za < za. for t < zd and w(zd) = a.

Lemma 1. P{zd = -Cal = 1. Proof. By virtue of the symmetry of the process w(t) (the

process - w(t) has the same distributions) we assume that a > 0. The event {z' < za} implies at least one of the events { max w(s) = a}, r= 1,2,...; m= 1,2, o585r/m

.

284

PROCESSES WITH INDEPENDENT INCREMENTS

Therefore to prove the lemma, it will be sufficient to show that for

a > 0, P{max w(s) = a} = 0 oss5t

for every t. It is easy to see that for t1 < t, P{max w(s) = a} < P{max w(s) = a}

asst

ossst1

-

+

P{w(t1) E dx}P{ sup w(s) - w(t1) = a - x} t15sSt

.

But P{supt15sst w(s) - w(t1) = z} can be nonzero at only countably many distinct values of z; that is, P{ sup w(s) - w(t1) = a - x} = 0 t15s5t

for almost all x (in the sense of Lebesgue measure). This means that P{w(t1) e dx}P { sup w(s) - w(t1) = a - x} t15s5t

(1f

L Zit 1

P{ sup w(s) - w(t1) = a - x}e-(x2/2t1)dx = 0 J¢

t Ss St

since the integrand vanishes almost everywhere. Thus P{max w(s) = a} < P{max w(s) = a} 0Ss5t

o5s5t1

that is, P{maxo5s5t w(s) = a} does not decrease as t 10; whereas, by

virtue of the continuity of w(t), for s > 0, P{max w(s) > s} 0Ss5t

0

as

t -+ 0 .

Therefore

P{max w(s) = a} < lim P{max w(s)

asst

do

05s5t

>

a 2

=0

This completes the proof of the lemma.

For our purposes in what follows, then, we shall not find it necessary to distinguish the instant of the first attainment from the instant of first crossing of the level a. We shall therefore let za denote both.

To study certain characteristics of the process w(t), we shall use:

Lemma 2. Let w(t) denote a process of Brownian motion, let a denote a number other than zero, and let v denote the instant of first crossing of the level a by the process w(t). We define a process w1(t)

5.

PROCESSES OF BROWNIAN MOTION

285

by wi(t) = w(t) for t < za and wi(t) = 2a - w(t) for t > va. Then the process wi(t) is also a process of Brownian motion.

Let us set

Proof.

w(n)(t) = LkSnt Wnk, and Win)(t) = Zk5nt (-l)FnkWnk, where

(i-) < 1

0 if sup ;5k-i

Snk =

a

wcn)(J ) 1

\\ n

if sup

a

j5k-i

>1

.

We note that the quantities (-1)Enkwnk for k = 1, 2, are mutually independent and identically distributed. Also, their distributions coincide with the distributions of the quantities wnk. This follows from the fact that wnk and -wnk are identically distributed and Wk is independent of snk, wn1, , wn, k-i Consequently the (-1)FnkWnk have a normal distribution with mean zero and variance 1/n. Therefore the finite-dimensional distributions of the processes w(n)(t) and win)(t) coincide. Since a process of Brownian motion is continuous, and therefor w(n)(t) --+ w(t) and w;n)(t) wi(t) with probability 1 as n -> - , the lemma is proved.

We shall now use this lemma to find the distribution of such characteristics of the process as max,,,ST w(t), minOS T W(t), and SUP05tSr I W(t)

Theorem 1. For a > 0, P{max w(t) > a, w(T) e [c, d]} 05t;5T

cmax[d,a]

1

J

27CT max[c,a]

Proof.

ed -

(.212T),4x

1

+

max[2a-c,a] - (x 2

1 ' 27rT max[2a-d,a]

e

/2T)dx .

(1 )

Let us use the relation

P{max w(t) > a, w(T) e [c, d]} = P{w(T) e [c, d] n [a, oo]} 05t5T

+ P{max w(t) > a, w(T) e [c, d] n (- oo, a]}

(The validity of this relation follows from the fact that the event {w(T) e [c, d] n [a, -]} implies the event {maxa5t5T w(t) > a.) Let us now find the probability P{max w(t) > a, w(T) e [c, d] n (- oo, a]} 05tST

.

PROCESSES WITH INDEPENDENT INCREMENTS

286

Let w,(t) denote the process defined in Lemma 2. Then the event {max w(t) > a, w(T) e [c, d] n(- oo , a]} OSt9T

coincides with the event {max w,(t) > a, _w,(T) a [2a - d, 2a - c] n [a, c -)l 05t5T

But the event

{w,(T) e [2a - d, 2a - c] n [a, oo)} implies the event {maxa5t5T wl(t) > a}. Therefore

{max w,(t) > a, w,(t) a [2a - d, 2a - c] n [a, .)I OStST

= {wl(T) e [2a - d, 2a - c] n [a, oo)} Consequently, on the basis of Lemma 2, P{max w(t) > a, w(T) e [c, d] n (- t", a]} 05t5T

max[2a-e,a]

1

2

e-(a /2T)dx y/27iT max[2a-d, a]

(2)

Furthermore,

P{w(T) e [c, d] n [a, -]} =

maxmax[e,a]V [d,a] 1

e-(=212T)dx

27eT

(3)

Proof of the theorem follows from (2) and (3). Corollary. For a > 0, P{max w(t) > a} = o5t5T

e_(x2I2T)dx 2 VT7FT-- a

This follows from Theorem 1 with (c, d) = (-=, oo). Theorem 2. Suppose that a, < 0 < a2 and [c, d] c [al, a2]. p{min w(t) > al, max w(t) < a2, w(T) e [c, d]} 05t5T

05t5T

1

Z/27cT

- exp { Proof.

Then,

i

ai))2k J"I

exp

2T

(x } 2k(a2 -

\

T (x - 2a2 + 2k(a2 - ai))2}]dx .

-2

J

(4)

Let W10 denote the event that the process w(t) defined

on the interval [0, T] crosses the level a; earlier than it does the level a; (where j # i and i, j = 1, 2) and then crosses the interval [al, a2] no fewer than k times (we are assuming that the function x(t) crosses the interval [al, a2] k times if the function sgn (x(t) - a) + sgn (x(t) - a2)

S.

287

PROCESSES OF BROWNIAN MOTION

The desired probability can

changes sign k times) and w(T) e [c, d]. be expressed as follows:

P{w(T) e [c, d]} - P{jol)} - P{Ko'}

To calculate P{ o°'}, let us find the probabilities y q-L j i, j = 1, 2) P{ k)} + P( W1} = P{'El(ks) U %W1}

As one can easily see, the %k' U Wk )1 is the event that the process

w(t) crosses the level a, prior to the instant T (though not necessarily before it crosses the level a;) and then crosses the interval [al, a2] no fewer than k times before, at time T falling into the interval [c, d].

Let v, denote the instant of first crossing of the level

a;, let v2 denote the first crossing of a; after the instant z-,, let z-,, etc. We set denote the first instant of crossing of a; after lPt)

2w(z-1) - w(t) wl(t)

w2(t)

12w,(z-,) (w2(t) 'N3(t)

- wl(t)

2w2(v3) - w2(t)

for for for for for for

t < z-1

,

t > zl t < 2.2

t> t < v3 t > z3

,

etc.

We note that the processes w,(t) are processes of Brownian motion since z, is the instant of first crossing of the level a; + (1 - 1)(ac - a,)

by the process w,_1(t). If the event Kki' U Wk+l occurs, the process wk+l(t) for t < T crosses successively the levels

aa,a.+(a;,-a;), ...,a;+k(a,-a;) and at the instant T falls in the interval [ck, dk], where

ck=c+(k+1)(ai-a;), dk=d+(k+l)(ati-a,) ck=2a;-d+k(a;-a.),

dk=2a:-c+k(a;-a3)

for odd

k,

I J

for even k .

Conversely, if wk+1(t) satisfies these conditions, the event k' U 'k+1 occurs. Since wk+l(t) is a continuous process that vanishes at t = 0,

for wk+l(T) to fall in the interval [ck, dk] it must beforehand cross Therefore the levels a; +ppl(a, - a3) for 1 = 0, , k. YY

P{Elk' U uk+1} = P{Wk+l(T) a [ck, dk]} = P{W(T) e [Ck dk]}

It follows from the continuity of the process w(t) that w(t) crosses, with probability 1, the interval [al, a2] finitely many times and hence,

PROCESSES WITH INDEPENDENT INCREMENTS

288

P{Wk)}--0 as k -->

Taking the limit in the equation

00 .

PJ%011)} + PIW011)} = (- l)n+l[p{1+1} + p{'Lln2+1}]

Y lyy + Jay E (- l)k(P{wkl)} l l + p{GLk2)} + _ P{'Llk1}1} + P{"Gl,k -1}) k=o\ we obtain

as n

2

p{ClI(01)l 1-L f +

l)k

k=0 i=1 2a1-c+2k(a2-a1)

l

(f k=0

-

5

(p{`^'kro)}

+ T p{LLk}1})

e (2I2T)dx +

52a2-+2k(a2--i1)

2a1-d+2k(a2-a1)

d+2(k+1)(a2-a1)

e-(x2/2T)dx

e-(x212T)dx

2a2-d+2k(a2-a1)

- 1 d+2(k+1)(a2-a1) e-(x2/2T)dxc+2(k+1)(a2-a

Therefore the desired probability is equal to d+2k(a2-a1)

1

l1

727' kfl

I

1

2a2 -d+2k(a2-al)

e-(a2/2T)dx -

c+2k(a2-al)

e-(x2I2T)dx

j2a2-c+2k(a2-a1)

If we set x - 2k(a2 - a) = u in the first integral and

2k(a2-a) +2a2-x=u in the second, we obtain formula (4). This completes the proof of the theorem. Corollary 1. The joint distribution of the quantities max w(t), and min w(t) 05t5T

05t5T

for al < 0 and a2 > 0 is given by the formula p{ min w(t) < a1, max w(t) < a2} 0
_

05t5T

a1[eXp _

V 27rf k '

_ 2T 1

(x

2k(a2 -

a1))2

}

- 2T (x - 2a2 + 2k(a2 - a,))2}]dx . 1

- exp

(5)

Corollary 2. For a > 0 and [c, d] c [-a, a], we have P{ sup I w(t) I < a, w(T) e [c, d)} o5t5T

d ckj_

1/ 27cT

6.

(-1)k exp(\

(x

2Taf dx )2

/

.

(6)

ON THE GROWTH OF HOMOGENEOUS PROCESSES WITH INDEPENDENT INCREMENTS

Let g(t) denote a random process with independent increments. We are interested in the rate of growth of the quantities supo5t5T e(t)

6. ON THE GROWTH OF HOMOGENEOUS PROCESSES

289

and sup,,,,, I E(t) I as T-. oc and the rate at which the quantities supo
lim g(t) = t" and limp(t) = 0 . t-.o

t-.o

We shall call the function g(t) an upper function for the process c(t) if p{limt_. E(t)/g(t) < 1} = 1 and we shall call it a lower function if p{limt- e(t)lg(t) > 1} = 1. If we replace g(t) with T(t) and analogous relations are satisfied as t -+ 0, cp(t) is called a locally upper function and a locally lower function for the process E(t) respectively. In what follows, we shall consider only regularly increasing upper and lower functions. Suppose that g(t) is an increasing continuous nonnegative function defined for t > 0. We shall say that g(t) is a function of regular growth if there exist functions k1(x) and k2(%) such that k1())g(t) < g(Xt) < k2(X)g(t)

for

X > 0, t > 0

- as X,-.

and if k2(X) -. 1 as X-1 and k1(X) We shall first look at upper and lower functions for symmetric homogeneous processes with independent increments. Then we shall consider upper and lower functions for I E(t) 1, where E(t) is not necessarily a symmetric process.

A process E(t) is said to be symmetric if the processes E(t) and -e(t) have identical finite-dimensional distributions. Lemma 1. Let E(t) denote a symmetric separable stochastically continuous process with independent increments. Then p{ sup E(t) > c} < 2P{E(T) > c}

.

OS t ST

(1 )

Proof. Let E1, E2, , E, denote independent symmetrically dis+ Ek. Then tributed random variables and define Sk = E1 +

p{ sup Sk > c} < 2P{Sn > c}

.

15k!9n

To see this, note that since 2P{S, - Sk > 0} > 1 we have P{sup Sk > c} = E p{ sup Si < c, Sk > c} k=1

k

ti5k-1

<-EP{supS;c}2P{Sn-Sk>_0} k=1

i k-1

=21P{SUpS;C, Sn-Sk>O}. i k=1

k-1

(2)

PROCESSES WITH INDEPENDENT INCREMENTS

290

The events {supti c, S. - Sk > 0} are incompatible and each of them implies the event {Sn > c}. Therefore 2P{Sn > c}

Sk>c, Sn-Sk>-o}>p{supSk>c}.

k=1

tick-1

k

Let us now apply formula (2) to the quantities k = e(k/nT) - e(k - 1/nT) and e, = E(T/n). Then This establishes formula (2).

UkEn

Taking the limit` as n , - and remembering that c(t) fails with probability I to have discontinuities of the second kind, so that

n/ In- 1
P{lim sup

we obtain (1).

= sup e(t)} = 1

,

OstsT

This completes the proof of the lemma.

Theorem 1. Let e(t) denote a symmetric homogeneous separable stochastically continuous process with independent increments and let g(t) denote a function of regular growth such that

1 t P{fi(t) > g(t)}dt < o . Then for arbitrary X > 1 the function Ag(t) is an upper function for the process E(t).

Let a denote a number greater than 1. Let ak denote the event that SUPOSts¢k S(t) > g(ak+l). Then on the basis of Lemma Proof.

But for t e [ak, ak+1] we have

1, P{'Ltk} < 2P{e(ak) > g(ak}1)}.

g(t) < g(ak}1)

2P{e(t) - e(ak) > 0} > 1

and

.

Therefore P( k) < 4P{e(ak) > g(t)}P{e(t) - e(ak) > 0} < 4P{e(t) > g(t)} Consequently, rak+l

P{Xk}dt

<

4¢k+1

p{e(t) > g(t)}dt

and

E P{Xk} <

In a

1

t

P{e(t) > g(t)}dt

It follows from the Borel-Cantelli lemma (Theorem 2, Section 3, Chapter III) that with probability 1, only finitely many of the events Ik occur; that is, for some (generally speaking, random) number, k0,

6. ON THE GROWTH OF HOMOGENEOUS PROCESSES

291

the events uk do not occur if k > k0. This means that p{lim k-.o 4(a

')

sup

e(t)

k-1--t5¢k

For t e [ak-1, ak] (where k >_ k0), fi(t) k2(a2)g(t)

<

1

sup

p k2(a2)g(ak-1) nk-15t,.k

g(t)

1

-

g(ak}1)

sup

e(t)

ak-15t5¢k

Therefore, for arbitrary a > 1 and X > k2(a2) the function Xg(t)

is

the upper function for e(t). This completes the proof of the theorem.

Theorem 2. Let E(t) denote a symmetric homogeneous process with independent increments and let g(t) denote a function of regular growth such that the series Z p{e(ak) > g(ak)} k=1

diverges for all a > 1. Then for arbitrary X < 1, xg(t) is the lower function for the process g(t). Proof.

We first show that: P

1.

(lim g(t)I > 1

(r < 1)

/

.

(3)

Suppose that lim p{e(t) > g(t)} >t-.co

Then there exists a sequence {tk} such that

p{e(tk) > g(tk)} > 2 -

'

Hence, by symmetry, p(I e(tk) I < g(tk)) < 1/k2, and so the series

jk=1P{I E(tk) j< g(tk)} < k=1 L

1

k2

converges. It follows from the Borel-Cantelli lemma that with prob-

ability 1, e(tk) > g(tk) from some k on. p

2.

m L EM > 1 {lik-.oo %9(tk)

This means that for X < 1,

-1

Suppose that there exists a S > 0 such that lim p{E(t) > g(t)} < 1 t__

Then for sufficiently large t,

PROCESSES WITH INDEPENDENT INCREMENTS

292

P{I e(t) I < g(t)} >- a Consider the independent events

0k = {e(ak+1) - S(ak) > g(ak+1) - g(ak)}

where a > 1. Then g(ak)

P{Bk} >

>

P{e(ak+1) - z > g(ak+1) - g(ak)}p{E(ak) e dZ}

_ J

9(¢k) J

g(ak+1)}

/

9(¢k)

P{e(ak) e LIZ} -9(¢k)

= P{e(ak+1) > g(ak+1)}p{I S(ak) I < g(ak)} > P{e(ak}1) > g(ak}1)}U

for sufficiently large k. Consequently the series Ek=1 P{ k} diverges.

Hence on the basis

of the Borel-Cantelli lemma, infinitely many of the events 0k occur with probability 1. We note that the event 8k implies one of the events

{-e(ak) > g(ak)}; {e(ak}1) > g(ak}1) - 2g(ak)}

.

Therefore, with probability 1, infinitely many of the events {I e(ak) I > g(ak) -

2g(ak-1)}

Given r < 1 choose a in such a way that

occur.

g(ak) - 2g(ak-1) = g(ak)rl LL

- 2g(ak-1) g(ak) J

> g(ak)[1 -

2 ] k1(a)

> Xg(ak)

(the possibility of such a choice of a is ensured by the regularity of the growth of g(t)). We see that {11He(t)I > 1} = 1

.

We now show that (3) implies that g(t) is a lower function for e(t).

C = lim t

E(t)

>1

Xg(t)

D = fri-m-

-e(t) >

1}

X90

Then it follows from (3) that P(C U D) = 1. We conclude from the symmetry of the process e(t) that P(C) = P(D). Finally from the 0-or-1 law (cf. Theorem 5, Section 3, Chapter III) it follows that p(C) and P(D) can be only zero or one. This means that P(C) = p(D) =1 since otherwise we would have p(C) = 0 and hence p(C U D) = 0

which contradicts equation (3). theorem.

This completes the proof of the

6. ON THE GROWTH OF HOMOGENEOUS PROCESSES

REMARK 1.

293

Suppose that in these two theorems we consider

a < 1 and instead of the function g(t) we consider the function p(t) = 1lg(1/t) where g is a function of regular growth. Without otherwise changing the proofs of the theorems, we see that the following assertions are true: a. If e(t) is a symmetric separable homogeneous stochastically continuous process with independent increments and if J1

t

P{fi(t) > p(t)}dt < o ,

then for ? > 1 the function ?p(t) will be a locally upper function for the process e(t); that is, p{lim, $(t)/xrp(t) < 1} = 1. If fi(t) is a homogeneous symmetric process with independent increments such that the series Ek=1 p{e(ak) > q (ak)} diverges b.

for every a < 1, then for ? < 1 the function ?q(t) will be a locally lower function for $(t); that is, p{lim, e(t)/xq(t) > l} = 1. Let us apply these results to a process of Brownian motion. Such a process is symmetric. By using the inequalities 1

1/ 27r

e-(u2/2)du

i -_e-

<

1

iz

1/ 27r

I f _e-cu2Mdu 1/ 27r z

>_

1

1/ 27r

u e-(u2/2)du

1

=

z

z

f +1e-(u2i2)du >

(z2/2)(Z

e

> 0)

27r

e-(z+1)212(z > 0)

1

1/ 27c

z

we see that e (J-tt }1)2/2 < p{w(t) > z} <

1

27r

1

e-(2212t)

z

Let us show that for arbitrary s e (0, 1) (1 + s) 1/2t In In t and (1 - s) 1/2t In in t are upper and lower functions respectively.

We

have

p{w(t) > (1 + s) V2t In In t}

<

I

1/27r(1 + s)22 In In t

(I + 6)22In In tl

exp 1

2

= O((ln t)-(,+f)2) and the integral dt/t(ln t)(1+f)2 converges for c > 1.

On the other

hand,

p{w(ak) > (1 - s)1/2ak In In ak} >_

e-(1/2)

1/ 27r

exp { - (1 - s2)[ In In all + V2 In In ak ] J}

C exp { - a In In ak} = C(ln ak)-a

PROCESSES WITH INDEPENDENT INCREMENTS

294

In ak)-1] < a < 1 (as will be the case for sufficiently large k). Consequently the series E P{w(ak) > (1 - s)1/2ak In In ak} diverges. Thus we have proved: Theorem 3. If w(t) is a separable process of Brownian motion,

if (1 - s2)[1 + (V2-1

then

w(t)

1/2tInInt

t

By using Remark 1 we can prove: Theorem 4. If w(t) is a separable process of Brownian motion, then

P lim t-o

w(t)

2t In In 1 t

Theorem 3 and 4 are called the "law of the iterated logarithm." In studying upper and lower functions for I e(t) I where e(t) is a process with independent increments, we use: Lemma 2. Let e(t) denote a separable stochastically continuous process with independent increments, for which there exists an a < 1 such that P{I e(T) - e(s) I > C} < a for 0 < s < T.

Then for every x > 0,

P{o5s5T supIa(s)I>C+x}< Proof.

1

I - a

P{Ie(T)I>x}.

(4)

It follows from Theorem 2, Section 4, Chapter III that

P{ sup 15k5n

e(nT) >x+

C}

<

1

1

aP{I$(T)I >x}.

Taking the limit as n , oa , we obtain proof of the lemma. Theorem S. Let e(t) denote a separable homogeneous stochastically continuous process with independent increments and let g(t) denote a

function of regular growth such that for arbitrary s > 0, lim p{I e(t) I > sg(t)} < 1 and

I J

P{I e(t) I > g(t)}dt <

6. ON THE GROWTH OF HOMOGENEOUS PROCESSES

295

Then for arbitrary X > 1 the function ? g(t) is an upper function for I e(t) I ; that is, I

Proof. event

I<1 =1

.

Let us choose ag(t) e (1, 2) and s > 0. Let %k denote the {sup I e(t) I > (1 + 2s)g(ak}1)} t
It follows from the hypotheses of the theorem that there exist a c > 0 and a TO such that P{I E(t) I > sg(t)} < 1 - c

for t > T0.

Since for TO < t < ak, P{I e(t) I > sg(ak}1)} < 1 - C

P{I e(t) I > sg(t)} < I - c , and

lim sup P{I e(t) I > sg(ak+1)} = 0 k-.oo t<7'0

it follows on theyy basis of Lemma 2 that for sufficiently large k, 1 P{I $(ak) I > (1 + s)g(ak+1)} .

P{'Ll.k}

c

Remembering that t - ak < (a - 1)ak < ak forte [ak, ak+1] and hence P{I e(t) - e(ak) I > sg(ak)} < 1 - c for sufficiently large k, we obtain P{%'} < 2 P{I e(ak) I > (1 + s)g(ak+1)}p{I c(t) - e(ak) I c

Eg(ak+1)} <

p{I e(t) I > g(ak+l)} <

P{I e(t) I > g(t)}

so that py

1

P{"Ik} < c2

c

ak+11

ak t P{I e(t) I > g(t)}dt ,

P{ Ik} < (ln a)-1c-2J at t P{ fi(t) > g(t)}dt .

This means that with probability 1, only finitely many events ak occur. By using the reasoning of Theorem 1, we see that the function X(1 + 2s)k2(a2)g(t) is, for X > 1, an upper function for I e(t) 1.

Since k2(a)

1 a,1 and since a> 1 ands > 0 are arbi-

trary, the assertion of the theorem follows.

296

PROCESSES WITH INDEPENDENT INCREMENTS

Analyzing the proof of Theorem 2, one can easily prove: Theorem 6. Let c(t) denote a homogeneous process with independent increments. If a function of regular growth is such that the series

E P11 e(ak) I > g(ak)} k=1

diverges for every a > 1, then for every 0 < X < 1 the function ?,g(t) is a lower function for I e(t) 1; that is, l

pflimlS(t)I >1}=1 The results of Theorems 5 and 6 can be reformulated for the case in which t 0, in a manner analogous to that used in Remark 1.

VII JUMP MARKOV PROCESSES

Let X denote an arbitrary space with fixed 6-algebra 0. Let us interpret X as the phase space of some physical system E and

let us denote the state of E at the instant t by c(t) (e X). Let us suppose that the time t varies in discrete amounts (t = 0, 1, 2,

).

Let us suppose that the change in the system E, from its state x at the instant t into another state at the next instant t + 1, is completely determined by the time t, the state x, and some random factor at that constitutes, for the different values of t, a sequence of independent random elements. Thus

(1 ) e(t + l) _ .f(t, e(t), at) , where f(t, x, a) is a function of the three variables t, x, and a, where t = 0, 1 , 2, , x e X, and a e A. Formula (1) enables us to express the state of the system E at an arbitrary instant s by at an instant t < s: starting with the state e(t) of the system e(s) = gt,s(c(t), at, at+1, ..., a8-1) .

(2)

We emphasize that e(t) in this equation is independent of the set at, at+1, ..., as-1. Let {S2, Cam, P} denote the probability space on which the random

elements at are defined. Let us suppose that for arbitrary fixed t and s (where s > t) the function gti8(x, at, at+1, , a8-1) is (0 x e)measurable. Then if the motion of the system Y, begins at the

instant t and its initial state e(t) = x is known, formula (2) enables us to determine the probability that E will fall in an arbitrary set A e 0 at the instant s > t. We shall call this probability the transition probability and we shall indicate it by p(t, x, s, A).

If XA(x)

denotes the characteristic function of the set A, then P(t, x, s, A) = MxA[gt,8(x, at, ..., a8-1)] .

(3)

Let u and v denote two numbers such that t < u < v. It follows from formula (2) (cf. Theorem 7, Section 4, Chapter IV) and the 297

JUMP MARKOV PROCESSES

298

independence of the random variables at,

, av_1

that

a.9 ... ,

P(t, x, v, A) = MxA[gu.

au, . . ., av_1)]}&=e(u)1 = MP(u, e(u), v, A) which may be rewritten p(t, x, v, A) _ P(u, y, v, A)p(t, x, u, dy), t < u < v

.

(4)

It expresses an important property of the systems that we are considering, namely Equation (4) is called the Chapman-Kolmogorov equation.

the absence of aftereffects: if we know the state of a system at a certain instant u, the probabilities of transition from that state do not depend on the motion of the system at previous instants of time. Systems enjoying this property are called Markov systems. They are

frequently encountered in equations of science and technology. Chapters 7 and 8 are devoted to a study of Markov processes. In this chapter, we shall consider systems whose motion can be characterized by the fact that the system E is immobile in phase space for some period of time and at a random instant its position changes by a jump. In the next chapter we shall consider systems whose states change continuously with time. 1.

TRANSITION PROBABILITIES

Let X denote an arbitrary space with fixed a-algebra of sets 0 and let Z denote a set of real numbers. Definition 1. A family of functions p(t, x, u, A), where t, u e Z,

t < u, x e X, and A e 0,

called a Markov process in the broad sense in the phase space X if the functions p satisfy the following is

conditi a. p(t, x, u, A) is, for fixed t, x, and u, a probability measure

on 0, b. for fixed t, u, and A, the function p(t, x, u, A) as a function of the variable x is 0-measurable, and c. for arbitrary t, u, v, x, and A (with t < u < v) the functions

p(t, x, u, A) satisfy the Chapman-Kolmogorov' equation

p(t, x, v, A) = f p(t, x, u, dy)p(u, y, v, A) .

(1 )

X

The functions p(t, x, u, A) are called the transition probabilities.

According to our interpretation of transition probabilities we naturally assume that

1.

TRANSITION PROBABILITIES

299

(2)

P(t, x, t, A) = XA(x) ,

where XA(x) is the characteristic function of the set A e 0. There are two families of operators connected with transition probabilities: 1. Suppose that the distribution of the position of the system in phase space is given at the instant to and suppose that po(A) = p{e(to) e Al where A e 0. Let the distribution of the system E at the instant t > to be denoted by pt(A). Then

(3)

5p0(dx)P(to, x, t, A)

!pt(A) =

Formula (3) defines an operator T[to.t] (for to < t), which maps the probability measure ,uo(A) into a new probability measure fct(A). This is true because (obviously) pt(A) is nonnegative, pt(X) = 5xpo(dx)p(to, x, t, X) =

5xpo(dx) = 1, and the countable additivity of

pt(A) follows from the countable additivity of the integral and the transition probability p(to, x, t, A). Furthermore, if instead of the probability measure ,uo in formula (3) we substitute an arbitrary finite charge Wo, the right-hand side of this formula remains meaningful and we obtain a transformation T[to.tI in the space W of all finite charges (cf. Section 1, Chapter II): W0(A) =

Jx

W0(dx)p(to, x, t, A)

.

(4)

If Wo = Wo - Wa , where Wa and Wo are the positive and negative variations respectively of the function Wo, then W (A)

Wo (dx)p(to, x, t, A) - Wo (dx)p(to, x, t, A) . J

(5)

Consequently, for the positive and negative variations of the charge WW(A) we have

Wt (A) <

Wo (dx)p(to, x, t, A)

`

Wr (A) < J

x

Wo (dx)p(to, x, t, A) ,

from which it follows that IlWW

=Wt(X)+Wt(X)<Wu(X)+Wo(X)=11WoH

Thus the operators T[to,tl map W into itself, they are linear, and IIT`tot'11<1.

The Chapman-Kolmogorov formula provides a simple law of

JUMP MARKOV PROCESSES

300

composition of the operators T[to,t). To establish this law, we need an auxiliary proposition. Let X and Y denote two spaces, let e, and c 2 denote a-algebras of subsets of X and Y respectively, and let p denote a finite charge on e, Suppose that v(B I x) is for fixed B e COQ a e, -measurable

function of the argument x e X, and that for fixed x it is a finite measure on COQ.

Suppose that

supv(YIx)
We define X(B) = xv(B I x)p(dx) . J

Then X(B) is a finite charge on CAE. Lemma 1. For an arbitrary bounded CAE-measurable function fly), Xp(dx)Jyf(y)v(dy I x) = 5f(y)X(dy).

(6)

Equation (6) can be written more expressively in the form I x)p(dx)

Jxp(dx) f(y)v(dy I x) =

(7)

It will be sufficient to prove the assertion of the lemma for the case in which p(A) is a measure. We then have X(B) < cp(X) and, by virtue of Lebesgue's theorem on the integration of a series of nonnegative functions (cf. Theorem 1, Section 5, Chapter II), X(B) is a measure on e.. Let K denote the class of functions for which (6) holds. Then K contains the characteristic functions of

the sets e, it is linear, and therefore it contains all simple funcFurthermore, K is closed under the operation of passing to the limit with respect to monotonic sequences. Consequently it contains all nonnegative and all bounded e, measurable functions. This completes the proof of the lemma. Let us now turn to formula (4). Suppose that to < u < t. By using equation (1) and the lemma just proved, we obtain tions.

W0(A) =

x W0(dx)Ja .P(u, y, t, A)P(to, x, u,

xOx =

J

P(to, x, u,

dy)

dy)Wo(dx))P(u, y, t, A)

WW(dy)P(u, y, t, A) ,

This equation can be written T[to,t] = TCu.t]T[to,u]

(to < u < t)

(8)

1.

TRANSITION PROBABILITIES

301

Consequently if we consider the set of operators T[to.t] as a function of the interval [to, t], it is in a certain sense a directed noncommutative multiplicative family on the interval. * 2. If we know that the system is in a state x at an instant t, the mathematical expectation of a function f(e(v)) of the state of the system at the instant v > t is given by the expression

ft(x) = f(y)p(t, x, v, dy),

t
(9)

where f(y) is bounded and 0-measurable. Here, (10) fv(x) = f(x) Formula (9) defines an operator S[1°v] defined on the normed

linear space B of all 0-measurable bounded functions with norm I

I f h= sup . E X ix) I. Obviously, If (x) 11 < If(x) 11, S [t w] (1) = 1, so that 11 S [t, v]

By using formula (1) and Lemma 1, we obtain for t < u < v, fe(x) = 5fiy)P(t, x, v, dy) _ f(y)JXp(u, z, v, dy)p(t, x, u, dz) = XP(t, x, u, dz)J f(y)P(u, z, v, dy)

=

fu.(z)p(t, x, u, dz) JX

or

S[t,v" = S[t,ujS1u.tJ

(t < u < v) ,

(11)

that is, the operators S[t,v] also constitute a multiplicative family on the interval, although it is directed in a different way from that of the family of operators T[t,v].

Definition 2. A Markov process is said to be homogeneous in time if the transition probabilities p(t, x, u, A) as functions of t and

u depend only on u - t: P(t, x, u, A) = p(u - t, x, A)

For processes that are homogeneous in time, the operators T[t,vI and S[t,v] depend only on the single scalar parameter v - t > 0. Here the Chapman-Kolmogorov equation becomes * For a noncommutative multiplicative family on an interval, the directedness characterizes one of the two possible orders of the factors corresponding to the given partition of the interval.

JUMP MARKOV PROCESSES

302

P(t + v, x, A) = J

x

P(t, x, dy)P(v, y, A) ,

and the relations (8) and (11) become Tt+r = TtTr

t>0,

St+r=SrSt,

(12)

(13)

z>0,

(14)

where Tt = 7'[t°.t°+t7,

St = S[t°'t°+t7

.

Here T° = S° = I, where I is the identity operator. It follows from equations (13) and (14) that the operators Tt (St) commute with each other. If Z is the infinite interval [a, - ), then Tt (St) constitute semigroups of operators. Markov processes can be classified according to their phase spaces. In this sense the simplest of them are Markov processes with finitely or countably many states, that is, processes for which X consists of a finite or countable number of points. In such a case it is sufficient to give the transition probabilities for one-point Specifically,

sets.

if we let the points of X be denoted by the

and if pij(t, u) = p(t, i, u, j) is the probability of transition into the set consisting of the single point j, then p(t, i, u, A) = Y, pij(t, u) (u > t) letters i, j,

jeA

The Chapman-Kolmogorov equation for the probabilities pij(t, v) is written Pij(t, V) = EP:k(t, u)Pkj(u, v),

t < u < v, i, j e X .

(15)

In the homogeneous case we write pij(t, v) = pij(v - t), and equations (15) take the form t > 0, r > 0 (16) Pij(t + v) = E P;k(t)Pkj(z), keX

In the following section the transition probabilities of a homogeneous process with countably many states will be considered in greater detail. 2.

HOMOGENEOUS PROCESSES WITH COUNTABLY MANY STATES

Let us consider a homogeneous Markov process in the broad sense with a countable set of states J and with transition probabilities {pij(t); i, j e J}. It follows from Definition 1 of Section 1 that the transition probabilities satisfy the following relations: a.

pij(t) >0,

t>0;

2.

HOMOGENEOUS PROCESSES WITH COUNTABLY MANY STATES

b.

303

Ep;;(t)=1;

ICJ

t>0,s>0. paj(t + s) _ keJ P:k(t)Pk,(s), To these three conditions we add a fourth C.

1

d.

limpaj(t) = P.j(0) =

Sa;,

a;; = {0

if i=j, if i = j

,

which we shall assume throughout to be satisfied. From the probability point of view condition d indicates the stochastic continuity of the process at t = 0 no matter what the initial state of the system was. Sometimes condition b is replaced with the weaker condition,

l;EJP:7(t) < I. The case E,E, p,,(t) < 1 can be interpreted as follows: A system

finding itself in the ith state at some instant of time will, with positive probability equal to 1 be absent from the phase space after an interval of time t. In other words, there are too few points in the phase space to describe all possible states of the system. Let us agree to call processes of this type improper

One can easily see that by adding a certain set of points to the phase space, we can extend the domain of definiMarkov processes.

tion of an improper Markov space and turn it into a Markov process

in the proper sense, without changing the given transition probThis can be done by simply adding to the phase space an "absorbing" "infinitely distant" state "oo". We define abilities.

J*=JUpi-(t) = 1 -EP,j(t) jcJ

p-M) = 0,

i e J,

p,,,,(t) = 1

It is easy to see that the set of transition probabilities {p,;(u)}; i, j e J* constitutes a Markov process in the proper sense. To prove

this it will be sufficient to show that condition c is satisfied. We have P.3(t + s) = EaEJ P.a(t)Paj(s) = EaGJ Pia(t)Pa,9(s) +P$-(t)P-9(s) = EaEJ* Ptia(t)Pa1(s),

P,,,(t + s) = 0 = EaEJP,a(t)Paj(s) + P--(t)p.,(s) = EaEJ* P-a(t)Pa,(s),

j e J,

p__(t + s) = 1 = Eap,, (t)Pam(s) + p--(t)pao-(s) E J// a EJ*

PO.(t)paoo(s) ,

i, jeJ,

JUMP MARKOV PROCESSES

304

pp(t)Pa-(s) p;-(t + s) = 1 - EaeJP;a(t + s) = 1 - EaeJEPeJ = 1 - EPeJ Pap(t) E Paa(s) = 1 - EPeJ p (t)(1 - PPm(s)) aeJ = Pa_(t) + E pp(t)Pam(s) = P.-(t)P--(s) PeJ

+E pp(t)Pp-(s) = EPeJ Pip(t)Pp-(s) PeJ

Thus improper Markov processes do not actually broaden the class of Markov processes. The general properties of transition probabilities of homogeneous Markov processes are also properties of transition probabilities in improper Markov processes. Furthermore, for improper processes the quantity p;-(t) = 1 - Eae, ptia(t) enjoys the same properties as the transition probability p;a(t) for a e J. It might be noted that the quantity p;-(t) is a nondecreasing function since as we have just seen, s>0 p;-(t + s) = p;-(t) + E Pip(t)Pp-(s) > P;-(t), PeJ

We shall therefore confine ourselves to a consideration of Markov processes in the proper sense. Theorem 1. The transition probabilities of a homogeneous Markov process with countably many states are uniformly continuous for t > 0 (with i, j fixed). Proof.

Suppose that h > 0.

Then

pij(t + h) - P.,(t) = EaeJ Ptia(h)Pa,(t) - P.,(t)

(1 - P..(h))P.,(t) +a E p;a(h)Pa,(t) eJ\[tif from which it follows that

- (1 -

pij(t + h) - P.,(t) < aEe J\{i} P:a(h) = 1 - p;.(h)

that is, I pij(t + h) - Pa,(t) I < 1 - p::(h) For negative h it follows from the preceding inequality that

IP.,(t+h)-P.,(t)I <1 -

I), (1) from which the uniform continuity of the function p;;(t) for t > 0 follows.

Let us look at the question of the differentiability of the transition probabilities pa;(t). Theorem 2. The limit q; = limh.O [{1 - p;;(h)}/h] < - always exists, and for arbitrary t > 0, {1 - p;;(t)}/t 5 q;. Proof. Let us set qi = limh.O [{1 - p;a(h)}/h]. If q; = 00 , then q;, exists and q; _ oo . Suppose that q; < oo.

2.

HOMOGENEOUS PROCESSES WITH COUNTABLY MANY STATES

305

Let us show that 1

- Pma(t)

(2)

qti

t

for arbitrary t > 0. Lets and t denote arbitrary positive numbers. There exists an h such that 1 - P44(h)

< qt + 2

and 1 - pa,.(s) < st/2 for all 0 < s < h. Let us set t = nh + s for 0 < s < h. It follows from condition c that p%;(t + s) > p;;(t)p;;(s). Therefore Pi;,(t) > [Pati(h)]nPja(s) > (1 >_

[1

- h(qa +

- nh(q; + 2)](1

>1-

2

-t(q,+

2

2

)/ '(

-6t

1

2)

)

2)= 1 -t(q;,+E)'

Taking the limit as s 0 in this inequality, we obtain inequality (2). From that inequality it follows that lim 1 - P%ti(t) < t

t-.4

qb

so that 1 - P.=(t) = Jim 1 - Pti:(t) = Jim 1 - Pas(t) q: = lim t-.o t-.o t

t

two

t

which completes the proof.

Theorem 3. If qi < -, then for all t > 0 the derivatives pi ;(t) for j e J exist and are continuous.

They satisfy the following relations:

t>0, s>0,

P'i,(t + s) = E P%k(t)Pk,(s) keJ

E Pik(t) = 0 ,

kEJ

keJ p k(t) I

Proof.

2q.

We set

t + s) = Pa,(t + S) - Pai(t) s

On the basis of condition (c) and inequality (2) we have

JUMP MARKOV PROCESSES

306

O;j(t, t + s) ? Pij(t) Pii(s) - 1 > -gipij(t) S

(6)

Summing this inequality over an arbitrary set M C J of indices, we obtain

EAij(t,t+s)> -qc

jeM

Ai;(t, t + s) = 0, we have

Since jeM

Aij(t, t + s) = -E Aij(t, t + s) < qi

jeM

Thus

E Aij(t, t + s) < gi

jeM

(7)

Let P denote the set of all j such that A,,(t, t + s) > 0. Then

EAi,(t,t+s) E IAij(t,t+s)I =jePE4j(t,t+s)+ jeP

jeJ

that is

IIoij(t,t+s)I <2qi jeJ

(8)

In particular, I Ajj(t, t + s) < 2qi; that is, the functions pi,(t) satisfy a Lipschitz condition. Therefore pi;(t) = 8i; + f pa;(u)du, where p;j(t) is a Lebesgue-integrable function and p,,(t) has a derivative p; ;(t) almost everywhere.

It follows from (8) that for t > 0, E jP:j(t)

2gi

jeJ

almost everywhere.

Using inequality (9), we obtain t

0

(9)

t

jeJ 0P!,(u)du = OieJ p',(u)du

so that EieJPzj(t) = 0 for almost all t > 0. Let us now consider the case in which t = v, for which inequality

(9) is satisfied and the derivatives p',(z) exist for arbitrary j. Let us begin by setting for t > v, A = EkeJP%k(v)Pkj(t - v)

and let us consider the difference Ai;(t, t + s) - A. Since Aij(t, t + s) = E Aik(Z, v + s)Pkj(t - v) keJ we have

2.

HOMOGENEOUS PROCESSES WITH COUNTABLY MANY STATES

307

Pij(t) < L Pik(v)Pkj(t - z) keJ

for almost every t by Fatou's inequality and (6). Now

0 = EjeJPij(t) = EjeJE keJ Pik(z)Pkj(t - Z.)

=0 and hence

Pij(t) _

keJ

Pik(z)Pkj(t - Z)

(10)

for almost every t. However, by (9), the right-hand side of (10) is uniformly continuous in t > z. Hence pij(t) is absolutely continuous with a derivative which is uniformly continuous on the complement of a null set; i.e., pij(u) is essentially continuous. Since pij(t) = Jzpij(u)du we conclude that pij(t) is differentiable for all pij(z) + t >-_ z, and that (10) holds. Since z can be chosen arbitrarily small, it follows that for arbitrary t > 0, the derivative pij(t) exists and

pij(t) = E pik(z)pkj(t - z) keJ

for arbitrary z < t

.

Furthermore, if relations (4) and (5) are satisfied for some t = v, it follows from (10) that they are satisfied for arbitrary t > T. To prove this we merely sum equation (10) over all j G J. Furthermore, it follows from (10) that the function pij(t) is continuous for all t > 0. This completes the proof of the theorem. Let us now show that finite derivatives pij(0) exist for j i. (Here we do not use the restriction qi < oo.) To prove this we need: Lemma 1. Suppose that for some i, j, H > 0 and s,

1 - Pii(t) < s , 1 - Pjj(t) < s , 0 < t < H . Then for nh < t < H, where n is an integer, Pij(h) < Pij(t) n

1

1-3s

.

Proof. Let us observe the states of the Markov system at the instants 0, h, 2h, , nh, where nh < t. We let pk denote the

probability that the system leaving the ith state at the initial instant

t = 0 and observed at the instant rh, where r = 1, 2, , k, falls in the jth state for the first time at the instant kh. We let p = Lk=,Pk denote the probability that the system leaving the state i

JUMP MARKOV PROCESSES

308

is in the state j at one of the instants kh, k = 1, 2,

, n.

We

have

>Pzj(t) > k=1 EPkpjj(t - kh) > (1 - s)P so that p 5 {s/(1 - s)}.

Let Qk denote the probability that the system leaving the ith state will return to it at the instant kh without entering the state Then , k - 1. j at the instants rh, for r = 1 , 2, Qk

->Pa: ( kh

)-

P:->1-s-

k-1

E

1-E

We now have n

kh)

AM) >- Ek=1Qk-1PiJ(h)Ptj(t

>- n(1 - s)(1 - e -

s

1

pz;(h)

(1 - 3s)np8;(h) ,

from which the assertion follows. Theorem 4. lim t-+o

The finite limits

Pi#) = 4a, < 0

t

i:,E j

always exist.

Proof.

Let us choose H > 0 so small that the conditions of

Lemma 1 are satisfied. Let t and h denote an arbitrary pair of positive numbers not exceeding H. We set nh 5 t < (n + 1)h. Then on the basis of Lemma 1, Pos(h) h

<

Pas(t)

1

= t-h 1-3s

By letting h approach 0 we obtain lira Ls(h) < Pas(t). h

h--.o

t

1

1 - 3s

<0

If we then let t approach 0, we get lim &(h) < lim P:;(t) h--.o

h

- -F-- -o

t

1 - 3s

< 00

Since s can be chosen arbitrarily small (though positive), it follows

that the limit superior and the limit inferior of the ratio p;s(t)lt for t > 0 are equal. This completes the proof of the theorem.

2.

HOMOGENEOUS PROCESSES WITH COUNTABLY MANY STATES

309

REMARK 1. In the case of a finite number of states it follows from the equation

1 - pii(t) =j E iPij(t) E \}1f

(11)

and Theorem 4 that the derivative p;,(0) = -qi exists and is finite, and that

qi =jEJ\{i} Z gij

(12)

If the number of states is infinite, Theorem 4 does not imply finiteness of qi. Also, finiteness of q;, does not imply equation (12). In the general case equation (12) is replaced with the inequality

q: ?j EJ\{i} E qij

(13)

which we obtain by using only a finite number of terms on the right side of (11), dividing the resulting inequality by t > 0, taking

the limit as t

0, and then letting the number of terms on the

right side of the equation approach co so that their sum ultimately coincides with the right-side of inequality (13). Equation (3) is proven for t > 0. If we set t = 0 this equation becomes formally P:j(s) = EkeJP:k(0)Pkj(s) or

i e J,

p:j(t) = -gipij(t) + keJ\{i} E gikpkj(t),

t>0.

(14)

Equations (14) are called the first Kolmogorov equations. Let us find conditions under which they hold. Theorem S. For the transition probabilities pij(s) to satisfy the first system of Kolmogorov equations for given i and arbitrary j e J, it is necessary and sufficient that

qi < °°, ing.

.EJ\{i}

qij = qi

(15)

Proof of the Necessity. If qi = 0, equation (14) has no meanSuppose that qi < o a. Let us sum (14) over all j e J. On

the basis of (4), 0 = - qi + E qik keJ\{i}

which proves the second of equations (15). Proof of the Sufficiency. Suppose that h > 0.

Cij(t + h) - Pij(t) h

- 1 - Pii(h)pi h

t)

+

E

keJ\{i}

We have Pik(h)Pk (t) h

JUMP MARKOV PROCESSES

310

Let us choose a finite set of indices J' c J (where i 0 J') so that

E qik < e .

keJ\J'

Then plo(t) 0
Pii(h)

1

_

h

el


keJ\J'\1i1

LL(h) h

k

Pik(h) h

as

> qi - kei, qik <

Consequently lhm -0

Pit(t + h) - Pit(t) h

- {q1p1,(t) + kE, gikpki(t)}

Remembering that the derivatives p!j(t) (for t > 0) exist (cf. Theorem

3) and taking the limit as s , 0 in this last inequality, we obtain equation (14).

We recall that in the case of a Markov process with finitely many states, equations (14) follow from the continuity of the transition probabilities. Furthermore, in this case the transition probabilities

satisfy yet another system of differential equations, known as the second system of Kolmogorov differential equations.

We have (for h > 0) pit(t + h) - pit(t) = pit(t)Ptt(h) -

1

+

ke

pik(t)Pkhh)

.

From this it follows that the derivative of the function pit(t) satisfies the equation

pik(t)gkt, i, j e J . dpi'(t) _ _pit(t)gt + 1 keJ\fj) dt

(16)

As we can see, the derivation of equation (16) remains valid for processes with a countable set of states if we assume that q, < oo

and that the relations lim Pkt(h) = qk t h-0

(17)

h

are satisfied uniformly with respect to k. This brings us to: Theorem 6. If the transition probabilities pit(t) are continuous

for t = 0, if qt < -, and if relations (17) are satisfied uniformly with respect to k, then the system of equations (16) holds.

The system (16) can be carried over to unconditional probabilities pi(t) for i e J. If the pk(0) are the "initial" probabilities (the probabilities that at the initial instant t = 0 the Markov system

3.

JUMP PROCESSES

311

is in the kth state), then at the instant t the system is in the state i with probability Pi(t) = EkeJPk(0)Pki(t)

Multiplying equations (16) by pi(0) and summing over all i e J, we obtain dpi(t)

= -P,(t)q, + keJ\til E p,(t)gk, .

dt

(18)

The validity of our termwise differentiation of the series is easily shown. A homogeneous process is stationary if pi(t) = const. It follows from (18) that the unconditional probabilities pi(t) = pi

for a stationary process satisfy the system of equations

E Pkgk, = P,q,

(19)

keJ\ii}

3.

JUMP PROCESSES

We might expect that under sufficiently regular conditions a Markov system with countably many states would behave as follows:

In the course of some random interval of time the system E is in the initial state, after which E changes over to another state with definite probabilities, where it stays for some random interval of time, and so forth. But processes of such a kind can be considered in an arbitrary phase space X. In this section we introduce a class of Markov processes that can be regarded as a generalization of a process with a countable number of states. Here we do not require that the process be homogeneous in time, and the phase space X is arbitrary, but we impose on the transition probabilities restrictions more stringent than those discussed in Section 2. When we speak of a Markov process here, we mean a Markov process in the broad sense. Let us consider a Markov process with phase space X, with fixed u -algebra 0. Let Z denote a finite interval of time, let p(t, x, s, A) denote the transition probabilities for s > t, where s, t e Z, x e X, and A e 0. For s = t we have p(t, x, t, A) = xA(x), where XA(x) is the characteristic function of the set A. A generalization of condition d in Section 2 is the condition that the transition probabilities be continuous at s = t: lim p(t, x, s, A) = X,,(x) . 81t

But we impose additional requirements.

Suppose that

JUMP MARKOV PROCESSES

312

p(t, x, s, A) - XA(x)

a.

s-t

--- q (t ,

x , A)

(1 )

uniformly with respect to (t, x, A), where t e Z, x e X, and A e 0, as s approaches t from above. b. For fixed (x, A) (where x e X and A e F8), the function q(t, x, A) is continuous with respect to t e Z and uniformly con-

tinuous with respect to (x, A). It follows from (a) that for some k > 0

q(t,x,A)
c.

for all

teZ, Definition 1.

xeX,

(2)

Ae0.

A Markov process in the broad sense that satisfies

conditions a and b is called a jump process. It follows from the definition of q(t, x, A) that it is an additive function of the set A e 0. On the left side of the relation (1) is a charge, depending on the parameter s, that converges to a limit uniformly as s approaches t from above. Therefore q(t, x, A) is a charge on 0 (cf. Section 1, Chapter II). Furthermore, it is obvious that q(t, x, X) = 0 ,

q(t, x, A) = lim p(t, x, s, A) >_0

s-t

81t

for x0A,

q(t, x, {x}) = -q(t, x, X\{x}) < 0 , where {x} is the set consisting only of the point x. Let us set q(t, x) = -q(t, x, {x}) = q(t, x, X\{x}) 11(t, x, A) =

q(t, x, A\{x}) q(t, x)

for

XA(X)

for q(t, x) = 0

q(t, x)

0, (3)

Then l1(t, x, A) is a probability on 0 for fixed (t, x). It is easy to obtain a probabilistic interpretation of the quantities q(t, x) and II(t, x, A). It follows from (1) that p(t, x, t + At, {x}) = 1 - (q(t, x) + s)Ot , where s -* 0 as At , 0. Thus, up to infinitesimals of order higher

than At, the quantity q(t, x)Ot is the probability that a system in state x at the instant t will not be in that state at the instant t + At (where At > 0). Furthermore, for q(t, x) z 0, H(t, X, A) = lim p(t, x, t + At, A\{x})

At-.o p(t, x, t + At, X\{x})

(4)

3. JUMP PROCESSES

313

which can be regarded as the conditional probability that the system

in the state x at the instant t and leaving that state at that instant will, as the result of a jump, be in the state A. This interpretation of the function II(t, x, A) will be proved in Section 7. The relation (1) can be rewritten in the form p(t, x, s, A) = XA(x) + q(t, x, A)(s - t)

+ r(t, x, s, A)(s - t)

,

(5)

where r(t, x, s, A) is some charge whose absolute variation approaches

0 uniformly with respect to t and x as s approaches t from above (where t e Z).

In particular, it follows from (5) that IIP(t,x,s,A)-XA(x)II
(6)

is the norm of the charge in the space W (cf. Section w 1, Chapter II) with convergence in variation and where kl is a constant not depending on (t, x) E Z x X. Let us now look at the smoothness properties of the operators

where

T[to,t] and S11°'] introduced in Section 1. Let us begin with the operator T[101. Suppose that to is fixed and that w is an arbitrary

finite charge defined on 0.

For t > t' > to we have

T[to,t]w - T{to,t']w = f wt,(dx)[p(t', x, t, A) - XA(x)] X

where wt(A) _

(7)

It follows from (6) and (7) that

w(dx)p(to, x, t, A). X

II (T[t°'t' - T[to,t'l)w II = 11 wt - wt, 11 < kl(t - t')II w II

Furthermore, it follows from (7) and (5) that (T[to'tJ - T[to,t'3)w(A)

_ (t -

t')hXq(t', x, A)wt,(dx) + Xr(t', x, t, A)wt,(dx)]

Let s denote a member of the interval (t, t') and define J r(t', t) = sup I r(t', x, t, A) I

The following estimate o b tains if w

wt(A)

- tt (A) -

Jxq(s, x,

I

t - t'

I

.

is small:

A)wa(dx) < II w r(t', t)

+ JX[q(t', x, A) - q(s, x, A)]w8(dx)) + Ox q(t', x, A)(wt(dx) - w8(dx))


(8)

JUMP MARKOV PROCESSES

314

where, by virtue of condition b in the definition of a jump process, s= sup Jq(t', x, A) - q(s, x, A) I --*0 as t'.->s, x,A

r(t', t) , 0 as t', t -- s by virtue of condition a, and wt - w$ --' 0 in accordance with (8). Theorem 1. As a function of t, the operators T[to,t7 of a jump process are uniformly differentiable and dT [ta,t7

dt

= QtT [tu t7 , Qtw = w(dx)q(t, x, A)

(9)

Formula (9) means that the distribution ftt(A) of the system Y

in phase space at the instant t is differentiable with respect to t and satisfies the equation dlit dtA)

=

q(t, x, A)lpt(dx)

(t > t0)

(10)

,

to which we still need to add the initial condition pto(A) = p(A). This is the so-called "second Kolmogorov equation." With the aid of the functions q(t, x) and II(t, x, A), equation (10) can be rewritten as follows (by setting q(t, x, A) = q(t, x)[(II(t, x, A) - XA(x)1). d d(A) _ -JAq(t, x)pt(dx) +

q(t, x)II(t, x, A)ljt(dx)(t > to) (11)

If we choose p(A) = XA(y(,), that is, if we assume that at the instant to we know precisely the state of the system E, then pt(A) = p(to, yo, t, A)

and we obtain the following corollary to Theorem 1: Corollary. The transition probabilities p(to, y, t, A) of a jump Markov process are differentiable with respect to t, and aP(to, y, t, A)

=

f xq(t, x, A)P(to, y, t, dx)

+

q(t, x)p(to, y, t, dx)

q(t, x)II(t, x, A)p(to, y, t, dx), t > to

.

(12)

Let us now turn to the operators S[t°°I with v fixed and t' e (t, v).

Using the notation of Section 1, we have

3.

JUMP PROCESSES

315

(S[t.v] - S[t/,v])f = f (x) - f ,(x) =

5[f(z) - f,,(x)]

x P(t, x, t', dz) = (t' - t)[f,(z) - f,(x)]q(t, x, dz) +

J

[f,(z) - f,(x)]r(t, x, t', dz) ,

(13)

from which we get I l.f (x) - ft,(x) I <

Furthermore, if

I

t' - t I Ilft,(x) II <

t' - t I

Il f(x)

I

.

(14)

t' - t is small with t < t, < t', we obtain I

S[t,v'(f) - SEt"NP _ f .fo(z)q(ta, x, dz)

t' -t

<

[ff ,(z) - .f o(z)]q(to, x, dz)

fto(z)[q(to, x, dz) - q(t, x, dz)

+ [ft,(z) - ft,(x)]r(t, x, t', dz)


supx,A I q(to, x, A) - q(t', x, A) I --> 0 as

t' --s to.

Theorem 2. As functions of t, the operators S[t,v] of a jump Markov process are uniformly differentiable and satisfy the condition d vl Qt = xf(z)q(t, x, dz) dt = Qt S, (15)

In particular, the function ft(x) _ entiable with respect to t and a attx)

_

f ft(z)q(t, x, dz),

f(z)P(t, x, v, dz) is differ-

x

t < v,

fv(x) _ fix)

(16)

t
(17)

or

a tt(x) = q(t, at

x)rft(x)

- 5ft(z)ll(t ,x, dz)], x

Equations (16) and (17) are called the "first Kolmogorov equations." They are integro-differential equations for determining the function f (x). The equation (18) A(x) = f(x) , together with the condition of continuity of the function ft(x) at t = v, plays the role of a boundary condition (that is, of a sup-

JUMP MARKOV PROCESSES

316

plementary condition on the endpoint of the interval of time for which equation (17) is applicable). Corollary. The transition probabilities p(t, y, v, A) are differentiable with respect to t and they satisfy the condition v, A)

aP(t,

- J xp(t

at

z, v, A)q(t, y, dz)

= q(t, y)[P(t, y, v, A) LL

P(t, z, v, A)II(t, y, dz)] (19) fX

for t < v. In the case in which the space X is countable (X = J), we introduce the probabilities p;;(t, v) = p(t, i, v, { j}) just as in Section 1. For a jump process, the limits Jim P:5(t' v) vlt v - t = q ;(t),

i#j,

lim 1 - P:ti(t, v) = qi(t) ,It v-t exist uniformly with respect to (1, j, t), the functions q;;(t) and q;(t) are continuous, and q ,(t) = E;E,\W q%;(t).

Equations (12) and (19) take the forms api_v, v) ___ , v) at

= EiPbk(t, v)q

= q;(t)P,1(t, v) -

-pa1(t, v)gt(v) ,

(20)

gik(t)Pk,(t, v)

(21)

k#i

For processes that are homogeneous in time, the systems of equations (20) and (21) coincide with equations (16) and (15) of the preceding section, but there they were obtained under broader assumptions.

Suppose that there exists a point x0 such that q(t, x0) = 0 for all t e [a, v]. Then p(t, x, v, A) = XA(xo) for t e [a, v].

This means

that during the interval [a, v], the point xo in phase space is an

absorbing state: If the system E is in the state xo at the instant t e [a, v], then it will be in that state up to the instant v. Let us now investigate to what extent the function q(t, x, A) defines a Markov process.

We shall find it convenient to deal with the two functions q(t, x) = - q(t, x, {x}) and q(t, x, A) = q(t, x, A\{x}) .

From our description of jump Markov processes, we note that these functions satisfy the following conditions:

3.

JUMP PROCESSES

317

a'. For fixed t e Z and x e X, the function q(t, x, A) for A e 0 is a measure and q(t, x) = q(t, x, X). b'. For fixed x e X and A e 0, the function q(t, x, A) is continuous with respect to t, uniformly continuous with respect to (x, A), and 0-measurable for fixed t e.Z and A e 0. c'. q(t, x, A) < k for (t, x) e Z x X and A e 0. Suppose that we are given an arbitrary pair of functions q(t, x) and q(t, x, A) and suppose that

q(t, x, A) = -q(t, x)x4(x) + q(t, x, A) Let us look at the question of the existence of solutions to equations (10) and (16). Theorem 3. If the functions q(t, x) and q(t, x, A) satisfy conditions a', b', and c', then equations (10) and (16) have unique solutions.

The first of these is in the class of all finite charges and the second is in the class of all bounded functions ft(x) that are 0-measurable with respect to x.

We introduce the space W of all finite charges on 0 with absolute variation of the charge used as norm (cf. Section 1, Chapter II):

11w11 = Iwl(X)=w+(X)+w-(X) The operator Qt defined by formula (9), wt(A) = Q'[w](A) = -

q(t, x)w(dx) + `x q(t, x, A)w(dx) ,

(22)

A

maps W into W. Here (23)

11wt11
We also note that if I q(t, x, A) - q(t', x, A) I < s for all (x, A) e X x 8, then I wt - w, < 2e 1 w 11; that is, the operator Qt is continuous with respect to t. In particular, for arbitrary A e 0, the function wt(A) as a function of t e Z is continuous. We introduce the space C"' of continuous functions w = w(t) I

defined on [to, v] into W. Equation (10) can be written

P = P. + DP

(24)

,

where

/2 = /2(t) = pt(A), /o = 1jo(A) D is a linear transformation in CW, and

Dfi =

t to

QrP(z)dz

,

318

JUMP MARKOV PROCESSES

Let D" denote the nth power of D. Then - Dp2 11 < 2kI I /

11 D p1

11 DIP,

P2 I I

(t - t0)

- D"P2 11 < (2k)1 - P21 (t - On n!

Thus, some power of the mapping D is a contraction mapping and on the basis of the principle of contraction mappings, equation (24) has a unique solution in CW (cf. Kolmogorov and Fomin). We may treat equation (16) analogously. The substitution fi(x) = exp (- \tq(T, x)dv)gt(x)

(25)

reduces equation (16) to the somewhat simpler form

- J gt(z) exp ([q(v, x) - q(v, z)]dv)

aga(x)

x q(t, x, dz), gv(x) = f(x) ,

(26)

or

gt(x) = f(x) + Ixge(z) exp ([q(v, x) - q(v, z)]dv)q(O,

x, dz)dO . (27)

We introduce the space B of all 8-measurable bounded functions f(x) for x e X with norm I I f I I = sup, Ex I f(x) and we introduce the space CB of continuous functions f (x) of the argument t e [to, v] with range in B and with norm 1f(x) 11 = suptosts II.ft(x) 11

The linear operator D defined by Df (x) = ft(x) + J tl xga(z) exp ([q(v, x) - q(z, z)]dr)q(O, x, dz)dO , where gt(x) is a fixed nonnegative function in CB, maps the set of nonnegative functions in CB into itself. Here I Dftl)(x) - D.ft2)(x) II < k1I I fi"(x) - fE2)(x) I I v - t I

k1 = ek(

II D"ft(l)(x) -

D"f(2)(x) II C

to) klllftl)(x)

- y2)(x) 11

1V-t n!

that is, some power of the operator b is a contraction operator. Thus equation (27) and equation (16) have a unique solution in CB. This completes the proof of the theorem. Corollary. The transition probabilities of a jump Markov process are uniquely defined by the function q(t, x, A). REMARK 1.

The solutions of equations (10), (16), and (17) can

4.

EXAMPLES

319

be obtained by the method of successive approximations. Applying this method to equation (27), we obtain for the solution of equation (16) the formula

ft(x) _

.ftn)(x)

(28)

n=0

where

fi11(x) = exp (_ J tq(T, x)dv) (x)+ ft

Ln

1)(x)

f fen (z) exp (- 'q(-r, x)dz)q(6, x, dz)d8 .

(29)

For the transition probabilities p(t, x, v, A), we obtain the expressions p(t, x, v, A) = Y, p(n)(t, x, v, A)

(30)

n=0

where

p(0)(t, x, v, A) = exp - q(z, x)dz)xA(x), p(tt}1)(t x, v, A) = Jt JXp(n)(B,

z, v, A) exp Ht q(z, x)dr)q(B, x, dz)dO

(n=0,1,...).

The functions p(n)(t, x, v, A) have a simple probability-theoretic interpretation, which will be discussed in Section 7. 4.

EXAMPLES 1.

Random Signals

Imagine a channel that can be in one or the other of two states at an arbitrary instant of time: Either a signal is being transmitted or a signal is not being transmitted. We let these states of the channel be denoted by 1 and 0 respectively. Suppose that transitions from one state to the other correspond to a homogeneous Markov process and suppose that p0jh) = X h + o(h)

pi0(h) = ih + o(h) The quantities q; and q;; (for i, j = 1, 2), introduced in Section 2, have the values q0 = qo, = ?, q, = q10 = The second system of Kolmogorov's differential equations is written

JUMP MARKOV PROCESSES

320

P110(t) _ -ap10(t) + pp11(t), pi1(t) = -/2p11(t) + XP1o(t)

p'00(t) = -xpoo(t) + ppo1(t); Po1(t) = -pP111(t) + XPoo(t);

Solving these equations with initial conditions p,,(0) = P,,(0) = 1 and p10(0) = p01(0) = 0, we obtain Poo(t) = Xoe-(1+p)t + /o Po1(t) = xo[1 - e-(x+p)tj

p1o(t) = Po(1 - e-11+P)t),

Pu(t) = Xo + .toe-(2+p)t

where X, = (x)/(x + p) and po = (fi)/(X + ,"). As t --> -, the probabilities p;;(t) approach the values lim p;o(t) = Po = flo

limp;1(t) = P1 = Xo

t__

(j = 1, 2) ,

which are independent of i. On the other hand, the probabilities Po = 1-1o and p1 = x,o coincide with the probabilities of the states 0 and 1 in a stationary mode. If v(t) denotes the state of the channel

in the stationary case, then

and the covariance function of the process v(t) is calculated as follows:

R(t) = M(v(t + z)v(v))

-

m2 ,

M(v(t + v)v(T)) = P1P11(t)

from which we get R(t) = Xopoe-`1+N>t .

2.

Processes of Pure Growth

As an example of a Markov process with countably many states, let us consider a so-called process of pure growth.

The pos-

sible states in this example are the numbers 0, 1, 2, , and the sample functions do not decrease with increasing t, but with each jump they increase by unity. A process of this kind can serve as a mathematical model of processes of registration of a phenomenon that takes place at random instants of time. For example, when there is successive radioactive discharge from an original radioactive substance (the parent substance), another radioactive substance (the first daughter substance) is formed; from that is formed a second daughter substance, and so forth. Let us consider one particular atom of the original substance. In the course of a random instant of time this atom is in the original state. Then it disintegrates, becoming an atom of the first daughter

4.

EXAMPLES

321

substance, and so forth. Here the probability of disintegration of the atom in the instant (t, s) does not depend on the time the atom

has "lived" up to the instant t, and each state of the atom has a definite mean life span 1k = 1/X,. Examples of such chains of successive radioactive discharge are the transformations of natural These terminate with the formation of stable isotopes of lead. isotopes of uranium and thorium.

Let us state mathematically our assumptions regarding the random process e(t) in the form of conditions on the transition probabilities.

for

It is natural to assume that p;,,i+1(t) = X,it + o(t), pi;(t) = o(t) j - i 0, 1, pii(t) = I - xit + o(t). The first system of Kolmogorov equations take the form

i>0,

(1 ) and the second system of Kolmogorov equations takes the form -Aipi1(t) + xipi+1,,(t),

Pa2(t) = -Pi1(t)X1 + Pao(t) = -Pio(t)X0 -

Pi,j-1(t)X,-1,

j>1

,

(2)

To these equations we need to add the initial conditions pi,(0) _ ai;. Let us solve the second system. If i > j, then according to the definition of a process of pure growth we set pij(t) = 0 which of course is also a solution of the system (2) and satisfies the initial conditions. The system (2), for fixed i is then recursive. First let us determine pii(t) from the equation

P i(t) _ -Xipii(t), Pii(0) = 1 , and then let us determine successively the functions pi,;+1(t) and pi,i+Q(t), for each of which the system (2) is an ordinary linear differential equation. For the first step we have pii(t) = e-41. Then pi;(t) = x;-1J exp {-X;(t - s)}Pi,A-1(s)ds .

One can easily obtain a solution in explicit form for the system (2) by the usual methods of operational calculus. To do this, consider the transforms of the functions pij(t): Tij(z) = z10e-atp;,(t)dt . Then 0

z(,pi;(z) - 3ij)

and when we shift to the transforms of the functions, the system (2) takes the form

JUMP MARKOV PROCESESS

322 ZIPi.i(Z) _ -((X799iAZ) +

J>is

Z(Tii(z)

from which we get m

rii(Z) - Z+

j> i

Ti.7

f

x

and cpij(z) = (IIk- Xk)z/ r(z), where ' r(z) = k( z + Xk) the numbers Xk are distinct, then 1

_i

If all

1

*(Z) - k=i (Z + By means of this formula, the expression for rpi;(z) can be written in the form Z

111 Xk1) ,Pi,(z) = \k=i /k=x (Z + Xk)r,(-Xk)

Since q)ii(z) = z/(z + Xi) is the transform of the function e-xit, we have if j
t"i9 (t) =

i

(H XkJE k=x k=i

e-Zkt

*-X k) pii(t) = e-lit

,

.l > Z

,

where r=i

(7'r - X k)

r#k

These expressions are solutions in explicit form of equations (2).

In the simplest case, in which Xk = k%, the corresponding process is called a process of linear growth, and

pi5(t) = i(i + 1) ... (j - 1))i-i e-lkt X

k=i Xi-i(i - k)(i + 1 - k) ... (-I). I ... (j _ k)

= e-i1tCj-i(1 - e-2t)7-i 7-1

The corresponding distribution is called a Yule-Furry distribution. We note that for (0) = i, Me(t) = ielt, De(t) = ielt(elt - 1). In this example of a process of pure growth it is easy to analyze the possibility that a particle executing a Markov process will go out "to infinity" or, in other words, the possibility that it will disappear from the phase space. Let us suppose that at the

instant 0 the particle is in the state 0. It will be in that state for some random interval of time zo, after which it will jump to

4.

EXAMPLES

323

the state 1 where it will remain T1 units of time, and so forth. If the series

(5)

LJTk k=0

converges with positive probability, after an interval of time e the particle will disappear from the space. In the sum (5) the quantities zk are positive, and by virtue of the definition of a Markov process they are independent. (This fact seems obvious but requires proof, which will be given in Section 7). A condition for convergence of this series is given by: Theorem 1. The series (5) converges with probability according as the series

1

or 0

converges or diverges.

To prove Theorem 1 we use Corollary 1 of Section 4, Chapter The quantity zk, the duration of occupancy of the kth state, has the distribution function III.

P{zk < x} = 1 - Pkk(x) = 1 - e-xkx and M2k = 1/%k. Define zk = zk for zk < 1 and zk = 0 for zk > 1. Then

E MTk < E, Mvk, k=0

k=1

E P{-r" > 1} < k=O

,

Mzk

k=1

and on the basis of the corollary mentioned, the series (5) converges with probability 1 if the series (6) converges. On the other hand,

',P{Tk> 1}=I,e-1k k=o

(7)

k=O

e- k

E MZk = E e-1k + E

(8)

and if the series (6) diverges but the series (7) converges, then the series (8) diverges.

Thus in the case of divergence of the series (6), one of the series (7) or (8) diverges and consequently the series (5) diverges with probability 1. This completes the proof of the theorem. It follows from Theorem 1 that if the series (6) converges, then after a finite interval of time the system moves out to infinity with probability 1 (or disappears from the phase space). With probability 1, a process of linear growth does not go out to infinity.

JUMP MARKOV PROCESSES

324 3.

Birth and Death Processes

Birth and death processes are homogeneous Markov processes in which transitions from with possible states 0, 1, 2, , n, the state n into the states n - 1 and n + 1 are possible. Accordingly we set

i-j>1

pi.i+1(t) _ X it + o(t), pi,j(t) = o(t),

;

Pi,i-1(t) = lAit + o(t), pii(t) = 1 - (Xi + ui)t + 0(t)

In the case of the first system of Kolmogorov equations, the differential equations for the transition probabilities take the form pi,(t) = -(A. + iui)Pi1(t) + Xipi+1, (t) + P pi-1,A(t)

i=0,1,2,...(lc0=0)

(9)

and in the case of the second system of Kolmogorov equations, they take the form pin(t) = -pi,(t)(X1 + !A,) + Pi.g-1(t)Xj-1 + pi,j+1(t)1l;+1

(10)

For the unconditional probabilities pi(t) we have the system of equations

Pi(t) = -pi(t)(X + 1-t) + pi-1(t)Ai-1 + pi+1(t)lAi+1

i=0,1,2,

(11)

,

p_1(t) = 0

Let us find a stationary distribution of the probabilities, that that a distribution of the probabilities pi(t) for i = 0, 1, 2, satisfies the system (11) and does not change in time: pi(t) = const. For such a distribution the system of differential equations (11) is,

degenerates into a homogeneous algebraic system:

- (xi + Pi)Pi + Xi-1Pi-i + ti+1Pi+i = 0,

-x0Po + piPi = 0 . Suppose that pk > 0 for k = 1, 2, and as we easily find by induction, Pk =

X0X1 ... Xk-1

.

Then p1 = (A0/i1)p0

Po

(12)

a0A1... X,,k 11

(13)

//

and

Ilk = p0(1 + k=0

k=1

[-1P2

('// 'k ff

Theorem 2. For a stationary probability distribution to exist in a birth and death process it is necessary and sufficient that the series

4.

EXAMPLES

325 XoX,,l//...Xk

k=1

1

(i'k>0,k_ 1,2,...)

(14)

P1("2 ... )lk

converge.

It is interesting to note the connection between stationary distributions of a process and the so-called "final probabilities" P k(°° ), where Pik(°°) = limt--p;k(t) Let us suppose that the final probabilities ptik(oo) exist. When

we integrate equation (11) from h to h + T, divide by T, and then

take the limit as T

-, we obtain

-pti,(o)(Xi + N) + p:,j-1( °° )X,-1 + -p.o(°O )7o + p:1(°°

p:,j+1( °° ))Ui+i = 0, )!11 =

j>0

0;

that is, the final probabilities p;; (=) coincide with the stationary distribution (for fixed i) and they are independent of i. In technology, physics, and natural science there are many problems that involve birth and death processes. Let us look at some of these. 1. The servicing of lathes. Suppose that in lathes are serviced by a crew of s repair men. When a lathe fails to function properly

it is repaired immediately unless all of the repairmen are working

on lathes that have already failed in which case the lathe must await repair. The lathes are repaired in the order in which they fail.

Let us make the following assumptions: For an individual functioning lathe, the probability of getting out of order during an interval of time (t, t + At) is independent of t and is equal to x(At) = xAt + o(At), independently of the "history" of its operation

(the length of time that it has been in use, the number of times that it has become out of order, and the length of service) up to the instant t. Analogously, if a lathe is being repaired, the probability of its being put back into operation during an interval of time

(t, t + At) is equal to te(At) = At + o(At) and is independent of the nature of its work and its length of service up to the instant t. The lathes are used, get out of order and are repaired independently of each other. Let E denote the state of the industrial process. Let us agree

to say that E is in the state Ek if at a given instant the number of lathes being repaired or awaiting repair (that is, the total number of lathes not in operation) is equal to k. Subsequent removal of a single lathe from service denotes transition to the state Ek+1, and

completion of the repair of one of the lathes indicates transition

JUMP MARKOV PROCESSES

326

into the state Ek_1. Thus we have a homogeneous Markov system with finitely many states Eo, E1, , E.. It follows from our assumptions that Pk,k+1(Ot) = (m - k)XAt + o(At), k = 0, pk,k-1(At) = kaAt + o(At) for 1 < k < s , Pk,k-1(At) = s fcAt + o(At) for m > k > s , Pk,ktr(At) = o(pt), r>2;

, in - 1

;

that is, we have a birth and death process with finitely many possible states. In the preceding notation,

k=0, 1, ...,in; Ilk=SP for s!
?k = (m -k)X,

^k=ku for 0
A stationary distribution always exists. By virtue of equations (12) and (13) it is given by the formulas

Pk=c

(X)k

k<s,

po,

Pk = C m k(k

- 1)sk-a ...

(s + 1) (X)kpo,

s
Po=[1+Ck()k

k=0

(

C,k(k - 1) ... (s + 1) (x)k]_l

+ k_a+1

sk-a

which can be used to calculate the most suitable state for the number of lathes and the number of repairmen under specific industrial conditions.

In many cases we have the same type of set-up as in the preceding example, but with m = o o. Such a situation exists in the case of a central interurban telephone station with s lines serving, to all intents and purposes, an infinite number of subscribers. The subscribers are the units that need to be served, and the connecting lines are the servicing units. The number of the state of the system I indicates the number of subscribers requiring service at a given instant. If this number exceeds s, the subscribers must wait their turn for a free line. In contrast to the assumptions of the preceding example, we now assume that the probability that a single subscriber will ask for service in the course of an interval (t, t + At) is equal to x(At) = XAt + o(At), and the probability 2.

Telephone networks.

that more than one subscriber will require service in that instant is equal to o(At). These probabilities are independent of the number

5.

BRANCHING PROCESSES

327

of subscribers requiring service up to the given instant. We keep the assumptions of the preceding example regarding the mode of service for the subscribers. We then have a,k = X, ,uk = kIe for k < s, pk = she for k > s . For a stationary distribution to exist, it is necessary and sufficient that the series

S=1+

.t

k=o k!

+k

k

i

)k

1

1 $! sk-8

, `

o0

converge, that is, that a. be less than sp. In this case, a stationary distribution is given by the formulas llk

Pk

( k!\ / S,

0
Pk

s.sk-a

CFt/

S'

<,s

k>s.

3. Textile threads. A textile thread is a sheaf of fibers, the number of which varies from point to point along the length of the thread. If we assume that fiber length has a fixed negative exponential distribution independent both of the number of fibers at any point of the thread and of their lengths, and if we assume that the probability that the end of a new fiber will appear in a segment (t, t + At) of the thread is equal to XOt + o(At) and is independent of the number of fibers and their lengths, then the number of threads v(t) at the point t on the thread is a birth and death process with parameters X. = X, p = n p, where ,u is the reciprocal of the mean length of a fiber.

5.

BRANCHING PROCESSES

An important class of Markov processes with countably many states is the class of branching processes. Let us suppose that we

are observing a physical system Z consisting of a finite number of particles either of the same type or of several different types. With the passage of time each particle can disappear or turn into a group of new particles, independently of the other particles. Phenomena described by such a scheme are frequently encountered in natural science and technology, and include, for example, showers of cosmic rays, the passage of elementary particles through a substance, the development of biological populations, and the spread of epidemics.

JUMP MARKOV PROCESSES

328

A precise definition of processes of this type within the framework of the theory of Markov processes brings us to the concept of a branching process. Let n denote the number of distinct possible types of particles. The state of the system Y, at the instant t is characterized by an integer-valued vector , va(t)}

v(t) = {vl(t), v2(t),

where vi(t) is the number of particles of the ith type that exist at the instant t. Let us identify the state of the system Y, at the instant t with the vector v(t). With regard to the nature of the evolution of the system E in time we make the following assumption: For every particle that exists at the instant t, its subsequent evolution is independent of when and how this particle first appeared and of the nature of the evolution of all the remaining particles in at the instant t. Let a, ,8, denote n-dimensional vectors with integer-valued nonnegative coordinates a = {al, a2, , a}, ,8 = {b1, b2, . introduce the transition probabilities pa1(t1, t2) of the

, b,,j. We

system E

from the state a that exists at the instant t, to the state l& at the instant t2: p p(t1, t2) = P{v(t2) = N I v(tl) = a} .

Let {i}, for i = 1, , n, denote the state of the system E that consists only of a single particle of the ith type. The assumption regarding the nature of the evolution of the system Y, in time can be described by the formula TaTi

Pap(ti' t2) _

T7ft7

n

ai

11 II P{ij P(ij)(ti, t2)

i=1 j=1

,

//

(

1

)

i=1 j=1

where the summation on the right is over all possible vectors 8(ijl with nonnegative integer-valued components (i = 1, . , n; j = 1, , aa) that together constitute the vector 8. Here if ai = 0 we assume ai

JJ P{flp(i7J(t1, t2) = 1

j-1

Thus a branching process is a Markov process whose space of all possible states N is the set of all integer-valued n-dimensional vectors with nonnegative coordinates, and whose transition probabilities satisfy equation (1). In what follows, we consider only homogeneous branching processes, that is, processes for which pap(ti, t2) = Pab(t2 - t,).

5.

BRANCHING PROCESSES

329

Let us consider processes with a single type of particle (n = 1). Since in branching processes every particle evolves independently

of the others, we may assume that at the initial instant only one particle exists. With the passage of time it either disappears or turns into k particles that are all of the same type and constitute the first generation. Every particle of the first generation "lives" independently of the other particles, and the same probability-theoretic laws obtain for it as for the initial particle. At some instant of time the particle either disappears or turns into second-generation particles, and so forth. The entire process is described by a single integervalued random function v(t) that is equal to the number of particles existing at the instant t. By our assumption, v(0) = 1.

The set of possible states of the process is the sequence of natural numbers 0, 1, 2, Here 0 is an absorbing state: If v(to) = 0 then at all subsequent instants of time v(t) = 0. If, with probability 1, v(t) = 0 at some instant t, the process is said to be degenerate. In other cases the quantity v(t) vanishes after a finite interval of time with probability less than one. This probability is called the probability of degeneration of the process. It is possible that the quantity v(t) increases without bound in time. In the case of a nuclear reaction this situation can be interpreted as an explosion. Thus in the theory of branching processes we need to ask the following questions: What is the probability of degeneration of the branching process and what is the asymptotic behavior of the quantity v(t)? Let pi;(t) denote the conditional probability that the system consists of j particles at the instant t + v if there were i particles at the instant v. To solve problems that arise in the theory of branching processes, we shall find it convenient to use the method of generating functions. We therefore introduce the generating functions fi(z, t) of the distributions {pi;(t)}, for j = 0, 1, . . .,

.

i(Z, t) _ ik=o Zkpik(t),

IZI

1

.

The probabilities pi,(t) with i fixed correspond to the distribution of i independent random variables all having the same distribution with generating function f (z, t): fi(z, t) _ V(Z, t)]i

.

(2)

To determine the function f,(z, t), we may use either of the systems of Kolmogorov equations (cf. Section 2).

JUMP MARKOV PROCESSES

330

According to the general theory, the derivatives

lim 1 - P11(t)

lim P1,(t) = b; j zf- 1, t

o

t

= b1

t

t-.o

Let us suppose that

exist.

b1 = bo + > b; < oo

.

=2

Then we have the first system of Kolmogorov equations dPlj(t) _ -b1P1j(t) + dt

k=o

(3)

bkPks(t)

k#1

Multiplying both sides of equations (3) by z' and summing with

respect to j from 0 to o, we obtain af1(z, t) at

_ -b1f1(z, t) + j bkfk(z, t)(I z I < 1) , k=0

k#1

or on the basis of (2), af(z, t) = -b1f(z, t) + j bkfk(Z, t) at

k=0 k#1

where we set f(z, t) = f1(z, t).

Finally we obtain the following

nonlinear differential equation (4)

at = u(f) where

u(z)=b(, -b1z+jbkzk k=2

z

1).

(5)

To equation (4) we must add the initial condition f(z, 0) = z . (6) The solution of equation (4) with initial condition (6) can be written in the form

P(.f) - P(Z) = t, P(Z) _

u(z) '

(7 )

Let us suppose that the conditions under which the second system of Kolmogorov differential equations may be applied are satisfied. From the definition of branching processes we get the formulas

pkk(t) _ (1 - b1t)k + o(t) = 1 - kblt + o(t) pk,k_1(t) = k(1 - b1t)k-'bot + o(t) = kbot + o(t) Pk.k-7(t) = o(t), j > 2 , Pk,k+j(t) = k(1 - b1t)b,+1t + o(t) = kb;+1t + o(t), j ? 1

S.

BRANCHING PROCESSES

331

from which it follows (in the notation of Section 2) that q99 = jbl, q,,,j-1 = jbo, 79,9-k = 0, k > 2 ; q9,9+k = jbk+1, k >

1

.

Thus for the functions p1j(t), the system (16) of Section 2 takes the form dpdt(t) = (j + 1)bop1,9+1(t) - jb1p19(t) + (j - 1)b2p1,9-1(t)

+ ... + b5p11(t)

(8)

.

Multiplying (8) by zi and summing with respect to j from 0 to 00, we obtain a new equation for the generating function f(z, 1),

<1),

af(z't) =uz) af(z,t) at

oz

(

(9)

where u(z) is defined by equation (5). To this equation we add the initial condition (6). The solution of equations (9) and (6) takes the form

f(z, t) = *(t + (Z)) ,

(10)

where ar(t) is the inverse of the function du u(z)

t=P(z)= This solution coincides with (7). Example.

Let us set

u(z)=p-(p+q)z+gz2. In this case a particle disappears with probability pt + c(t) after an interval of time t, divides into two particles with probability

qt + o(t) at the end of the same interval, or keeps its original state with probability 1 - (p + q)t + o(t) for that interval. The probability of dividing into more than two particles is equal to o(t). The process in question coincides with a linear birth and death process (cf. Section 4, Example 3). ,P(z)

dz

We have

dz

= u(z) - p - (p + q)z + z2 1

q -p

In

z-1 z

p

(p

q)

q

For the inverse function z = i/r(t) we obtain the expression

JUMP MARKOV PROCESSES

332

1 - Rest(l-P)

z=fi(t)-

R=

1

P q

from which we get

z - ,3 - R(z -

1)e4t(1-P)

ftz+ t) = o + (P(z)) = z - R - (z Expanding f(z, t) in a series of powers of z, we get et(9-P)) -let(4-P) (1 - R)2(1 (Z, t) = 1 - et(v-P) + 1)eg1(1-fi)

1-1

[R -

n=1

et(9-P)

z

et(9-P) l n +1 J

R

which leads us to the formulas

1 - et(a-P)

Plo(t) _

(1 1)

,

1 - 1 et(v-P) R

et(vP)) _

pin(t) = (1 - 81)2 (1

P)

]n+4-P)

-

n = 1, 2, ...

,

,

(12)

Let us now consider the case in which p = q, which we have excluded up to this point. If p = q, then dz

R

'P(z) = o p(1 - z)2

-

1

z

z = 'i/r(t) = 1 -

p1-z

from which we get

Pt + Pt

+

z-.1

1-z

f(z, t) _ 'i/r(t + P(z)) = 1 -

1 + pt(1 - z) (pt)n-1

n=1 (1 + pt)n+'

Consequently plo(t) =

1

+t pt

(13)

,

p1n(t) _ (1(+tpt)n+1

n = 1, 2, ...

(14)

The following asymptotic relationships as t - are immediate consequences of formulas (11) to (14): If q < p, then 0 for n > 1, t plo(t) -- 1, pen(t)

if q > p, then plo(t) ---' R(R < 1), pin(t) -' 0

for

n > 1, t ---p o .

If q < p, the branching process degenerates with probability 1; that is, all particles eventually disappear. If q > p, the probability of degeneration of the process is equal to 8 = p/q < 1. However if

5.

BRANCHING PROCESSES

333

the particles do not disappear the number of them increases without bound in time, because P{v(t) > N I v(t) > 0} = 1 -

E P1k(t) --> 1 P10(t) k=1

for every N.

Let us look at the question of asymptotic behavior as t --> 00 of a branching process in the general case. In what follows we shall need the moments of the quantity v(t). Since we are using generating functions we shall find it convenient to introduce factorial moments and use them instead of ordinary moments. We define Mk(t) = M[v(t)(v(t)

- 1) ... (v(t) - k + 1)]

.

It is not difficult to set up linear differential equations that are satisfied by the factorial moments Mk(t). Let us suppose that Then for I z I < 1 the differential equation (4) can be differentiated with respect to z, and this gives us Ek=1 kbk < o o.

aM1(z, t) = u'(f)M1(Z, t), M1(Z, 0) = 1 at

(15)

,

where we set M1(Z, t) = a f(Z, t) =

Mv(t)z"(t>-1

u'(z) = -b1 + j kbkzk-1 k=2

It follows from (15) that M1(z,t)

exp

(u'fdt) (I z i < 1) 0

As z --> 1, u'(z) --> - b, + >k=2 kbk = m1 < - and f(z, t) --> 1, increasing monotonically and hence uniformly with respect to t. Consequently lim M1(z, t) = emit. Lebesgue's theorem,

On the other hand, by virtue of

lim M1(z, t) = lim Mv(t)z1(t)-1 = Mv(t) x11

x11

_

kpk(t)

.

k=1

Thus M1(t) = emit

(16)

.

This result is immediately generalized to factorial moments of higher orders. Lemma 1.

Suppose that

m,,=Y, n(n- 1)...(n-r+ 1)bn< oo, n=1

r= 1,...k. (17)

JUMP MARKOV PROCESSES

334

Then the factorial moments Mr(t), for r = 1, 2, , k, of the variable v(t) are finite and they satisfy first-order linear differential equations with constant coefficients.

Proof.

Let us differentiate equation (4) successively with respect

to z (for I z < 1). Setting Mk(z, t)

ak azk t)

= M(v(t)(v(t) - 1) ... (v(t) - k + 1)zl("-k)

,

we obtain aM2(z, t) at

= u'(.f)M2(z, t) + u'V)M'(z, t) (18)

aMk(z, t) = u'(f)Mk(z, t) + Fk(z, t) at

where Fk(z, t) is a polynomial in M1(z, t),

t) with coef-

, u(k'(f). To these equa-

ficients that depend linearly on u"(f), tions we add the initial conditions Mk(z, 0) = 0

, Mk_1(z,

(k = 2, 3, ...)

.

The solution of equation (18) is of the form 0t0

Mk(z, t)

= exp

u'(f)dt)\0 0Fk(z, z)

Using the same considerations as in the case of k = 1, we obtain by induction t

Mk(t) = lim Mk(z, t) = emit F,(1, z)e-m1rdv 2-1

0

Here Mh(t) obviously satisfies equation (18), in which we have set z = 1.

In particular,

M2(t) = m2 (em11 - l)emlt

0)

,

(19)

(m1 = 0) .

(20)

(m1

m1

M2(t) = m2t

In the study of the asymptotic behavior of a branching process,

an important role is played by the function u(z) (cf. (5)), which we now consider for real values of z. We note that

u(0)=bo>0,u(1)=bo-bl+bk=0, k=2 u"(z) > O

for z>0.

Here we assume that not all the bk (for k > 2) are equal to 0. Thus u"(z) is convex downward for z > 0 and hence has no more than one zero in the interval (0, 1). Let us turn to the definition

S.

BRANCHING PROCESSES

335

of the probability a of degeneration of the branching process v(t). Since the events v(t) = 0 constitute an increasing class of events, we have a = P{lim v(t) = 0} = lim P{v(t) = 0} = lim plo(t) t-.oo

t-.co

.

t-.co

Theorem 1. The probability of degeneration of a branching process coincides with the smallest nonnegative root of the equation u(x) = 0. If

ul(l)=m, =-b,+Ekbk<0 k=2 then for u'(1) < 0 the probability of degeneration is a = 1; if u'(1) > 0,

then a < 1. Proof.

If plo(t) = f(0, t), it follows from (4) that dpio(t) = u(pio(t)), Pm(0) = 0 .

dt

(21)

If bo = 0, then plo(t) = 0 is a solution of equation (21) and the theorem is trivial. Suppose that bo > 0. We note that if xo is the smallest positive root of the equation u(x) = 0, then plo(t) < xo for all t > 0, because if p,o(to) were equal to xo for to > 0, then by virtue of the uniqueness of the solution of equation (21) we would have plo(t) - xo, which is impossible. Furthermore since the limit a = limt__plo(t) < 1 exists, it

follows from (21) that the limit limt...p o(t) = u(a) also exists. But this implies that u(a) = 0 because otherwise the quantity t

Po(t) =

po(t)dt + po(to)

to

would increase without bound. Thus we have shown that a = xo.

If x0 < 1, then the function u(x) is an increasing function at the point x = 1 and the derivative u'(1) > 0 if it exists. On the other hand, if xo = 1, then u'(1) < 0. This completes the proof of the theorem.

Let us now investigate the asymptotic behavior of the proboc for degenerating processes (a = 1). Theorem 2. If m, = u'(1) < 0 and m2 = u"(1) < co, then 1 - plo(t) kerit for m, < 0

ability plo(t) as t

and

1 - plo(t) Proof.

Let us define

2

m2t

for m,=0.

JUMP MARKOV PROCESSES

336

q(t) = 1 - plo(t) The function q(t) satisfies the equation

dq = -u(1 - q(t)), q(0) = 1 Using the formula for finite increments, we obtain

= -u(1) + q(t)u'(E) =

dq

where a lies between plo(t) and 1. Since u'(x) is an increasing function and $ 1 as t -, it follows that u'(e) = u'(1) - e(t),

where e(t) > 0, and lim, e(t) = 0. Thus dq = q(t)(ml - e(t))

from which we get q(t) = exp (mit

-

o

We note that for e < < 1,

-)

0 < e(t) = u'(1) - u'(e) = u"(C)(1 < u"(l)(1 - Plo(t)) < m2emlt Therefore the integral for m1 < 0, q(t) sto kemlt

From this it follows that

e(t)dt is finite. where

k = exp (_°(t)dt)

Consider the case m1 = 0. We have

dq = -u(1 - q(t))

-u(1) + q(t)u'(1)

where 1 is a number in the interval (plo(t), 1). as t -4 00 , we have q 22 t) Wt

where e(t) - 0 as t m2t +

Since u"(e) ---> u"(1)

(mz + e(t))

From this it follows that 2

q(t) =

-

Jo

(2)d2 + 2

=+0\/. m2t

t

This completes the proof of the theorem. We shall supplement Theorem 2 with h-a result dealing with the asymptotic behavior of the probabilities plo(t) for degenerating

5.

BRANCHING PROCESSES

337

processes.

Since lim,-- p1,(t) = 0 (for n > 0), we have limt_ f(z, t) = 1. We define q(z, t) = 1 - f(z, t). For z = 0 we have q(0, t) = 1 - f(0, t) = 1 - P1u(t) = q(t) kem1t . We may assume that the same is true of the rate of decrease of the function q(z, t) at z 0 also. In connection with this we define q (z, t) = qq(t)) 1

f

(,

(22)

t)

We note that the function p1 (t) zn

.f*(z, t) = 1 - q(z, t) _

(23)

can be regarded as the generating function for the conditional distribution of the number v(t) of particles under the hypothesis that

it is nonzero up to the instant t. Theorem 3. If m1 = u'(1) < 0 and m2 = u"(1) < oo then as t --> oo the conditional distribution of the number of particles v(t), under the hypothesis that the process has not degenerated (v(t) zt- 0) up to the

instant t, approaches a definite limit, the generating function f*(z) of which is equal to

f*(z) = 1 - exp (mi__).

(24)

u Z)

Proof. Let us consider the function q?(z, t). It follows from (4) that cp(z, t) satisfies the equation (25) u(1 - q(t)p) + T u(1 - q(t)) Expanding the right-hand member of the equation obtained in

at

q(t)

accordance with Taylor's formula, we get

a

q(t)

[u(1) - q(t)q u'(1) +

(q()2 (u"(1) + -`1)]

q2(2 t) (u"(1) + E2)] + q(t) [u(1) - q(t)u'(1) + where s1 = u"(E1) - u"(1) and s2 = u"(e2) - u"(1), the number e1

(resp. e2) lying in the interval (f(z, t), 1) (resp. (f(0, t), 1)).

As t---

the functions s1 (for i = 1, 2) approach 0 uniformly in an arbitrary region I z I < p < 1. The preceding equation can be written in the form

ap at

q(t)T2 (m2 + s) + q(t)y (m2 + s2) 2

2

(25')

JUMP MARKOV PROCESSES

338

Beginning with some sufficiently large t, we have

at

< q(2t)2

p

(Mz +

2z) =

4mzq(t)q

so that t

q )(z, t) < q(z, to) exp (

The convergence of the integral

remains bounded as t written in the form Op

=

at

q(v)dr) 4 mz

.

q(z-)dr implies that the function q)(z, t)

Therefore equation (25) can be re-

q(t)y [m2(1 2

- q) + s] ,

q)(z, 0)

1-z

,

where s = sz - q)s1-. 0 as t -. co . Representing the solution of the last equation in the form t)

fq(z,

z) exp (tq(r)[m2(l - q.(z, r)) +

s]d-r)

,

we see that the limit lim, p (z, t) = K(z) exists. Furthermore, it follows from (25') that lim, dq /dt = 0. Since q(z, t) is an analytic function inside the disk I z I < 1 and since all the limit relationships that we have used hold uniformly inside every disk I z I < p < 1, it follows that the function K(z) is also analytic inside the disk and that lim ap(z, t) = dK(z) t__

az

dz

uniformly inside an arbitrary disk I z < p < 1. To determine the function K(z), we may use equation (9). Setting f(z, t) = 1 - q(t)q(z, t)

in that equation, we obtain u(z)q(t) ap(Z, t)

- q'(t)p(z, t) - q(t) a at' t)

Dividing this equation by q(t) and letting t approach oo , and remembering that q'(t)/q(t) - m1 (cf. proof of Theorem 2), we obtain

Here, K(0) = lim,

m1K(z) = u(z) ddzz) p(0, t) = 1. Thus

K(z) = exp

(z))

1 - f(z, t) ' q(t)K(z) = exp (mitt +

Jo u() z ))

5.

BRANCHING PROCESSES

339

which completes the proof of the theorem.

Let us now derive an expression for the mean number of particles at the instant t under the hypothesis that the process has not degenerated up to that instant. We have m*(t) = M{v(t) I v(t) > 0} =

(26)

,

9(t)

from which (keeping Theorem 2 in mind) we get the following asymptotic relations as t -->

00: 1

for m1 < 0 ,

k m*(t) P&

met

for m1 = 0 ,

2 emit

for m > 0 . 1

1-a

For m1 > 0, the number of particles v(t), under the hypothesis

that v(t) > 0, increases without bound. We define (t) _ "(t) m*(t)

Let us study the limiting behavior of the quantity v*(t) as t --> oo. As we might expect, the limiting distribution of the quantity v*(t) under the hypothesis v(t) > 0 will, if it exists, be a continuous Then M{v*(t) I v*(t) > 0} = 1.

distribution on the halfline [0, a), and therefore it is convenient for us to shift over from the generating functions to the characteristic functions. For the characteristic function g(X, t) of the random variable v*(t), under the hypothesis that v(t) r> 0, we have the value iXn

t)

_ n=1 exp (m*(t))

Pi(t) _

ff eXp \m*(t) ), t} - f 0, t) q(t)

g(t)

or

1 - f{exp ( g ('

9

t) -- 1 -

Consider the case m1 = 0.

m*(t))

tI (27)

q(t)

1

Theorem 4. If m1 = 0 and m2 < -, then lim p 2v(t) < x v(t) > 0} = t-

mgt

1-e

x .

(28)

Proof. Defining * (z, t) = 1 - f(z, t), we obtain from equation (4)

JUMP MARKOV PROCESSES

340

-

as = _u(l

[u"(1)

*r(z, 0) = 1 - z .

+ 6(t)],

2

Since the process is degenerate for m1 = 0, we see that * (z, t) -- 0 uniformly in the region I z I < 1 as t From this it follows that s(t) also approaches 0 uniformly with respect to z, for I z I < 1, 00 as t Integrating the last equation, we obtain *(Z1 t)

1

-.

1z

o(t)dt1 ,

_2_[ met

so that

* jexp ( m*(t)), t} +

m2 q(t) 2

q(t)

= lim

q(t)

t-

1 - exp

t

+

t

m*(t)

ix +

g(X) = lim g(A,, t) _

1

1 - ix

The function g(?) is the characteristic function of the distribution F(x) = 1 - e-z for x > 0, F(x) = 0 for x < 0. This completes the proof of the theorem. In the case m1 > 0, the quantity q(t) approaches the nonzero limit 1 - a. Therefore normalization with the aid of the function q(t) or shifting to the conditional mathematical expectations under the hypothesis that v(t) > 0 cannot play a significant role. Theorem 5. If m1 > 0 and m2 < =, then the quantity

v(t)e-m1t

converges in the sense of mean-square as t , 00 to the random variable 7) = l.i.m. v(t)e-""1t, whose characteristic function g(X) satisfies the functional equation

g(X)) exp (-

1

u(vu(v)(vl(v 1) ) dv)_

(29)

To prove the convergence of v(t) = v(t)e--1t in the sense of mean square, we use Cauchy's criterion. Suppose that t < t'. Then M(i (t) - v(t'))2 = MD(t)2 + Mi (t')2 - 2M(i (t)i (t'))

On the basis of formulas (16) and (19), MD(t)2 Gv' m2 m1

as

t -- 0

Using the definition of a branching process and the fact that it homogeneous, we obtain

is

S.

BRANCHING PROCESSES

341

M(v(t)V(t')) = MP(t)M{v(t') I v(t)} = Mv2(t)Mv(t' - t)

,

from which it follows (also on the basis of formulas (16) and (19)) that m2

M(v(t)v(t'))

m1

Thus M(v(t) - v(t'))2 --+ 0 as t, t' ---+ co and the limit 2 = l.i.m.t v(t) exists. Writing equation (4) in the form

df

u(f) - m1(f - 1) df = mldt

f- 1

u(f)(f- 1)

and integrating with respect to t from 0 to t, we obtain In (1 - f) - f u(v) - ml(v - 1) dv = m1t + In (1 - z)

.

u(v)(v - 1) Here if we set z = eaaim*(t) and let t approach oo, we obtain the equation JZ

In (1 - g)

i) 1)dv

- gu(v)- m1(v

= In(-i?)

from which (29) follows. This completes the proof of the theorem.

The theory of branching processes with particles of several types is analogous but more complicated. We pause only for the basic relationships in that theory. Just as in the case of particles of a single type, it is convenient to use the method of generating functions.

Let M denote the set of all possible states of a process, that is, the set of all vectors a = (a1, a2, , an) with nonnegative integral components. Let us agree to denote n-dimensional vectors and their components with the correby Greek letters a, 6, a, sponding Roman letters.

We define the generating functions

F:(t, Q) = F;(t, s1, s2, ... , sn)

of the transition probabilities p{;, p(t) F;(t, a) = FF(t, Si,

.

the relations

bby+

// , sn) = L p{i} (t)sl1sa2 ... sin,

9eM

(a = {sl, s2 .., sn}, $ _ {b1, b2, .. , bn}) .

We recall that {i} denotes the vector

{i} = {8ti1, 8;2, , bin}. functions F;(t, a) are analytic functions of the variables s1, s2,

in the region I s; I < 1 (for i = 1,

F;(t,a) < 1 for

.

(30)

The

, S.

, n). Also,

s; < 1,F;(t, 1, , 1)

= 1, FF(0, Q) = s;

(31)

.

If we define the n-dimensional vector-valued function ID(t, a) = {F1(t, a),

,

a)}

,

JUMP MARKOV PROCESSES

342

it follows from (31) that

c(0, v) = a .

(32)

Let us now find the equivalent of the Kolmogorov-Chapman formula for branching processes expressed in terms of the generating functions. We have pfofi(t + T) = E P(i)a(t)PaA(T), t > 0, T > 0 . aeM

If we substitute into this equation the value of pap(z) corresponding to formula (1), we obtain n

aeM

ai

E H H P(kldck,j)(z) .

P(.la(t + z) = E pr}-(t) kj

k=1 j=1

If we multiply both sides of this equation by s,11 ... Snn and sum over all 8, we obtain the relations: Fi(t + T, a) TnT

E P(i}(t)peM E aeM

kj

a7k

E 11 11 fi(k,j)=P k=1 j=1

Sb, (k,j)

7n7 7ak

aeM

Plila(t)H 11 E k=1 j=1 (01heI

Sx(k,j)1

/

7n

aeM

P(ila(t) k=1 11 Fkk(T, Q) ,

from which it follows that Fi(t + r, a) = FF(t, F1(r, or), ..., FF(T, a)),

i = 1, ,s,

or

4)(t + r, a) = 4)(t, O(r, a)) .

(33)

Theorem 6. The system of generating functions of a branching process satisfies the system of functional equations (33) and the initial condition (32).

Let us derive for the generating functions the differential equations

corresponding to the first and second systems of Kolmogorov equations for the transition probabilities. Suppose that

lim LL A!-) = bia ($

{i}),

lim 1 - Poi}lil(t) = bii

and

bii =deM,F#(iI E bia < ,

I = 1, ..., n

Then the transition probabilities p(i)p(t) satisfy the first system of Kolmogorov equations (cf. Section 2):

6.

THE GENERAL DEFINITION OF A MARKOV PROCESS

dp(i,a(t) dt

343

= -biip(i(a(t) a+e M,a#(i( E biapaa(t)

Multiplying this equation by sll, noting that equation (1) implies

, snn, summing over all 8, and

+

[F:(t, a)]ai Li paa(t)S°1sP ... snn = II beM i=1 (this equation expresses the independence of the evolution of the

particles that exist at a given instant of time), we obtain

- -biiFi(t, a) +

aFi(t, a) at

a.

n

aeM,a#(i(

bia H [Fi(t, a)] . i=1

or

a1 (t, a) at

i = 1, ...,n ,

= ui(F1(t, a), ..., FF(t, a)),

(34)

where ui(s1, ... , s,,) _ - biisi +

a E M,a#(i}

bia sa, ... San, i = 1 , ... , n . (35)

The functions ui(s1, , s,) are the generating functions of the systems of quantities { - bii, bia, a e M, a # {i}}.

To obtain the second equation, let us suppose that I si I < 1 (for i = 1, . , n). Then I Fi(t, a) I < 1 (for i = 1, , n) and let us differentiate equation (33) with respect to z.

then setting r = 0, we obtain aipat a) =

uk(a)

acp(t, ask

Differentiating and

a)

(36)

Equation (36) is a system of the same type of equation for the generating functions Fi(t, a), aFi(t, a) at

_

k=1

uk(a)

aFF(t, a)

i=1

ask

which must be solved under the initial conditions (31). Thus, we have obtained.

Theorem 7. The system of generating functions Fi(t, a) for I si I < 1, where i = 1, , n, satisfies the system of ordinary differential equations (34), the partial differential equation (36), and the initial conditions (31).

6.

THE GENERAL DEFINITION OF A MARKOV PROCESS

At the basis of the concept of a Markov system (process)

is

the concept of a system whose future evolution depends only on

JUMP MARKOV PROCESSES

344

the state of the system at the given instant of time (that

is, its

future evolution does not depend on the behavior of the system in the past). Let {U, C5, P} denote a probability space on which a random process e(t) is defined with range in a complete metric space X.

We shall call the space X the phase space of the system and we shall call e(t) the state of the system at the instant t e Z, where Z is a finite or infinite interval of the real line. We let 0 denote the algebra of Borel subsets of X. The hypothesis of absence of after-effect is most easily written with the aid of conditional probabilities:

(1 ) for arbitrary A e 8 and t1 < t2 < . . . < t < t. Since the conditional probability regarding a random variable can be regarded as P{e(t) e AI e(t1), e(t2), ... ,

P{e(t) P_ A I e(tj} (mod P) ,

a function of that variable, we set (s < t)

P{e(t) G A I e(s)} = P(s, $(s), t, A)

.

(2)

It follows from formula (26), Section 6, Chapter III that for

t, < t, <

< t the equation

M{g(e(tl), $(t2), ..., e(t.)) I e(tl)} P(t11 e(tl), t2, dx2) I p(t2, x2, t3, dx3) ,,.dx%)g(e(tl),

J

...

1 p(t%-1, x,-1, tn,

x2, ... , x,)

(mod P)

holds for an arbitrary bounded Borel function g(x1, x2,

(3) ,

(where xk e X for k = 1, 2, , n). In particular, if we set g = XA(x3), where XA(x) is the characteristic function of the set A e 0, it follows from (3) that with probability 1, p(tl, ee(tl), t3, A) =

E

p(t2, x2, t3, A)p(tl, S (tl), t2, dx2)

This equation appeared in Section

1

I

(4 )

as the Chapman-Kolmogorov

equation, and it served there as the basis for the definition of a Markov process in the broad sense. We now give the axiomatic definition of a Markov process within the framework of the general definitions of Chapter IV. Definition 1. A random process e(t) (for t e 1) with range in X is called a Markov process if a. equation 1 is satisfied for arbitrary t1 < t2 < . . . < t < t,

where each tk (for k = 1,

, n) and t belong to Z, b. the conditional probabilities P(s, x, t, A) are 0-measurable

6. THE GENERAL DEFINITION OF A MARKOV PROCESS

345

functions of x for fixed s, t, and A, and are probability measures on

0 for fixed s, x, and t, and c. the Chapman-Kolmogorov equations are satisfied for all ee(t1) = x

X.

Thus, by definition, the family of conditional probabilities (1) is regular and the process e(t) is independent of the "past". We shall call the property of the process expressed by equation (1) the Markov property or the absence of after-effect.

Let us show that the Markov property implies stronger assertions. Again using formula (26), Section 6, Chapter III and equation

< tm < -, En + m), the equation

(1), we obtain for t1 < t2 < (for kJ = /1,

< to+m, where tk e

M{g(e(tm+1), S(tm+2) ... e(tn+m)) I e(t1), e(t2), ..., e(tm)} P(tm, e(tm), tm+1, dx1) ...

P(t%+m-1, xn-1, t,t+m, dxn)

x g(x1, .. , x%) = M{g(se(tm+1),

., e(tn+m)) e(tm)}

(mod p)

If we set g(x1, , x,) = x4(n)(x1, , xj, where AW is a Borel set in X'", we get the following equation generalizing the Markov property of a process: p{[Wm+1), ... , W.+.)] e A`n' I SE/ (t1), ... ,// $(tm)} = p{[e(tm+1), ..., e(tm+n)} e A(") I e(tm)}

(mod p)

< t,,+m( e Z), n and in. Let U{
and st} denote the or-algebra e(s), where s generated by the quantities 6 (s), where s e Z and s > t. Then with probability 1, for an arbitrary cylindrical set C e or{> t}, P{C I e(t1), ..., e(tn)} = P{C I s(tn)}, t1
(5)

Let A denote the class of events for which equation (5) is valid. On the basis of the properties of conditional probabilities (cf. Theorem

6, Section 6, Chapter III), A is an algebra and a monotonic class of events. Therefore ADC{>t}. On the other hand, let 9 denote the class of events N such that for arbitrary S e v{> t}, p(S I o'{
N

(6)

On the basis of (5), 9Z contains all cylindrical sets in a{
JUMP MARKOV PROCESSES

346

Theorem 1. For arbitrary S e Q{>t}, P(S I o{
(mod P)

(7)

Equation (7) shows that the conditional probability of an arbitrary event S, defined by the behavior of a Markov process in the "future" with completely given "past," depends only on the "present."

Let X denote a complete separable metric space and let Z denote some interval of the real line [a, b] or [0, -). Let us show how one can construct from a given Markov process in the broad sense, {P(t, x, z, A), x e X; t, z e Z}, a Markov process in the narrow sense whose conditional probabilities (2) coincide with the given transition probabilities p(t, x, z, A).

We introduce some initial distribution {X, 0, uo}, and for an , xn) of n variables for

arbitrary bounded Borel function f(x1, x2,

xkeX,andforarbitrary tkeZ(for k= 1, ,n;a
Ft1,t2,...,t,y[f}

...

x2,

.

(8)

. ., xn)p(tf_L, x,,_L, tn, dxn)

and

Pt1...tn(An) = Ft1,t2,...,tn[XA(n)]

(9)

,

where XP) is the characteristic function of the set A(n) e 0(n), where

fi(n) is the nth power of the a-algebra 0. arbitrary Borel function f1(x1, x2, ... , xn_1) =

x2,

,

We note that for an

the function

x2, ... , xn)P(t, xn_1, s, dxn)

is also a Borel function, since the integral is the limit of integrals of simple functions and these simple functions are Borel functions of the variables x1, x2, . . . , xn_1. By virtue of the properties of an integral, pt.....,t(A(")) is a measure on fi(n). Obviously the family of measures pt1,...,tn(B) satisfies the compatibility conditions and by virtue of Kolmogorov's theorem (Theorem 3, Section 2, Chapter III) has a representation of the form in, p}, where 12 is the

space of all functions w(t) for t e Z, with range in X. Let e(t) denote an arbitrary process stochastically equivalent to {12, P}. Let us show that P{E(t) e A I e(t1), (t2) ... (tn)} = P(t,,, E(tn), t, A)

(mod P)

,

that is, that E(t) is a Markov process with given transition probabilities. To do this, it will be sufficient to prove that

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

347

A(n)p(tnf xn, t, B)ptl,t2,...,n(dx,, dx2, . .., dxn)

-

Pt1,t2,...,tn,t(A(n) x B)

for arbitrary Ann) e 01n), B e 0, and tl < t2 < t3 < t < t. But this follows immediately from formula (8) and Lemma 1 of Section 1.

If X is separable and locally compact, the process In, p} is stochastically equivalent to the separable process fi(t) (by Theorem 2, Section 2, Chapter IV). In what follows we shall consider only separable Markov processes. In many cases, the following theorem characterizes the properties of regularity of Markov processes. Let e(t) denote a separable Markov process and suppose that a(s, 3) = sup {p[t, x, S, SE(x)]; t < s; t, S G Z; I t - s j < 3; x e X}, where S,(x) is the complement of the sphere S,(x) of radius s with

center at the point x. Theorem 2. (a) If a(s, 6) then the process fi(t)

0 as 3 --, 0 for arbitrary s > 0,

does not have discontinuities of the second kind.

(b) if a(s, 6) , 0 as 6 --, 0 for arbitrary E > 0 and if for an , mn} of the interval

arbitrary sequence of partitions {tnk, k = 0, 1, [a, b], L p{P( (tnk), E(tn,k-,)) k=1

s} --> 0

as

n

then the process e(t) is continuous.

(b') If a(s, 3) = o(3) for arbitrary s > 0, then the process fi(t) is continuous.

Proof. Assertion (b') is a particular case of (b). Assertions (a) and (b) follow from Theorem 1 of the present section, Theorem 2 of Section 4, and Theorem 4 of Section 5, Chapter IV. 7.

THE BASIC PROPERTIES OF JUMP PROCESSES

Jump Markov processes were introduced in Section 3. Let us now look at them in greater detail. Let p,(A) denote the distribution of e(t). Lemma 1. A jump Markov process is stochastically continuous. Proof. If t' > t, then on the basis of conditions a and c of

Definition 1 in Section 3, p{e(t)

(t')} = 1 IA,(dx)p(t, x, t', X - {x}) r

(q(t, x) + s)(t' - t) ftt(dx) < (k + s)(t' - t)

,

348

JUMP MARKOV PROCESSES

from which the assertion follows. Lemma 2. If e(t) is a separable jump process, then

P{e(z) = x for all z e [t, s] I e(t) = x} = exp (- 8q(z, x)dz)

\

Proof.

(1 )

t

Let M denote the set of separability of the process

E(t) on the interval [t, s]. It follows from the stochastic continuity of the process and Theorem 5, Section 2, Chapter IV that we may take for M any countable set that is everywhere-dense on [t, s]. It follows from the separability of the process that P = p{e(z) = x for all z e [t, s] I e(t) = x} = p{e(z) = x for all z e M c(t) = x}. For the set M, we can take the set of points of the form tnk = t + kh/2n, for k = 0, 1, . , 2n and h = s - t. We note that p = lim p{e(tnk) = x, k = 1 , , 2" 1 e(t) = x} n-_

since the events A. = {c(tnk) = x, k = 1 , , 21} constitute a decreasing sequence and fln=, A. = {e(z) = x, z e M}. Furthermore, pn = P{e(tnk) = x, k = 1, 2, ..., 2-1 e(t) = x} 72%

2%

= [f p{tn,k-1, x, tnk, {x}}, lnpn = E In P{tn,k-1, x, tnk, {x}} k=1

k=1

Let fn(t) denote a piecewise-constant function that on the interval [tn,k-1, tnk) is equal to 1/Atnk In p(tn,k-1, X, {x}), where It follows from the definition of a jump process Atnk = tnk - tn,k-1' (cf. Definition 1, Section 3) that p(tn,k_1, x, {x}) --+ 1 uniformly with respect to t. Setting p(t,n,k_1, /x, tnk, {x}) = 1 + ank, we see that In (1 + ank) + q(tnk, x) = In (1 + ank) ank + q(tnk, x) --.0 At uniformly with respect to t.k. Therefore fn(z) -q(r, x) uniformly

on [t, s] and

lnpn =

fn(z)dz

x)dz

J

This completes the proof of the lemma. Lemma 3. A separable jump Markov process has no discontinuities of the second kind. This is true because a(s, 3) <_ sup p(t, x, v, X - {x}) t5r
xcX

< sup 1 - exp (-

t+d

t

&, x)dz) < k3

and the assertion of the lemma follows from Assertion (a) of Theorem

2, Section 6.

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

349

From this lemma and Theorem 3, Section 4, Chapter IV we get Corollary 1. For a separable jump Markov process e(t) there exists an equivalent process $'(t) whose sample functions are continuous from the right with probability 1.

Corollary 2. If E(t) is a separable jump process and e(t) = x, then with probability 1 there exists an interval of time (t, t + h) dur-

ing which $(z) = x for z e (t, t + h). Proof. The event A that such an interval does not exist can

be represented in the form A= nn-, An, where A. is the event that fi(t) = x and that for at least one value of z, e(r) # x on the interval (t, t + 1/n). Therefore,

+1/n

p(A I e(t) = x) = lim P(An e(t) = x)

lim {1 - exp (Jt

q(z, x)dz)} = 0

Since an arbitrary separable jump process is equivalent to a

process whose sample functions are continuous from the right (mod p), we can confine our study to these. In subsequent discussions in this section we shall always assume that fi(t) is a jump process that is continuous from the right (that is, a process whose sample functions are with probability 1 continuous from the right on Z), and where 1 _ [0, oo). The process is necessarily separable

and we can take as the set of separability an arbitrary countable set that is everywhere-dense on Z. Suppose that we know that e(to) = x. On the basis of Lemma 2 there exists a z1 such that either z1 = oc and E(t) = x for all t to (as will be the case when t to), or e(t) = x for t e [to, to + z-,) and (to + z1 - 0). Let us find the probability of the

q(t, xo) = 0 for all

(to + event

B={z1St,e1eA},

x0A.

Define B(n)(r)

_ Nt. +

x, 1 < j < r,

to

+ r n 1) e Al

and [nt]

B(n) = U B(n)(r) r=1

If the sample function of the process right, then

B=unB(k). n=1 k=n

fi(t)

is continuous on the

JUMP MARKOV PROCESSES

350

For brevity in writing, we define Pn,(x) = (t0 + J - 1 , x, to + n

I n

and

Pni(x, A) = P(to +

.x, to +

n

,

{x} )

jn

/

,

A)

Then P(B(n)(r)

e(to) = x) _ j=1 fl

A)

As shown in the proof of Lemma 2, H j=1

P.9(x)

= exp t

\

-

to+(rl n' q(6,

x)dOJ + En

to

where sn , 0 uniformly with respect to r < nT (where T is a fixed number) as n oo . It follows from the definition of a jump process that pn,r+1(x, A) = (q(to +

r n

, x, A)

(2)

+ sn') 1 n

where sn' also approaches 0 uniformly with respect to r < nT as n -± oo , Since

P(B I E(to) = x) = lim p(BW I e(to) = x) , n-,m

it follows from the preceding formulas that P(r1

=

t, set e A I Se(to) = x)

o(-\

(3)

q(B, x)de)q(s, x, A)ds . t to

Here, let us set A = X\{x}.

Remembering that q(s, x, X\{x}) _ q(s, x), we obtain the distribution function of the quantity z1 under the hypothesis that E(to) = x: V(to, x, t) =

1 - exp (-

to+t

(4)

q(O, x)dO) . to

Equation (3) can be written in the form P(v1 < t, 1 e A I (to) = x) = J

o

q(to + 0, x, A) Y(to, x, dB) q(to + 0, x)

(5)

From this it follows that P(E1 e A I Z1

= t, (to) = x) = g(to + t, x, A) q(to + t, x)

(q(t, x)

0) .

Thus the interpretation of the quantity q(t, x, A) _ II (t, x, A), q(t, x)

xvA

(q(t, x) # 0) ,

(6)

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

351

of which we spoke in Section 3, is rigorously justified: fl (t, x, A) is the conditional distribution of the state $(t + 0) under the hypo-

thesis that at the instant t the state of the process changes and $(t - 0) = X. Consider the event k-i

for

B = J$(t) _ $k-1

k

z; < t < t0 +

to

;=1

j=1

k=1,2,...,m, (t0 +

j=1

z; J = em, Ck-i # Ck' Ck E Ak I

zl < t1, ... , zm < tmJ

Generalizing formula (5) let us find the conditional probability , tm; A , Am I t0, x) of the event B under the hypothesis that e(t0) = x. To do thus,we introduce the events F(t1, t2,

B(n)(r1, r2, .. ., rm) = {e(t0 + nI ) = x, 1 < j < r1

se(t0+ J) =E1,r1+ 1 < j
1 E A1, ... ,

to + I) = Cm-1, r1 + ... + rm_1 n

+ 1 < j < r1 + ... + rm, m-1 EIAm_,

+rm+ 1)EAm }

e(t0+ r1+

n

/

and we define B(n)

=

D

U

ll (n)(r1, r2,

. .., rm)

m]+1

The continuity from the right of the sample functions of the process implies that with probability I the events B and limn- B(n) coincide. Using the definition of a jump process, we obtain p(B(n)(r1,

., rm) I e(10) = x)

, x)dO)Ln =exp (_t0+)q(o to

x exp

(-

to + f

to+ ((r1+r2) f,t

J t0+r1/n

x ... eXp (-

f

q(O, x1)dO) J A2 n

n

x, dxl)

(t, + rl + r2 x1, dx2) .. . n

t0+((r1+...+rm)/n1

x n q(tl + r1 +

q(O, xm-1)dO

n+ rm

,

Am)

+ n s;,

,

(7)

JUMP MARKOV PROCESSES

352

where s' - 0 uniformly with respect to r1, , rm as n oo (where r;,ln < T for i = 1, , m). From this it easily follows that F(t1, t2, ... , tm;A1, A2, ... , Am\ to+t1

I

t0,, x)

exp (- 31q(6, x)dO Jds1J q(s1, x, dx1)

/

for

Jt0

Al

x)dO)1 x I s1+t2 exp (_S2q(o, sl sl Sm 1+tm

x J J

exp

(-

sm-1

sm J sm

7

q(s2, x1, (Lx,) .

. .

A2

q(O, xm-1)d0)q(sm, xm-1, Am)dsm

(8 )

1

The expression we have obtained shows that the joint conditional distribution of the variables v1, Z2, .. , Vm, e1, e2, , em can be expressed simply in terms of the separate conditional distributions of the variables Zk and sk. Specifically, formula (8) can be rewritten in the form F(tl, t2, ... , tm; Al, A2, ... , A. I t0, x) t

l,V(to, x, ds) 0

Al

II (1, + s1, x, dx1)

t2

X

'Y'(t0 + s1, x1, ds2) 0

. .

.xJ

A2

11 (t, + S1 + s2, x1, dx2) X

tm,,Y(to

+ sl + ... + Sm-i, xm-i, dsm) 0

X II(to+s1+ ... +Sm,xm-1,Am) It follows from this last relation that

(9)

P{Cm E A. I x0, e1, ... , Cm-l' TI, ... , V,,}

_

(to + Z1 + ... + Zm, em-v A)

(10)

and

P(Zm < t I x0, Co, ... , Cm-1, Z'l, Z2, ... , r--l)

_(t0 + Z1 + ... + Zm-1, Cm-J, t)

(11)

These formulas appear to follow immediately from the absence of after-effect in Markov processes. However, this is not quite the case since the absence of after-effect here is used at random instants of time depending on the progress of the process C(t). Therefore formulas (10) and (11) do require proof. The intuitive simplicity of the relations obtained becomes especially clear in the case of a homogeneous Markov process. Then 'T(s, x, t) and rj (s, x, A) are independent of s, and formulas (9) to (11) take the forms

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

353

F(ti, t2, ... , tm; Al, A2, ... , A. I t0, x)

=

ti

tm

...

0

0

... Al

yr(x, dsl)

(x,, ds2) ... yr(xm-i, dsm)

A,m

x II (x, dxi) II (xi, dx2) ... II (xm-i, dxmm)7 , P{Cm e Am I x0,

P{2. < t

(91) Am) ,

11

=

I xo,

(10')

t)

(ill)

Theorem 1. Let C(t) denote a homogeneous jump process whose sample functions are continuous from the right with probability 1. Then

a. The sequence S i,2, of distinct states of the system , sum, is connected in a simple homogeneous Markov chain with the transition

probability I (x, A) = q(x, A)/q(x), b. the conditional distribution of the time of existence in the state Cm+i for given x, $1, ... , Cm, Zi,

, rm depends only on Cm.

Let us return to the general formula (9).

Corollary 1. In a jump Markov process that is continuous from the right, after a finite interval of time only finitely many transitions from one state to another are possible. Proof.

Let us obtain a bound for the quantity P{Zi + ... + 'r,m
We have

P{zi+

... +r
2k)

C(t0) = x}

(12)

for arbitrary N > 0. Suppose that N > 2K, where K= sup q(B, x) . X e X,O E [t0,t0+tJ

Since e-N3

(t, x, ds) = o e-N3q(t

o

\tt

+ s, x) exp (-

< K f o e-3Nds =

K<

+3

q(t + s, x))ds

2

it follows from formulas (11) and (12) that P(v0 +

+ 2< t

C(to) = x) <

()meNt -, 0 as m

The last inequality proves the finiteness of the number of jumps of the function C(t) on a finite interval of time [t0, t0 + t] with probability 1.

JUMP MARKOV PROCESSES

354

Corollary 2. The sample functions of a jump process that is continuous from the right have with probability 1 the following structure: There exists a sequence of positive random variables z1, z2, , z, such that m k=1

with probability 1 as m --+ oo and E(t) = Ek-l for k-1

k

;=1

;=1

is a sequence We should also note that: If {Z-k} for k = 1, 2, of positive random variables, if for k = 0, 1, , is a sequence

of random elements with values in X, if all the finite-dimensional distributions of the quantities zk and k are given, if !__,zk = 00 k=1

with probability 1 and if E

)7(t)=Sk

for

Zz;
k+1

;=1

;=1

then fi(t) is a random process defined for all t > 0, the sample functions of the process 7)(t) are continuous from the right, and all finite-dimensional distributions of the process are uniquely defined.

Let E(t) for t > 0 denote a jump process that is continuous from the right, and suppose that Vk and Ek are the same as in the preceding paragraph. We can regard the process E(t) as a function of the infinitely many variables t, Eo, zl, E1, z2, E(t) = f(t, Eo, zl, E1, z2, . . . )

_ Ek,

if

.

Then

7k < t C 7k+1 ,

k=0, 1, 2, .. , where 70 = 0 and 7k = ;=1 z; Let SZ denote the space of all sequences w = {xo, t1, x1, t2, } for xk e X and tk > 0, and let denote the oa-algebra generated by cylindrical Borel sets in S2.

It is easy to see that the function f(t, (o) for w e f2 is a Borel function of the variables t and w; that is, for arbitrary A e 0, It, co; f(t, (o) e Al e 6{01 x g-} ,

where 01 is the o'-algebra of subsets of the half-line t > 0 and o{%LJ, x 3} is the product of the o-algebras 01 and g- (cf. Definition 1, Section 8, Chapter II). Specifically, the set it, w; f(t, w) e Al is the sum of countably many cylindrical Borel sets

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

tj < t <

fx, e A,

355

k = 0, 1, .. .

t;+11, 9=1

.9=1

Let us now consider the random process

t>0

E'(t) = E(t + 7m),

Since E'(t) =

Sk+m

for 7k+m

f(t, cem, Zm+1, Sm+i, Zm+2, .

- ym < t < yk+m-1 - ym, we have

. .) and for arbitrary fixed t > 0, '(t) is a

Borel function of m, zm+1,

,

that is, a random element of X.

Theorem 2. Suppose that P'(t, x, s, A) = P{E'(s) e A I E'(t) = x, 7m = T, E'(0) = z} Then

P'(t,x,s,A)=p(t+T,x,s+T,A),

t<s.

In other words, for fixed ym = T and E'(0) = E. = z, the transition probabilities for the process E'(t) coincide with the transition probabilities of the Markov process E(t + T), where t > 0 and E(T) = z. Proof. The theorem follows easily from formula (9), by virtue of which P(zm+i < t1, Em+i E Al, ... , 2'r < tr

=

i

(t0 + 7m, Em, ds)

10

,

7--

1 ,1 (t0 + 7m + Si, Em,

dx) x

Al

+ 7m + Sl + ... + sr_l, xr_l, dsr)

X 0

X 11 (t0 + 7m + s1 + ... + sr, xr_1, Ar)

that is, the conditional distribution of the variables zm+i, , Em+r

Em+i,

for given E0, g1, ... , gm, zi,

, zm+r,

, zm depends only on Em

and 7m, and it coincides with the distribution that is obtained if we consider the Markov process E"(t) = E(t + T), T = 7m with fixed initial state E"(0) = Em. Since the joint distribution of the variables Zm+19 Zm+E,

, Zm+r, Em+1,

, Em+r ,

where r is an arbitrary positive number, uniquely determines the finite-dimensional distributions of the process E'(t), the theorem is proved.

Let us now give an important generalization of this theorem. Let a = a(u) denote a nonnegative Cs-measurable function that is finite and defined on some Q e e.

In the present section we shall call a a random variable, although this description is not exact since the function a(u) is not defined for all (mod P) elementary events.

JUMP MARKOV PROCESSES

356

Definition 1.

The function a = a(u) is said to be a random

variable independent of the future (with respect to the process E(t))

if for arbitrary t > 0, {u; a(u) < t} e a{ < t}

.

Thus a(u) is a quantity independent of the future if all we need to do to know whether the event {a(u) < t} occurred or not is to observe the sample function of the process fi(t) up to the instant t. Let o denote the class of all events Bee such that for arbitrary t > 0, B n {u; a(u) < t} e a{
Obviously u. is a a-algebra of events.

If a(u) = to' = const for u e U (obviously this constant is a random variable independent of the future e(t)), then a,, = 6{
of occurrence of an event, then the a-algebra a. consists of all those events that are determined by the progress of the process e(t) up to a random instant of time a. We are interested in the question of how the process E(t + a) where a is a random instant of time independent of the future, behaves. It is natural to expect that the conditional transition probabilities of the process (t + a) with respect to a coincide with the transition probabilities of the process e(t) (displaced by an appropriate amount along the time axis). Let us prove this for jump processes. Let a denote a random variable that is independent of a future process $(t). Suppose that e(a) = Eo (which is meaningful only on

a set Qc U).

Let 'y denote the earliest instant at which (where y > a). Define v = 71 - a and define 7; inand ductively as the earliest instant at which m = g(ym) (for m = 2, 3, ). We define the events m

for yk-1 C t yk, D= k=1 n 1 e A'k-1 g J ( = Sm a A' } k ek-, 0 < yk - 7k-, < tk} n {e(7m) k

and

j, ... , jm) _

B{ JOn- 1

<
<

jo n

Jk

nn m

Ck=1 i=5k-1

i

n

= Ek-1 e Ak-, 6', # S k) n {(Jm± 1 /J = n

Since the set of numbers of the form j/n is a set of separability, and the sample functions of a jump process on a finite interval of time assume a finite number of distinct values and are continuous

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

357

from the right, it follows that for the event D = B n {a < s} n D, we have

D=Bn{a5s}nD = lim

Dn(j°, j1, ... , jm)

U

n~OO 0<j05[8n1+1,0
and

P(D) = lim n-.oe

E

P(Dn(j0, j', ... , jm))

0<j0<[en]+1 0<7 k-j k-1< [tin]+1, k =1, , m

Since

Bn(j°n 1


we have P{Dn(j0, j1, ..., j.)} jk

P{km1

1n

Bf11(j0-1)/nza5(jp)+4)f

S k-1 E

Ak-1, ek-1

1

j-n

l 1

em

n

E AIn

e(n°)}dP

.

(13)

The principal part of the integrand is found in accordance with formula (7).

We now define the function m(s, A) = P{B n (a < s) n (e(s) E A)} .

This function is a measure on 0 for fixed s and it is a monotonic function with respect to s for fixed A. In accordance with the general theorems on the extension of set functions we can construct from the function m(s, A) a measure on the Cartesian product of the a-algebras {01 x 0}: p([O, s] x A) = m(s, A) for ,u < 1. Let F(t) denote the generalized distribution function of the variable a, F(t) = P{a < t}. Since a is absolutely continuous with respect to F, it follows from the Radon-Nikodym theorem that there exists a function g(s, A) that is 81-measurable for fixed A, that is a measure

on 0 for fixed s, and that satisfies the equation ft([O, s] x A) = m(s, A) =

g(t, A)dF(t) .

(14)

Applying formula (14) and the expression (7) to the principal part of the integrand in (13), we may rewrite this formula (13) in the form:

JUMP MARKOV PROCESSES

358

-J.)J

P{D(jo, j1,

x

1 q( n \n

exp

q(B, x)dO)

X

j'q(O, 91I n) I n)

, x, dx1) exp

/

x1)d6A

(50/n)

_g(J, x1, dx2) ... exp (_ ` (Jmin)

x

n

/

n

J (Am_Iln)

q(B, xm-1)de)

dx)dF(t) + n sn

X n g(in , X.,

,

(15)

where s;, approaches 0 uniformly with respect to jo, j1, , jm for jk < Cn, where k = 1, , m and C is arbitrary. Here we shift from integration with respect to the variable u in the space {U, Cam, P}

to integration with respect to the variables a and $(j/n) (cf. Theorem 3, Section 6, Chapter II), and we use Lemma 1 of Section 1. It follows immediately from formula (15) that P{D} _ edF(t) f g(t, dx) ti ir(t, x, dt1) f Ij (s + t1, x, dx1) x X Al 0

08

x J1 t'm !\'s + t1 + . . . + t.-,, x., , , dxm) 0

x II (s + t1 + ... + tm, xm_I, A;)

(16)

From this it follows that

P{Bn{a<s}(1DIa=s,e(a) =x} "W(s, x, dt)` II (s + t1, x, dx1) x Ai

0

. . .

)( tm 7!(S 'Y + L

t1 + . . . + t,n_I, X,m_I, dxm)

0

x IjT(s + t, + . . . + tm, xm-1, Am)

(17)

Formula (17) shows that the times at which the system remains in fixed states after a random instant a and the probabilities of transitions from certain states to others under the hypotheses that a = s and E(a) = x have the same conditional joint distributions as those for the process E(t + s) for t > 0, with the hypothesis $(s) = x. Since these joint distributions uniquely determine the conditional finite-dimensional distributions of the process, we have proven Theorem 3. Let $(t) denote a jump Markov process and let a denote a random variable that is independent of the future of E(t). Suppose that B e o'.. Then if 0 < t1 < . . . < tm,

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

359

P{B,a<s,$(t1+a)eA1, P{e(t, + t) a A1, ..., (tm + t) e A. I e(t) = x} x g(t, dx)dF(t) , g(t, A)dF(t) = p{B, a 5 s, e(a) e Al.

where J

Corollary 1. If the conditions of the preceding theorem are satisfied and the process e(t) is homogeneous, then

P{B, a < s, E(t, + a) e Al, ... , (tm + a) e A. I e(a) = x} = P{B, a < s}p{e(t1) e A, ... , E(tm) e A. 1 (0) = x} . Corollary 2. Let e(t) denote a homogeneous Markov jump process with countably many states, let a denote the time the system first

falls in the sth state (a = z1 + - - - + zk if k = s. Suppose that P(a < -) = 1. Then the process c'(t) = g(a + t) for t > 0 is a Markov process with the same transition probabilities as the process e(t), it satisfies the initial condition e'(0) = s, and it is independent of the o'-algebra of the events a,,.

For an example of the application of these results, consider the problem of determining the distribution function of the time of first transition from the sth state to the rth state in the case of a jump birth and death process. We recall (Example 3 of Section 4) that a birth and death process is a homogeneous Markov process with countably many states (0, 1 , 2, ) for which q(n) = Xn + jam,, q(n, In + 11) = X.,, and q(n, In - 11) = u,,, for n = 0, 1, 2, .. , where po = 0 and {n} is the set consisting of the single element n. Thus in a jump birth and death process, transitions from the nth state are

possible only into the adjacent (n - 1)st and (n + 1)st states with probabilities (cf. Theorem 1)

7c' = H(n, in + 1}) = X. + /^m

7r' = II (n, In - 11) =

112

X. + Suppose that s > r and zar(t) is the length of the interval of time

to the first instant the system falls in the rth state if the system is in the sth state at the instant t. Then Zar(t) = Ta,a-1(t) + Ta-1,r(t + ra,a-1(t))

.

(18)

It follows from Corollary 1 to Theorem 3 that the terms in the right-hand member of equation (18) are independent and that the second term has the same distribution as za_i,r(t).

Let the distribu-

JUMP MARKOV PROCESSES

360

tion function of the variables r,,,.(t) be denoted by F,r(x); that is, F,r(x) = p{z,r < x}. Here we do not assume in advance that F,,.(+ oo) = 1. We introduce the Laplace-Stieltjes transform of the function F,r(x):

e -dFsr(x) ,

(P,r(Z) = J

Re z _> 0 .

0

On the basis of what we have just said, it follows from (18) that q)sr(z) =

(19)

Setting cp,,,_1(z) = (p,(z), we obtain from (19)// ,P .,(Z) - 7'a(Z)7's-i(Z) .

.

. 7'r+1(Z)

(20)

.

Since the function cp8T(z) determines F,r(x) uniquely, the problem is reduced to determining rp,(z). Let z,(t) denote the length of the

interval of time from t to the instant when the system leaves the sth state, which it is in at the instant t. Then with probability acs' we have z,,,_1(t) = z,(t), and with probability i8, re,8-1(t) = r8(t) + r,+1,s(t + ra(t)) + r8,8_1[t + r8(t) + r,+1,8(t + v8(t))]

(21)

The Laplace-Stieltjes transform be(t) of the distribution function of the variable z,(t) is easily found. Using Lemma 2, we obtain qi(z) =

e-Zxd(1

-e

(8

)=

q(s)

q(s) + z

o

It again follows from Corollary 1 of Theorem 3 that p8(Z) = X:'Ip:(z) + or

q,8+,(z)

= q(s) + z X8

-

f_eN

(22)

(p8(Z)

Successive applications of formula (10) lead to the representation

of the function p,+1(z) in the form of a continued fraction that is a rational function of pk = Iik/X'k and q1(z). Let us find the function p1(z) from the following considerations.

Define

poo(r) = p{e(t) = 0 1 M) = 0} .

Let r' denote the length of the kth interval of time in the course of which the system finds itself in the state 0, and let zlo' denote the time spent by the system after leaving the state 0 for the kth

time until the next return to

0.

The probability pou(t) is the

probability sum un=o En of incompatible events En, where E0 is the event z('' > t and E. is the event (n > 1)

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

361

x

x

E (Z(k) + Z10

k=1

- t < k=1 (Z(k) + Z10)) + z(x+1)

It follows from the independence of the variables z(k) and z(o) that (for p(z(1) > t) = e-1o') Pl

k=1

t<

(Z(k) + Z(10)) t

e-ao(t-x)dF(x)(x)

(7-1k)

k=1

+ Z10,) + 2(x+l'}

,

0

(k) + Z10)),

where F(x)(x) is the distribution function of the sum EL .1 F(1)(x) = 0xe-1o(x-t)dF1o(t)

,

and F(x)(x) is the n-fold convolution of the function F(1)(x) with itself. Taking the Laplace transform of both sides of the equation Foo(t) =

we obtain p0(Z) =

x0 + z +

e-lot

+j

te_2o(t-x)dF(x)(x)

x=1 J0 J

i

( '- )'q(z) 1 x0 + Z X0 + z

1

z + X0(1 - T1(Z))

from which we get TP (Z)

o

Z

o

po(Z) =

XoPO(Z)

c0poo(t)e-ztdt

.

(23)

Formulas (20), (22), and (23) give the general solution of the problem.

We use analogous considerations to evaluate the distribution of

the time necessary for a transition from a state S to a state R without entering the state N (where R < S < N). This problem is a particular case of the preceding one. Specifically, together with the given process E(t) we introduce a new process e'(t), where

e(t) = e'(t) for t < r, ,(O) and e'(t) = N for t > Z,,(0). The process e'(t) is also a jump birth and death process with the same values of X. and px as the process e(t) for n < N. All the preceding formulas remain valid for this case. However, the function pou(t) now corresponds to the auxiliary process e'(t) and we may not assume that it is given. In the present problem it is easy to find the distribution of the variable N_1,N-2: t (1 (ZN_1,N-2 < x) = e(2N-1+PN-1)x) 1'N-1 P

j/

XN-1 + -N-1

from which we get

-

JUMP MARKOV PROCESSES

362

'P'-1(Z) _

f N-1 q(N-

1) + z

.

Using (22), we obtain the representation of the function p,(z) in the form of a continued fraction that is a rational function of z: 'P5(z) Ps

1+ps+z-

Pa+l PS+2

+Ps+1+z-

1

(24) PN-1

1+

PN-1

+

z )'N-1

Previously, we have always started with a given jump process. Let us now suppose that the function q(t, x, A) is given. Can we construct a Markov process with transition probabilities P(t, x, s, A) connected with the function q(t, x, A) by the relation lim P(t, x, s, A) - XA(x) = q(t, x, A)? B-.c

s-t

(25)

If the the answer is affirmative and the process constructed is a jump process, then the sample functions of the process can be constructed by using the preceding results. This remark forms the basis of

the solution of the problem posed. Let X denote a complete separable space, let 0 denote the u-algebra of Borel subsets of X. Suppose that the function y q(t, x, A) = -q(t, x)XA(x) + q(t, x, A)

is defined for all t > 0, x e X, and A e B and that it satisfies the following conditions: a'. For fixed t > 0 and x e X, the function q(t, x, A) is a finite

measure on B and q(t, x) = q(t, x, X). b'. For fixed x e X and A e 0, the function q(t, x, A) is continuous with respect to t and uniformly continuous with respect to A on every finite interval of variation of t. We note that conditions a' and b' are more general than those that the function q(t, x, A) of a jump Markov process must satisfy (conditions a-c of Section 3). Consider the space fl, introduced above, of the sequences }, where ek E X and zk > 0. On the algebra of v , e1, o) cylindrical sets in SZ we introduce the measure P(C) as follows. If

C = {Ck E Ak, k-0, 1, , m; r'; < t,, j = 1, , m},

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

363

then we set p(C) =

F(t1, ... , tm; A1, ... , A. 10, x) fto(dx)

,

J Ao

where F(t1,

, t,,,; A1,

, A. 10, x) is given by formula (8) and 1-1o

is an arbitrary "initial" distribution on 0. In accordance with Kolmogorov's theorem (Theorem 3, Section 2, Chapter III), the measure P(C) can be extended as a complete measure {, p}, where is the complete u-algebra generated by the cylindrical sets in f . On Sb we define the function e(t) = f(t, w) by k

(/ f(t, w) = xk, if 7k G t < 7k+1{ 7p = 0, 7k = j Tj J j=1

If 't- _ Ek=1 Zk = =, then fit, w) is defined for all t > 0 for given w.

Let N denote the set

Nw;7-=LV <°°} k=1

which is %-measurable in 0. If p(N) = 0, then e(t) is a random process defined with probability 1 for all t > 0, and its sample functions are continuous on the right in the discrete topology on X. This will be the case, in particular, when the function q(t, x) is bounded.

To see that this is so, note that the proof of Corollary 1 of Theorem 1 depends only on formula (8) and the boundedness of the function q(t, x), so that P(N) = 0 under this restriction upon the process in question. To determine the random process e(t) for all t > 0 for p-almost-

all w in the case in which p(N) > 0, we proceed in a different manner. The simplest method is as follows: We add to X a single point "«c ." The extended space is denoted by X'; that is, X' = X u {-}. Let us assume that e(t) = cc for t > 7-. The process thus constructed is denoted by e,(t). Other extensions of the process e(t) can be obtained as follows: Let S (for k = 1, 2, ) denote

a sequence of spaces that can be considered as distinct copies of the space f2. On S2k let us consider the measure {g,,, Pk} defined in the same way that {%, p} is defined on fl, but with measure {,u, 0} as the initial distribution. Let Nk denote the set in S2k analogous to N in fl: Nk = 1 w(k) w(k) E 1l

j=1

J

are independWe shall assume that the a-algebras of the events ent. Fort > 7_ we obtain e(t) = fit - 7,,, 01)). If p(y2) < -) = 0,

JUMP MARKOV PROCESSES

364

then e(t) is now defined for all t for almost all (w, w(1'). On the other hand, if p(7((1) < oc) > 0 we set fi(t) = f(t - 7_ - 7_1), (o(2)) and so forth. We note that if the function for t > Y- + q(t, x, A) is independent of t, then the variables 7_k) (for k > 1) are identically distributed and independent, so that the inductively defined process y(k), w(k+l))

fi(t) = f(t - 700 -

t<

0)

is defined for all t with probability 1. Let us look at the process E0(t) in greater detail. It follows from (11) that

/

m-1

Z. > t - k=1

P1 Zm > t + x \\

Zk

x)

from which we get the important conclusion PwlTm

e , 7,=+, < tm+r, Sm e Am,

- T < tm, z,.+, < tm+1,

m-1

Z1, .. . , 7-11 E Zk < k=1

T < k=1 i Zk

F(tm, tm+l, ... , Am, ... , Am+r T,

(26)

But this means that the conditional probabilities of the events in 6{do(t), t > T} with respect to the u-algebra a{e(t), t < T} depend only on o(T) = Em-1 (or on o(T) = 00, which is obvious); that is, eo(t) is a Markov process. One can easily show that the same conclusion holds for the process e(t). Theorem 4. Let q(t, x, A) satisfy the conditions a' and b' (page 362).

Then

a. eo(t) is a Markov process and its sample functions are, with probability 1, continuous from the right; b. the transition probabilities of the process eo(t) are defined by the relations:

P(t x v A) = E Pcn)(t x, v, A) n=o

v > t, A E

,

(27)

where

Pco)(t x, v, A) = qt(t, x, v)XA(X)

,

/

p(n+1)(t x, v, A) = V p(n)(0, Y, v, A)

tX

''I1(t, x, v)

= exp (_5Vq(o, x)dO)

,

(t, x, O)q(O, x, dy)d8 ;

n = 0, 1,

(28)

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

365

and p(t, x, v, {00 }) = 1 - p(t, x, v, X) ;

c.

the function P(t, x, v, A) (satisfies the first Kolmogorov equation: ap(t,

at-

- J xP(t, z, v, A)q(t, x, dz)

, v, A)

(29)

and the boundary condition lim p(t, x, v, A) = XA(X) ttv

d.

equation (25) is satisfied uniformly with respect to

t (for

0 < t < T where T is an arbitrary number) for fixed x and A. Proof. b.

a.

This was proved before the statement of the theorem.

Note that Theorem 2 holds for the process e0(t) since its

proof was based only on formula (8) and the fact that the first state 0 changes by a jump. Let p(")(t, x, v, A) denote the conditional probability that with the hypothesis e(t) = x # oo , the function E0(v) e A, and the function e0(t) has exactly n jumps on the interval [t, s]. Then it follows from (26) that P10)(t, x, v, A) = XA(x). Furthermore, if 6(0) = x for t < 0 < t + r, and (t + v1) then

on the basis of Theorem 2, P("+')(t, x, v, A) = M{p(")(t + zl, " v, A) I e(t) = x}

=

P(")(0,

f0

tX

y, v, A)q'(t, x, 0)q(0, x, dy)dO .

Therefore the function p(t, x, v, A) defined by equation (27)

is

the

probability of falling at the instant v into the set A after a finite number of jumps, after leaving the point x at the instant t. This completes the proof of (b). c. It follows from (27) and (28) that P(t, x, v, A) = ''V(t, x, v)XA(x)

+

fx 0(6, y, v,

x, 0)q(6, x, dy)dO .

(30)

Jt

From this it follows that lim P(t, x, v, A) = XA(X) ttv

and the function P(t, x, v, A) is continuous with respect to t. It

follows from the boundedness and continuity of the function p(t, x, v, A) with respect to t and the continuity in t of q(t, x, A) uniformly with respect to A (that is, the continuity of the variation of the measure q(t, x, A) as a function of t) that the integral P(B, z, v, A)q(B, x, dz) X

JUMP MARKOV PROCESSES

366

is continuous with respect to 0. Consequently equation (30) can be differentiated with respect to t. Thus op(t, x, v, A) at

-

= q(t, x)'I`(t, v, x)X A(x)

P(t, y, v, A)'T(t, x, t)q(t, x, dy) ,

+ t X P(0, y, v, A)q(t, x)q(0, x, dz)d0 = q(t, x)P(t, x, v, A) -

J JX

p(t, y, v, A)q(t, x, dy)

,

which completes the proof of (c). d. It follows from the continuity of the function q(t, x) with respect to t that lim

'I'(tl, x, t2)

t2 - tl = q(t, x) uniformly with respect to t (for 0 < t < T). In view of (30), it tlr t,tzlt

then follows that lim It2>titlo

1-

1 - p(tl, x, t2, {x}) = q(t, x) t2 - tl

It follows from (30) that p(tl, x, t2, A) is a continuous function of tl and t2. Keeping the continuity (mentioned above) of the inner integral on the right-hand side of formula (30), we obtain, for x A, p(t1t x' tt' A) 2

_ Xp(0, Z, t2, A)'(t1, x, 0)q(0, x, dz) ,

(tl <

0 < t2)

1

Therefore the limit lim p(tl, x, t2, A) = q(t, x, A)

tl
t-t 2

(x V. A)

1

exists and is uniform with respect to t (for 0 < t < T). This completes the proof of the theorem. REMARK 1. One can show in an analogous manner that the transition probabilities of the process e(t) also satisfy the first Kolmogorov equation and equation (25). From this it follows that if P(t, x, v, X) < 1 for some v > t > 0 and x e X, then the first Kolmogorov equation and the problem of constructing a Markov process from the given function q(t, x, A) have a nonunique solution. Corollary. The sample functions of the process eo(t) have with probability 1 a finite number of jumps in the course of a finite interval

of time if and only if

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

367

v>t>0,

p(t,x,v,X)= 1,

xeX.

(31)

The question of whether or not a Markov process has infinitely many jumps in a finite interval of time is of great interest. However Equation (31) is not convenient for determining this. We can

obtain a more suitable condition by confining ourselves to the homogeneous case, that is, by assuming that the function q(t, x, A) _ q(x, A) is independent of t. In the homogeneous case, P(n)(t,

x, v, A) = p(n)(v - t, x, A)

P(°)(t> x, A) = e-tq(,)xA(x)

exp {-0q(x)}P(n-1)(t - 0, y, A)q(x, dy)d9

P(n)(t, x, A)

,

where q(x) = q(x, X) and P(t, x, v, A) = P(v - t, x, A). If we set K(t, x) = 1 - P(t, x, X), then K(0, x) = 0 and it follows from equation (29) that aK(at x)

= q(x)K(t, x) -

dy) x K(t, y)q(x,

(32)

In this equation let us shift from the function K(t, x) to its Laplace transform: z(X, x) =

e-'tK(t, x)dt

x

.

We obtain (X + q(x))z(X, x) =

X

z(X, y)q(x, dy)

Let f(x) denote an arbitrary solution of the integral equation (? + q(x))f(x) =

JX

f(y)q(x, dy) ,

(33)

and let f(x) satisfy the condition sup,,, if(x) I < 1. Lemma 4.

For X > 0,

-z(X, x) < f(x) < z(X, x) Proof.

We have z(X, x) = limn. z(n)(X, x), where z(n)(X, x)

= x -e-'t(1 o

\\

-

P(k)(t, X, X ))dt k=o

//

Define Q(-)(X, x) = x fo J

Then

e-1t0(n)(t,

x, X)dt

JUMP MARKOV PROCESSES

368 Q(O)(X, X) _ X + q(x)

Q(k-1)(X, y) 4(x, dy)

Q(k)(X, X) / = z(n)(7

'

X+q(x) Q(k)( x)

, x) = 1 k=0

=1-

X

X + q(x)

-

f

x

(1 -

z("-1)(X, y)) 4(x, dy)

X + q(x)

from which we get (? + q(x))z(R)(a x) _

z("-1)(a

y)q(x, dy)

(34)

Since -z(0)(X, x) = - 1 < f(x) < 1 = z(') (X, x), by combining

(33) and (34) we obtain by induction -Z(")(X,, x) < f(x) 5 z(")(%, x)

and taking the limit as n, -, we obtain the desired assertion. Theorem 5. For p(t, x, X) to be equal to 1 for all t > 0 and x e X, it is necessary that equation (33) have no nontrivial bounded

solutions for arbitrary ? > 0, and it is sufficient that this condition be satisfied for some ? > 0.

If P(t, x, X) = 1, then z(X x) = 0 for arbitrary X > 0, and on the basis of Lemma 4 every bounded solution of equation (33) is identically equal to 0. On the other hand, if equation (33) Proof.

has no nontrivial bounded solutions for some X, then z(%, x) which is a bounded solution of equation (33), is identically equal to 0 and P(t, x, X) = 1. This completes the proof of the theorem. Example. Suppose that e0(t) for t > 0 is a birth and death process that is defined by the sequences {?,"} and J u.1 (for n = 0, 1, 2, ), where e0 = 0, X0 > 0, and X. > 0 for n > 1 (cf. p. 324).

Under what conditions do the sample functions of the

process remain bounded with probability 1 in the course of a finite interval of time? Obviously, this will occur only when these sample functions have finitely many jumps in a finite interval of time. Let us use the preceding theorem. Equations (33) in this case become an infinite system of linear algebraic equations (X + A 0)f(0) = XJ(l)

,

X,(.f(n + 1) -.f(n)) = a"(ff(n) - f(n - 1)) + Xf(n) , n > 1 . (35) If X0 # 0, then equations (35) determine all the f(n) for n 1 up to an arbitrary factor equal to f(0). On the other hand, if ? = 0

7.

THE BASIC PROPERTIES OF JUMP PROCESSES

369

we can set f(0) = 0. Then all the f(n) for n > 2 are uniquely determined up to the factor f(1). Since f(1) > f(0) > 0 (for X > 0),

we obtain by induction the result that f(n + 1) > f(n). Let us rewrite the system (35) in the form Jf (n + 1) - fn) = 7n(fin) - f(n (n - 1)) + Snf(n) 7

n>

and let us show that the sequence f(n), where f(0) = 1, is bounded if and only if E (Un + 7A-1 + 7n7,-1(ln-2 + n=1

.. . + 7n7n-1 ... 7201 + 7n7n-1 ... 71) < 00

(36)

.

We have ff(n + 1) -.f(n) = Unf(n) + 7nun-1.f(n - 1) + ... + ynyn_1 ... 7281f(1) + 'y11( (1) ff(n)(an + 7 A-1 + 'Y

yn Y

1)

... 71)

= p(n)f(n)

where

p(n) = an + 7,yUn-i + ... + 7n7n-1 On the other hand, ff(n + 1) - .f(n) ? p(n) (f(l) - 1)

.. '1'1

.

Thus

f(n) + pn(f(l) - 1) < f(n + 1) < f(n)(1 + pn) from which it follows that n

n

f(1) + (.f(1) - 1) k=1 1 pk c .f(n + l) c .f(1) 11 0 + pk) k=1

Since the series Ek=1 pk and the infinite product Hk=1 (1 + pk) converge simultaneously, we see that condition (36) is necessary and sufficient for the sequence f(n) to be bounded. Theorem 6. For the sample functions of the birth and death process do(t) to have with probability I finitely many jumps on an arbitrary finite interval of time, it is necessary and sufficient that n=1

+

(-iunhln-1 X flX fl-1X %_2

+ [ pn-1 ... //2 + X%Xn-1

X2X1

f//n ... )U21U1

'nX _1

X 2X 1

= 00

.

VIII DIFFUSION PROCESSES

In this chapter we shall consider continuous Markov processes with range in m-dimensional Euclidean space R(-1; Up to this point we have not completely described such processes. We shall now study an important class of these processes, the class of so-called diffusion

processes which, as the name suggests, can serve as a probabilistic model of the physical process of diffusion. In Section 5, Chapter VI, we considered a process of Brownian motion as a probabilistic model of diffusion in a homogeneous medium. Using a similar construction in the case of a nonhomogeneous medium, we arrive at the concept of a general diffusion process. Let us clarify the basic concepts of diffusion processes by giving an example of a one-dimensional process.

Let xt denote the coordinate of a sufficiently small particle suspended in a liquid at an instant t. Neglecting the inertia of the particle, we may assume that the displacement of the particle has two components: the "average" displacement caused by the macroscopic velocity of the motion of the liquid, and the fluctuation

of the displacement caused by the chaotic nature of the thermal motion of the molecules of the liquid. Suppose that the velocity of the macroscopic motion of the liquid at the point x and the instant t is equal to a(t, x). We assume that the fluctuational component of the displacement is a random variable whose distribution depends on the position x of the particle, the instant t at which the displacement is observed, and the quantity

At, which is the length of the interval of time during which the displacement is observed. We assume that the average value of this displacement is equal to 0 independently of t, xt and At. Thus the displacement of the particle can be written approximately in the form here, 370

xt+ot - x, = a(t, xt)Ot + t,st,at ; (I ) 0. If a(t, x) is equal to 0 and the distribution of

1.

DIFFUSION PROCESSES IN THE BROAD SENSE

371

t,xt,at is independent of x and t, as we assumed when we were considering Brownian motion (cf. Remark 1, Section 5, Chapter VI), then

Since the properties of the medium are naturally assumed to change only slightly for small changes in t and x, the XAt.

process is homogeneous in the small. Therefore, we may assume that ,xt,at = 6(t, xt)et,at

where c(t, x) characterizes the properties of the medium at the point x at the instant t, and t,at is the value of the increment that is obtained in the homogeneous case under the condition that 6(t, x) = 1. Thus St,at must be distributed like the increment of a process of Brownian motion: w(t + At) - w(t). Consequently, for the increment xt+at - xt, we can write the approximate formula

xt+at - xt s a(t, xt)Ot + 6(t, xt)[w(t + At) - w(t)] . (2) To make this formula precise, we replace the increments, as one frequently does in mathematical analysis, with differentials. When we do this we obtain the differential equation for xt, dxt = a(t, xt)dt + o'(t, xt)dw(t) , (3) which we may take as our starting point in determining the diffusion process.

Let xt denote a multidimensional process with range in R(m). Then equation (1) remains meaningful if a(t, xt) is a function with range in Rcm, and t,xt,at is a random vector in Rcm>. In this case we assume that Et,xt,at can be represented in the form m

St,xt,at =k=1 L bk(t, xt)[wk(t + At) -

Wk(t)]

where the bk(t, xt) are functions with ranges in R(m) and the wk(t) are independent one-dimensional processes of Brownian motion. Such a representation corresponds to a nonisotropic medium: the displacements in the different directions have, in general, different distributions.

The equation for the variable xt in this case takes

the form vn

dx, = a(t, xt)dt + Y, bk(t, xt)dwk(t)

(4)

k=1

We note that we cannot as yet give a precise meaning to equations

(3) and (4).

The difficulty lies in the fact that the quantity w(t + At) - w(t) At

where w(t) is a process of Brownian motion, has a normal distribution

DIFFUSION PROCESSES

372

with mean zero and variance l/Ot, and hence this quantity does not have a limit in any probabilistic sense. Since w(t) does not have a derivative, the usual definition of the differential dw(t) has no meaning. We shall give a precise meaning to equations (3) and (4) when

we introduce the concepts of a stochastic integral and stochastic differential in Section 2. In Section 1 we shall define a diffusion process, beginning with the properties of transition probabilities.

In Sections 3 to 6, we shall study the solutions of equations (3) and (4) from the point of view of their existence and uniqueness, and of the properties that will enable us to determine the distributions of the basic characteristics of the process.

1.

DIFFUSION PROCESSES IN THE BROAD SENSE

Let us first consider the one-dimensional case of diffusion processes in the broad sense. Let e(t) denote a Markov process in the

broad sense and defined on [0, T] into

R(11.

This means that the transition probabilities p(t, x, s, A), of a process are given and satisfy conditions a-c of Section 1, Chapter VII. A process e(t) is called a diffusion process if the following conditions are satisfied. For every x and every s > 0,

a.

p(t, x, s, dy) = o(s - t)

(1 )

Ix-YI>E

uniformly over t < s; b. there exist functions a(t, x) and b(t, x) such that for every x and every s > 0, (y - x)P(t, x, s, dy) = a(t, x)(s - t) + o(s - t) 1--Yl_<-

(y - x)ZP(t, x, s, dy) = b(t, x)(s - t) + o(s - t)

,

(2) (3)

I

uniformly over t < s. The function a(t, x) is called the coefficient of transfer and it is an interpretation of the function a(t, x) defined above; the function b(t, x) is called the coefficient of diffusion. A connection between a and o on the one hand and the coefficients of diffusion and transfer

on the other will be established in Section 3. One can show that conditions (1) to (3) are insufficient for a

1.

DIFFUSION PROCESSES IN THE BROAD SENSE

373

unique determination of the transition probabilities of the process.

Let us show that under certain assumptions regarding the transition probability p(t, x, s, A) of the diffusion process, it is completely determined by the coefficients a(t, x) and b(t, x) if these coefficients are such that the Cauchy problem for the equation

-

au

at

= a(t, x) au + ax

1

2

a-u

b(t, x)

axe

u(s, x) _ fi(x)

;

in the region x e R"', t e (0, s) has, for every s e (0, T], a unique solution for all initial functions q(x) belonging to some class of functions that is everywhere-dense with respect to the metric of uniform convergence in the space of all continuous functions. To do this, let us prove: Theorem 1 (Kolmogorov). Let T(x) denote a continuous bounded function such that the function u(t, x) = p(y)p(t, x, s, dy) has bounded continuous first and second derivatives with respect to x and let the functions a(t, x) and b(t, x) be continuous. Then u(t, x) has a derivative au/at and in the region t e (0, s), x e R1', which satisfies the equation au

au

at - a(t, x) ax

I

+

2

b(t,

a2

(4)

x) ax

and u(t, x) satisfies the boundary condition lim,1. u(t, x) = p (x). Equation (4) is called the first (converse) equation of Kolmogorov. Proof. Validity of the boundary condition follows from the equation u(t, x) _ p(y)p(t, s, x, dy) = 9D(x)

+ +

(y) - T(x)]p(t, s, x, dy) = q)(x)

[

171-x15e

[

(y) - 9,(x)]p(t, s, x, dy) + o(s - t)

,

the continuity of the function T, and the arbitrariness of s > 0. To derive equation (4), we note that it follows from condition 3, Section 1, Chapter VII, that u(tl, x) = p(t1, x, t2, dz)u(tz, z)

(t1 < t2 < s)

Using the Taylor expansion of the function u(t, z) to get

u(t, z) - u(t, x) = 2

+

x)

az

at, x) (z - #(l + as)

,

1z-xI<6

DIFFUSION PROCESSES

374

where I a, <_ supt,iz-x1s (a2u/ax2)(t, x) - (a2u/ax2)(t, z) 1, we may write

u(t x) - u(t2, x) = [u(t2, z) .- u(t2, x)]p(t x, t2, dz) 1z-x15e

M4, Z) - U(4' x)]p(t x, t2, dz) + o(t2 - t1)

au (t2, x) (z - x)p(t1, x, t2, dz) Ix-zI5c ax

+I

a2u(t"

x)2p(t,, x, t2, dz)(1 + aE) + o(t2 - t,)

x)

By using formulas (2) and (3), we obtain u(t1, x) - u(t2, x) t2 - t1

= a(t, x) au (t, x) + ax

b(t1, x)

a2

2

axe

u(t, x)(1 + a.) + o(t2 - t1) t2 - tl

}

from which we get equation (4) by taking the limit as t2 - t1 0 and a 0; This completes the proof of the theorem. If the solution of equation (4) is unique with initial function

q(x), by solving it we find the unique function u(t, x), and the integrals p(y)P(t, x, s, dy) ,

as p ranges over some class of functions that is everywhere-dense in the space of continuous functions, uniquely determine the measure P(t, x, s, A).

In the case in which the transition probability p(t, x, s, y) has a density, that is, when there exists the function for which p(t, x, s, A) =

p(t, x, s, y)dy , JA

the Kolmogorov-Chapman equation takes the form p(t1, x, t3, y) = P(t1, x, t2, z)P(t2, z, t3, y)dZ

(t, < t2 < t3)

(5)

If p(t, x, s, y) is sufficiently smooth as a function of s and y, we can derive from (5) an equation for this density, known as the Fokker-Planck equation or the second (direct) equation of Kolmogorov.

Theorem 2. Suppose that relations (1) to (3) are satisfied uniformly with respect to x for the process e(t) and that there exist continuous partial derivatives ap(t, x, s, y)/as, (a/ay)(a(s, y)p(t, x, s, y)),

1.

DIFFUSION PROCESSES IN THE BROAD SENSE

375

y2 (b(s, y)p(t, x, s, y)) ,

Then, p(t, x, s, y) satisfies the equation ap(t,

s, y)

ay (a(s, y)p(t, x, s, y))

as

+

1

a2

(6)

_- (b(s, y)p(t, x, s, y)) 2 -ay2

in the region s e (t, T), - co < y < oo. Proof.

Let g(y) denote an arbitrary function that

is twice

continuously differentiable and that vanishes outside some finite interval.

Just as in the proof of Theorem 1, we can show that

lim 1 h-o h

x, s + h, y)dy - g(x)]

(7)

= a(s, x)g'(x) + 2 b(s, x)g"(x) uniformly with respect to X. Using (5) with tl = t, t2 = s, t3 = s + h and (7), we obtain as

JP(t, x, s, y)g(y)dy = o

= lim h h-,0

x, s, y)g(y)dy

x, s + h, y)g(y)dy - p(t, x, s, z)g(z)dz]

= lim 1 P(t, x, s, h-.o

p(t,

h

z)[Jp(s,

z, s + h, y)g(y)dy - g(z)]dz

= P(t, x, s, z)[a(s, z)g'(z)dz + 2 b(s, z)g"(z)dz] Integrating the last integral by parts, we obtain asp(t, x, s, y)g(y)dy =

[__(a(s

,

z)p(t, x, s, z))

2

s2 (b(s, z)p(t, x, s, z))]g(z)dz

+ 2 If instead of g(z) we take the functions ne- -uI and take the limit as n -, we obtain proof of the theorem (by using the continuity of the derivatives, referred to in the hypothesis). 1

REMARK 1. The reader who is familiar with the concept of a generalized function will easily see that equation (6) is always satisfied (under the assumption that the diffusion process has a

376

DIFFUSION PROCESSES

density) if the right and left members of this equation are regarded as generalized functions on the space of infinitely differentiable functions having compact support. REMARK 2. Let Fo(x) denote the distribution function of E(0). Writing equation (6) for p(0, x, t, y) and integrating it with respect to x, we obtain the following equation for p(t, y), the distribution density of g(t): atp(t, y)

(8)

y (a(t, y)p(t, y)) + 2 aye (b(t, y)p(t, y))

If instead of the density p(t, y) we consider the distribution function F(t, y) of the variable E(t), we obtain the following expression for it from equation (8): at

F(t, y) _ -a(t, y) aFat, y) + y

2

ay

(b(t, y) a F(t, y))

(9)

y

To equation (9) we must add the initial condition F(0, y) _ F0(y)

In the multidimensional case a process e(t) is called a diffusion process if relations (1) and (2) are satisfied for the process (a(t, x),

x, and y are now vectors in the space R(m containing the range of e(t)), and if there exists a symmetric linear nonnegative operator B(t, x) such that for all x, z e RI-1 and all s > 0,

(z, y - x)2p(t, x, s, dy) = (B(t, x)z, z)(s - t) + o(s - t)

(10)

IX-Ylfg

uniformly over s > t. The operator B(t, x) is called the diffusion operator of the process , em denote an orthonormal basis in R'"`), let e(t). Let ell e2, , y"a denote the coordinates of the vectors x and xl, . , x"`; yl,

y in R(-), and let b%,(t, x) denote the elements of the matrix of the operator B(t, x) in that basis. alent to the condition x-,JSE

Then condition (10) is equiv-

(y' - x`)(y' - xj)p(t, x, s, dy)

= bti,(t, x)(s - t) + o(s - t)

,

i, j = 1,

.

,m.

For multidimensional diffusion processes, Theorems can be reformulated as follows:

1

(11)

and 2

Theorem 3. Let f(x) denote a continuous function such that the function u(t, x) = f f(y)p(t, x, s, dy) has continuous partial derivatives

1.

DIFFUSION PROCESSES IN THE BROAD SENSE

a u(t, x)

377

u(t, x)

az

,

axtiaxj

axti

for i, (j = 1, , m), and let the coefficients ati(t, x) and btij(t, x), for i, j = 1, , m, be continuous. Then for x e R("z) and t e (0, s), the function u(t, x) has a derivative au/at which satisfies the equation -au(t' x) = E ati(t, x)au(t, x) + - E btij(t, x) aEU(t' x) (12) at

2 ti,j=t

axti

ti=1

axtiaxj

and u(t, x) satisfies the boundary condition limt,e u(t, x) = f(x).

If the transition probability p(t, x, s, y) has a density p(t, x, s, A) =

p(t, x, s, y)dy

,

A

we have:

Theorem 4. If relations (1), (2), and (10) are satisfied uniformly with respect to x and if there exist continuous derivatives as

p(t, x, s, Y) ' aZ

axtiaxj

axti

(ati(s, y)p(t, x, s, y))

(btij(s, y)p(t, x, s, y))

then p(t, x, s, y) for y e R(m and s e (t, T) satisfies the equation ap(t, as

s'

Y)

- E axti (ati(s, y)p(t, x, s, y)) (btij(s, y)p(t, x, s, y)) +1 2 ti,j=1 axax'' a2

(13)

The proofs of Theorems 3 and 4 are analogous to the proofs of Theorems 1 and 2. If we integrate equation (13) with respect , x', we obtain (as in Remark 2) an equation for the to x', unconditional density of the distribution of the variable E(t). In what follows, when verifying whether or not conditions (1) to (3) are satisfied, we shall find it convenient to use: REMARK 3. For the process e(t) to be a diffusion process, it is sufficient that for some a > 0,

y-x

2+3 p(t, x, s, dy)

= o(s - t)

,

(14)

and that relations (2), (3), and (10) are satisfied for s = +co. This is true because if equation (14) is satisfied, then for k = 0, 1, 2, Iy - x Ikp(t, x, s, dy) 1

< s2 + a - k I y - x

z+ap(t, x, s, dy)

= o(s - t)

DIFFUSION PROCESSES

378

2.

ITO'S STOCHASTIC INTEGRAL

Let w(t) denote a process of Brownian motion. Suppose that a set of a-algebras `z t defined for t > 0 on the basic probability space {U, C25, P} is related to w(t) as follows: (1) gtl c vt2 for tl < t2; (2) w(t) is measurable with respect to 2 t for every t; (3) for every h, the process w(t + h) - w(h) is independent of any of the events of the a-algebra ,. Let Ut2[a, b], where 0 < a < b, denote the

set of functions f(t) = fit, u) that are measurable with respect to the set of variables (t, u), that are defined for t e [a, b] and u e U, and that satisfy the two conditions: (a) f(t) is measurable with respect to the a-algebra 2gt for every t in [a, b] and (b) the integral I fit) I2dt is finite with probability 1. For all functions in M 2[a, b], the integral f(t)dw(t) will be defined below.

A function f(t) is called a step function if there exists a parti-

tion of the interval [a, b], a = to < tl < . . . < t,. = b, such that , r - 1; In what follows f(t) = fits) for t e [ti, is+1) for i = 0, we shall often use: Lemma 1. For every function f(t) e T22[a, b] there exists a sequence of step functions fa(t) e T 2[a, b] such that with probability 1, lim aI.f(t) - fn(t) n--n-- a

2dt=0.

Proof. If fit) is a continuous function with probability 1, the assertion of the lemma is obvious (since we can set fa(t) = f(k/n) for k/n S t < (k + 1)/n). If f(t) is a measurable function bounded by a constant independent of u, then there exists a sequence {spa(t)}, where all the pa(t) are bounded by a single constant, such that {q (t)} converges to f(t) for almost all t with probability 1. This means that with probability 1, ba

lim

f(t) - cpa(t) 2dt = 0 t

.

(For example, we may set pa(t) = n (en(8-tf(s)ds.) Finally, an arbitrary function in J322[a, b] can be approximated in the mean-square by a bounded function with any desired degree of accuracy. Thus the step functions are dense in the sense of the convergence that we are considering in V2[a, b]. This completes the proof of the lemma. Let us define the integral af(t)dw(t) for the case in which the a

2.

ITO'S STOCHASTIC INTEGRAL

379

function f(t) is a step function. Suppose that f(t) = f(ti) for t e [ti, ta,+1), where a = to < tl < ... < tr = b is a partition of the interval [a, b]. We then define f(t)dw(t) = E0 f(tk)[w(tk+l) - w(tk)] k=

a

We mention a few properties of this integral. 1. If M( f(t) I ¢))/ <{- for t e [a, b], then with probability 1, I

Ml 5 bJ (t)dw(t)

a1=

I

a

0

( 1)

This is true because M(f(tk)[w(tk+) - w(tk)] ff

= M[M/'(.f(tk)[w(tk+l)

CG 7

w(tk)] I Z)tk)ccI U¢]

M[f(tk)M[W(tk+) - w(tk) I

tk]

I Ua] = 0

-

by virtue of the fact that w(tk+1) w(tk) is independent of tk and has mean zero. 2. If M(I f(t) 2 a) < 00 with probability 1 for t e [a, b], then I

M\LJaf(t)dw(t)]2a/

- aM(f2(t) I ga)dt

(mod p)

.

(2)

This is true because b

M([\ f(t)dw(t)]2 + 2 j
7-0

M[f(tk)2(W(tk+1) - w(tk))2 I M[f(tj)f(tk)(w(tj+1) - w(tj))(w(tk+1)

r-1

C

E M{f(tk)2M[w(tk+) - w(tk))2 I

Utk] 1!U aJ

k=O

"t)a]

- w(tk))

+ 2 i
tk)

I

a]

I `-a)dt E M(f(tk)2 I Ua)[tk+1 - tk] = J bM(f(t)2 a k=0

.

Let us show that for a sequence of step functions ff.(t) such that b[ f(t) - ff(t)]2dt .- 0 in probability, the sequence b fa(t)dw(t) ¢ Ja converges in probability to some limit. To do this, we shall need the following property of integrals of step functions: 3. If q(t) is a step function, then for every N > 0 and c > 0, a99(t)dw(t)

> c} < N +

q(t) I2dt > N}

.

(3)

To see that this is true, suppose that T(t) = cp(tz) for tQ <- t <

DIFFUSION PROCESSES

380

ti+l, where a = to < t, < . . . < tr = b. Since cp(t) is measurable with respect to 2t. for t e [ti, ti+,), it follows that the quantity ti+i

1 T(t) 12dt is also measurable with respect to 2ati. ti+l1

Let us define

q(t) 12dt 5 N, where t;+, is such that t; < t < 'P'(t) = p(t) if tt+l ti, and let us define 4 t 2dt > N. N t = 0 for t e [ti, ti+l if

The function p,(t) is a step function belonging to

t2[a, b].

Since

TN(ti) 12(ti+l - ti) < N ,

2dt = E

it follows that I CgN(ti) 12
2= JbcpN(t)dw(t)

M

b

5

M I p,(t) 12dt < N.

Finally,

l jf N} > o} = P{ a c(t) 12dt > Using the last two relations, we obtain b

P{suP 19)N(t) - q(t) I

P{ f aacp(t)dw(t)

=

P{

b

J

< P{

b

J

>

.

c}

b('P

2N(t)dw(t) +

(t) -

p,(t)dw(t) > c} + P{

PN(t))dw(t)

> c} b[T (t)

gN(t)]dw(t)

> 0}

M 'q,,(t)dw(t)

<

a

+

C2

p(t) 12dt > N}

which completes the proof.

Now let I f.1 denote a sequence of step functions such that b

0 in probability. Then b I fa(t) - fm(t) I2dt also a approaches 0 in probability as n and m approach oo . Consequently, a for every s > 0, I f(t) - fa(t) 12dt

(11b

lim P{J If n(t) - J fm(t) 2dt > F} = 0

Using property 3, f/we can write, for arbitrary s > 0 and 6 > 0,

im P {

n' M--

E d

J aJ

a(t)dw(t) -

+ a,m limw

J a m(t)dw(t) ff

> o}

b

P{ J¢

I .fa(t) - fm(t) Wt > s} 2

2.

ITO'S STOCHASTIC INTEGRAL

381

so that because of the arbitrariness of s > 0 we have

slim Pj

fn(t)dw(t) -

fm(t)dw(t)

2>a}-0

J

for every 8 > 0. It follows from this equation that the sequence of the random variables f (t)dw(t) converges in probability to b

some limit. This limit is independent of the choice of the sequence {fa(t)} for which bI f(t) 0. (If there are two such sequences {fa(t)} and {fa(t)}, by combining them into a single sequence

-

we see that with probability 1, the two sequences have the same limit.) Let us define 6f(t)dx,(t)

= P-lim a

We shall call this limit Ito's stochastic integral of the function f(t). The definite integral defined in this manner is a homogeneous additive functional of the function f(t) on W2[a, b]. Furthermore,

for a < c < b, 'f(t)dw(t) +

f(t)dw(t) = f bf(t)dw(t)

.

a

a

(4)

Proof of these properties is obvious for step functions and it carries over to the general case by means of a trivial limiting operation. Applying the limit for step functions obtained in (3) to arbitrary functions in E2[a, b], we see that property 3 is valid for all f(t) e M2[a, b].

With the aid of this property we can show that: 4.

If f(t) e U2[a, b], b If

e T22[a, b], and

(t) - f(t) I2dt _ 0

in probability, then p-lim f(t)dw(t) Ja

¢

Finally let us prove a property that generalizes properties

1

and 2.

2'.

If the function f iss% such that bM(I

\t) I2

I g'a)dt < o0

with probability 1, then b f(t)dw(t)

M(J a

19-a)

= 0 (mod p)

(5)

DIFFUSION PROCESSES

382 and

M([ .f(t)dw(t)]z z`-a) = a

a

(mod p)

M(j.f(t) III

.

(6)

To prove these properties, let us show that in the case in which

a)dt < -, there exists a sequence of step functions such that bM(I f(t) - fn(t) I2 Ia) 0 in probability. Let {f-(t)} bM(I f(t) 11

I

fa

denote the sequence of step functions in E2[a, b] such that p-lim

I f(t) - fn(t) I2dt = 0

.

a

Let us set g,(x) = x for I x I < N and g,(x) = Nx/I x I for Since I g,(x) - g.,,(y) I < I x - y I, we have

I x I > N. rbI

g,(f(t)) - DN(f.(t)) 2dt in probability.

0

(t) - n(t) 12dt

a a

Since the quantities rbI Ja

g:\'(f(t)) - gN(fm.(t))

2dt

are bounded by the number 4N2(b - a), and since their sequence converges to 0 in probability, it follows on the basis of Lebesgue's theorem that M0a g (f(t)) - gN(ffa(t)) 2dt I

a)

bM(j a

gN(f(t)) - gN(fn(t))

12 I

z`

a)dt

0

in probability.

On the otherhand, bM(I g.(f(t))

- f(t)

I2 z`

I

a)dt

0

in probability (again by Lebesgue's theorem), since I f(t) - gN(f(t))

12

I f(t)

12

and

bM(If(t) 12 12 a)dt < 00

Therefore we can choose a sequence {Nn} such that a

M(If(t) - gN,y(fn(t)) 12 Z-a)dt

- pN,y(f(t))

+ in probability.

12 I

"Ua)

2\bM(I g N,y(f(t)) - gN (fn(t)) 12 I a-a)dt .--> 0 a

2.

ITO'S STOCHASTIC INTEGRAL

383

The functions gN,a(fa(t)) are step functions and can be chosen for the functions fa(t), whose existence we are proving. Substituting into (1) the functions fa(t) that we have constructed and taking the limit as n --. co, we obtain (5). To obtain (6), we note that if the sequence in probability and if m I a - c,a 11 .__, 0 as n, m - 9 oc, then M 1$ fore

0 and hence Min

f(t)dw(t)]2 g-a) = lim M([f afa(t)dw(t)]2

M\LJ

lim yM(I fa(t) I2 t)-Jdt = JaM(I f(t) 12 n-w J

ME2.

There-

a/ g_a)dt

.

I

Let us now consider the stochastic integral as a function of Let *2(s) denote the function that is equal for s < t and equal to 0 for s > t. If f(s) e 1U 2[a, b], then to f(s),frt(s) e l2[a, b] for every t e [a, b]. We define the integral f(s)dw(s) for all t by the upper limit. 1

f(s)dw(s) a

f(s)1Jrt(s)dw(s) a

It follows from the definition of a stochastic integral that this integral is defined probabilistically. Therefore, as a function of the upper limit, the integral is defined up to stochastic equivalence (cf. Section 1, Chapter IV).

In what follows we shall always assume

that the values of the integral as a function of the upper limit, for different values of t, are compatible in such a way that C(t) = \f(s)dw(s) is a separable process. The possibility of doing this follows from Theorem 2, Section 2, Chapter IV. Let us note the basic properties of the function C(t) _

f(s)dw(s) a

5.

If f 'm(j f(s) I2 Z-a)ds < oc, then a

PlQupb

f(s)dw(s)

aJ

`

J

G

>c

(7) C2

and

sup

P

a;Vigb

f(s)dw(s)

2

JM(I f(S) I )gds.

.

(8)

It will be sufficient to prove inequality (7). Let us choose a partition of the interval [a, b]: a = t, < t, < ... < to = b. We define bk = tkf(s)dw(s), and we define Xk = 1 if C; I < c for i < k Ja

DIFFUSION PROCESSES

384

and I x I > c, and xk = 0 otherwise. Obviously Ek=1 xk < 1 and Xk is measurable with respect to 5t5tx, and since I Ck I2xk > C2xk, we have y bn

++ / = bnk=0 L xk

y

yy

k=0

yy yy b kXk + 2k=0j bk(bn - byyk)/L.k yy

y + yy / > C2 n xk + 2 nj byk(Sny - bk)xk + Lk=0G. - bk)2 xk k=0 k=0 yy

While taking the conditional mathematical expectation of both sides with respect to Z-a, we note that on the basis of (5), M(bn - bk 19tk) = M(fs)dw(s)

tk)

tk

= 0.

Therefore yyk)/lk

M(bk(bn

b

I

"-a) =

bk

I

m) = 0

and then using equation (6), we obtain M(Cn aa)

? C2M(I xk i a)

We note that D=0 xk is equal to 1 if sup I Ck I > c and equal to 0 otherwise.

Therefore

M (E k=0

xk

a)

= PS SUP ISkI > c 15k5n

Thus we have proved the inequality P to

> C 12 aI <

SUP [

C

aM(I f(s) I2

)ds

,

from which we easily obtain property 5 by using the separability of the process f(s)dw(s). t 6. A separable process C(t) _ f(s)dw(s) is continuous. If f(t) is a step function, the continuity of C(t) follows from the continuity of w(t) and the formula defining C(t). Suppose that the function f(t) e 9Y2[a, b] is such that bM If(s) I2ds < -. Let { fn(t)} denote a sequence of step functions such that

-a

Jim bM If(s) - fn(s) 12ds = 0

By virtue of property 5,

Pttsup

f(s)dw(s) -

fn(s)dw(s)

162 'M If(s) a

>s}

- fn(s) I2ds .

Choosing a sequence {ak} (-0) and {nk} such that

2.

ITO'S STOCHASTIC INTEGRAL

385

j 1 'M I f(t) - fnk(t) I2dt < 00 k=1$?

,

a

we see that

i

P{ asupa

Jaf(s)dw(s) - ¢f k(s)dw(s) > Sk} <

and on the basis of the Borel-Cantelli lemma (Theorem 2, Section 3, Chapter III) we have with probability 1, a

sup f f(s)dw(s)

-

f,k(s)dw(s)

sk

a'

beginning with some number k. Jrtf(s)dw(s) is with probability 1 the uniform limit of a Thus a sequence of continuous functions. Therefore, with probability I this limit function is also continuous. Suppose finally that f(t) is an arbitrary function in 7t2[a, b]. Let us define ,(s) = f(s) if

,,(s) = 0 if

I flu) I2du < N, and P{a!ltp6

at f(s)dw(s)

-

I f(u) I2du > N.

> o} <

Then

P{J¢I f(s)

ta

t

Since \fN(s)dw(s) is continuous and since the probability in the right-hand member of the last inequality can be made arbitrarily small, the process at f(s)dw(s) is continuous in the general case. a

We shall need the following inequality regarding the fourth moment of a stochastic integral. 7. If f(t) is a member of Pt2[a, b] such that M f(t) 'dt) < -, J

then

M

f(t)dw(t)

"

< 36(b

a

- a)J6M I f(t) I4dt

(9)

.

a

Suppose first that f(t) is a step function for which f(t) = f(ti) for ti < t < ti+1, where a = to < t1 < . . . < tr = b is a partition of the interval [a, b]. Then 4

M

af(t)dw(t) 4 = M

Eof(ti)[w(ti+) - w(ti)]

M k=U L1 f(t)I4[w(tk+) - w(tk)]4 r-1

k-I

2

+ 6 k=0 EM E f(ti)[w(ti+1) - w(ti)] 4=0

I f(tk)

I2[w(tk+) - w(tk)]2

r-1

3E M I f(tk) k=0

r--+I

+6

kk-1

I4(tk+, - tk)2 2

Mi E f(ti)[w(ti+1) - w(ti)] ( f(tk) I2(tk+1 - tk) , k=1 6=0

386

DIFFUSION PROCESSES

since the mathematical expectations of those terms for which the w(tk) with highest index is raised to an odd increment w(tk+l) power are equal to 0, and for m = 1, 2, , we have*

-

M([W(tk+1) - w(tk)]294

2atk) = (2m - 1 )! I (tk+l - tk)' -

For an arbitrary step function f(t) we may assume that the intervals [tk, tk+1] are chosen in such a way that maxk [tk+, - tk] is arbitrarily tk] -- 0 in small. Therefore, by taking the limit as maxk [tk+1 the last relation, we obtain

-

4

=

.f(t)dw(t)

M

65

(10)

M af(s)dw(s) 2I.f(t) I2dt

By applying the Cauchy-Schwarz inequality, we 'may write b

1

f(s)dw(s)

M

f(t)12dt 4

bM I.f(t) 4dt JbM a

dt

It follows from formula (10) that M

J

af(v)dw(v)

increases with increasing t.

6 ta M

I

a

.f(s)dw(s) 21 f(v) 12dv

Hence,

¢M at f(s)dw(s) 4dt < (b

- a)MI ` f(s)dw(s) 4

Thus, M

f(t)dw(t) 4 < 6 11(b

- a)[

b M

(f(s) 14ds]M

f(s)dW(s) 4

Formula (9) then follows for step functions f(t).

Proof of this

formula in the general case can be obtained by constructing for f(t) a sequence of step functions f (t) so that Jim bM I f(t) - fn(t) 14dt = 0

(Such a sequence can be constructed by means of a slight modification of the device used in proving property 2.) We now introduce the concept of a stochastic differential. Let C(t) denote a process that is measurable for every t with respect to fit. Supposed that there exist b(t) e M2[a, b] and a(t), which is measurable for every t with respect to zit and has a finite integral I a(t) I dt with probability 1, such that, for all a < t1 < t2 < b, b

Translator's note: The symbol n!!, common in Russian mathematical literature,

indicates the product of all natural numbers <_n that are congruent to n mod 2; thus 8!! = 8.6.4.2 and 7!! = 7.5.3.1.

2.

ITO'S STOCHASTIC INTEGRAL

387

+ Jt2b(t)dw(t) b (t2) - b(t1) = Jt2a(t)dt t1 tl

Then, we say that a(t)dt + b(t)dw(t) is the stochastic differential of the process C(t) and we write do(t) = a(t)dt + b(t)dw(t)

We shall now establish an important property of a stochastic differential, namely, Ito's formula for differentiating a composite function. 8. Suppose that the process Z(t) has a stochastic differential a(t)dt + b(t)dw(t) and the function u(t, x) (nonrandom) is defined for t e [a, b] and x E R(1); suppose that this function is continuous, and that it has continuous derivatives u,(t, x), u'(t, x), and u" (t, x). Then the process ij(t) = u(t, C(t)) also has a stochastic differential

and dra(t) = [u(t, C(t)) + u'x(t, C(t))a(t)

+ 2 u'l (t, C(t))b2(t)]dt + u'(t, C(t))b(t)dw(t) .

(1 1)

Let us first prove formula (11) for the case in which a(t) and b(t) are independent of t. Suppose that t, = yyt(0) < ... < t("") = t2. Then "p/(t2) - "/(t1) = u(t2, S (t2)) - u(tl, b (t1)) [u(t(k+1), y

b(t(k+1)))

- u(t(k), b(t(k)))]

/

k=o

Using Taylor's formula, we can write u(t(k+1), y b(t(k}1))) - U(t(k), b(t(k)))

ut(t(k) + Okk(t(k+1) - t(k)),

+

b(t(k+1)))(t(k+1)

- t(k))

ux(t(k) b(t(k)))[C(t(k+1)) - Y(t(k))] + 2 Ulf (t(k), y S(t(k))

+ Bk[S(t(k+1)) -

C(t(11))])[S(t(kk+1))

- b(t(k)]2

where 0 < Ok
- b(t(k))]) u,, (t(k), b(t(k))) < R

u" (t(k), b(t(k)) + Since

01

a'

DIFFUSION PROCESSES

388

L[

(t(k}1)) - b(t(k))]2 G 2a2 Y, (t(k}1) - t(k))2

+ 2b2 E

Lw(tk}1) - W(t(k))lJ2

and

M E LW(t(k+1))

- w(t(k))I2 = tz . - t

1

,

we see that + Bk(t(k +1) _ t(k)),

(t(k)))(t(k+1)

- t(k))

b

L ut(t(k)

- L Ut(t(k), b(t(k)))Atk I= 0(a) I E uxx(t(k), b(t(k)) + 01 [C(t(k}1)) L uxx(t(k), n

-- N

k=1

[

y

S(t(k)))[C(t(k+1))

(t(k}1)) - y

I

yy

yy

1)) - b(t(k))]2

b(t(k))l)Lyy

S(t(k

- b(t(k)fl2

I

-s 0

in probability as max (t(k+ 1) - t(k)) 0. If we replace C(t(k+')) -C(t(k)) with the expression a(t(k 11) - t(k)) + b(w(t(k}1)) - w(t(k))) and remember that n-1

a2 Y,

(t(k+1) _ t(k))2

0

k=0

and n-1

b F, (W(t(k-1-1)) - W(t(k)))(t(ky1) - t(k)) '0 k=0

in probability as max (t(k+1) - t(k)) --> 0 k

(the latter in view of the continuity of the process w(t)), we obtain, n-1

"/(t2) - v(t1)

max(tP

n-1

+ Lk=0ux(t(k),

yy

llm

(k))-,0

ut(t(k), b(t(k)))(t(k, 1) - t(k)

[No

S(t(k)))a(t(k+l) - t(k))

n-1

+

yy

k=0

ux(t(k), S(t(k)))bLw(t(k, 1)) - W(t(k))

7

b2 n-1

11k) C(t k) ))(t k+1) - (k) + -2 Ek=ouxx(t, t

+

b2

(t(k+l) E1 uxx(t(k), b(t(k)))L(W(t(k}1)) - W(t(k)))2 _

2 k=0

/

t(k) )1]

1

Obviously, the limits of all summations except the last are equal to the corresponding integrals. To prove formula (11) in the present case, it suffices to show that n-1

E uxx(t(k), b(t(k)))LLW(t(k 1 k=0

1))

- W(t(k))]2 - (t(k 11) _ t(k))] J

2.

ITO'S STOCHASTIC INTEGRAL

389

converges to 0 in probability. We define

- (t(k}1) - t(k)

Sk = [W(t(k+1)) - w(t(k))]2

and we let ,ykN) denote the characteristic function of the event /L {1 e(t(ry))

I < N for i < k} .

Then kN)Sk12

MCK1 uz(t(k), S(t(k)))h k=0

/)

n-1

L

yy(t(k) + MU11 xx(t(k) , b ))

SUp k _ t,IxISN

k=O

3 sup

I

(t(k+1) _ t(k))2

I U11 xx (t, x) 12 E

utt (t, X) 12 xx(+ ---3

2

MEk

0

t,IxISN

and n-1 P

lk= 0

uxx(t(k) b(t(k)))2(1 - ^hkN)) s

as N -+ a o.

0


N0

b(i)

This completes the proof of formula (11) for constant

a and b. It is easy to show that this formula remains valid for step functions since the domain of definition of a step function C(t)

can be partitioned into finitely many intervals, throughout each of which a and b are constant. In the general case, we may choose sequences {a,a(t)} and {b,(t)} of step functions such that with probability 1, bI a(t) - an(t) I dt ---> 0 , `61 bn(t) - b(t) 12dt --> 0 a

and the sequence of the processes

Cn(t) = C(a) + ta,t(s)ds + Jb"`(s)dw(s) a J¢

converges uniformly, with probability 1, to C(t). Then the sequence of processes ?),a(t) = u(t, c (t)) also converges uniformly with prob-

ability 1 to

Taking the limit as n , o,:, in the formula

7).(t2) - "7%(t1) =

(t, bm.(t)) + u¢(t, Sn(t))an(t)

+ 1 u'xx(t, C.(t))bn(t)]dt + tlux(t, C%(t))bn(t)dw(t) J

we obtain proof of formula (11) in the general case. Let us also consider the integrals bF(t)dw(t)

,

DIFFUSION PROCESSES

390

of the functions F(t) with range in Rcm>. If fl, , fm are the coordinates of the function F relative to some basis, then bF(t)dw(t)

d

is a random variable with range in R(m) with coordinates b f ti(t)dw(t) J a

for i = 1, ,m. Properties 1 to 8 remain valid for stochastic integrals of vectorvalued functions if by I F we understand

fl l2 + ... + V-11

.

(Formulas characterizing these properties were deliberately written in such a form that they remain meaningful for integrals of functions with ranges in Rcm>.)

Let us suppose that k mutually independent processes of Brownian motion wl(t),

, wk(t) are such that condition (2) regard-

ing the connection with the a-algebra, is satisfied for each of them and, for each h, the processes w,(t + h) - wv(h), for i = 1,

,

k,

are all independent of gk. Then we can define the integral af(t)dw;(t)

for every i = 1, , k and every function f(t) e ITt2[a, b] with range in R(m). Analogously to what we did in the one-dimensional case, we can define the differential: k

do(t) = a(t)dt + E fi(t)dwti(t) a_1

Let u(t, x) denote a function defined for t e [a, b] and x e R(m), Let x', , xm denote the coordinates of the

with range in Rcm>.

point x relative to some basis, and let fj, , fp denote the coordinates of the function fj. 8'. If the function u(t, x) is continuous and has continuous partial derivatives a

at

for i, j = 1,

u(t, x)

a ,

axti

a(t, x) ,

aQ

a(t, x)

axiaxj

, m, and if the process r(t) has a differential do(t) = a(t)dt + E fi(t)dwi(t)

then the process i)(t) = u(t, '(t)) also has a differential and

3.

EXISTENCE AND UNIQUENESS

391

au(, C(t)) + Ei ax; u(t, r(t))ai(t) at

dv)(t)

+

12 m,;,a E ax'ax' u(t, Vt))fi(t)fi (t)]dt a2

+' (_ au f'(t))dw2(t) j=i ax'

i=1

(12)

//

The derivation of formula (12) is analogous to that of formula

3.

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

Consider the stochastic differential equation de(t) = a(t, e(t))dt + a(t, e(t))dw(t)

,

(1 )

whose solution, it is natural for us to expect, is a diffusion process with coefficient of diffusion a°(t, x) and coefficient of transfer a(t, x).

Let us assume that a(t, x) and 6(t, x) are Borel functions defined for x e R(1) and t e [to, T].

Equation (1) is equivalent to the equation

fi(t) _ (to) +

t

to

rt a(s, e(s))ds +

Q(s, ee(s))dw(s) J to

(2)

and it is solved under the condition that e(to) is given. For the integrals in (2) and hence the differentials in (1) to be meaningful, we need to introduce the a-algebras of events fit. In what follows, the quantity e(to) will always be assumed to be independent of the process w(t) - w(to) and by the a-algebra `t we shall understand the minimal a-algebra with respect to which the variables (to) and w(s) - w(to) for to < s < t are measurable. We shall consider e(t) to be a solution of equation (2) if e(t) is (fit-measurable, if the integrals in (2) exist, and if (2) holds for every t e [to, T] with probability 1. We note that property 3 of the preceding section implies that for stochastically equivalent processes f (s) and f2(s), the stochastic integrals

5t)

,

It

f2(s)dw(s)

coincide with probability 1, since fl (s) = f2(s) with probability 1 for every s and hence

DIFFUSION PROCESSES

392

to

Ifl(S) - f2(s)

gds>0}=0.

From this it follows that every process that is stochastically equivalent to a solution of equation (2) is itself a solution of the same equation. Since the right-hand member of equation (2) is stochastically equivalent to the left-hand member and with probability 1 is continuous, it follows that for every solution of (2), there In what exists a continuous solution stochastically equivalent to it. follows, we shall consider only continuous solutions of equation (2).

Theorem 1. Let a(t, x) and u(t, x) for t e [t0, T] and x e RM denote two Borel functions satisfying the following conditions for some K:

a.

For all x and y e k1>,

a(t, x) - a(t, y) I + I u(t, x) - 6(t, y) < K I x - y 1 b. For all x, I a(t, x) I2 + I u(t, x)12 < K2(1 + I x 12)

,

.

Then, equation (2) has a solution. If e1(t) and c2(t) are two continuous solutions (for fixed Eo(t)) of equation (2), then P{toupT I sil(t) - e2(t) I > 0} = 1

.

Proof. Let us first prove that a continuous solution is unique. Let $1(t) and e2(t) denote two continuous solutions of equation (2). Let XN(t) denote the random variable that is equal to 1 if I e1(s) I < N and 1 e2(s) I < N for all s e [to, t], but equal to 0 otherwise. = XN(t) for (1s < t, we have Since XEN(t)XN(s)

XN(t)[S1(t) - U01 =

XN(s)[a(s, $1(s)) - a(s, E2(s))]ds o

+

t J to

X N(s)[u(s, $1(s)) - 6(s, ee2(s)]dw(s)

J

'

Since

XN(s)[I a(s, e1(s)) - a(s, $2(s)) I + 16(s, E1(s)) - 6(s, e2(s)) J] < KXN(s) e1(s) - $2(s) I < 2KN ,

the squares of the integrals on the right-hand side of the last equation have mathematical expectations.

Applying the inequality (a + b)2 < 2(a2 + b2), Cauchy's inequality, and property 2 of the preceding section, we obtain the inequality

3.

EXISTENCE AND UNIQUENESS

393

2(t)]2 MXN(t)[E1(t) - Oto

< 2MXN(t)XN(s)[a(s, + 2MXN(t)

e1(s)) - a(s, e2(s))]ds)1

Ot to

[a(s, e1(s)) - a(s, $2(s))]dw(s))

/

to)t

< 2(T -

MXN(s)[a(s, e1(s)) - a(s, $2(s))]2ds to

11rt

+ 2J MXN(s)[a(s, S1(s)) - a(s, Us))]2dw(s) . tt

to

Taking into consideration condition a, we see that there exists a constant L such that MXN(t)[e1(t) - e2(t)]2 <

L

t

(3)

MXN(s) I e1(s) - e2(s) I2ds .

to

We now prove an auxiliary proposition that will often be useful when we are seeking inequalities that provide bounds: Lemma 1. Let a(t) denote a nonnegative integrable function that is defined for t p- [to, T] and that satisfies the inequality

a(t) <

(4)

a(s)ds + ,Q(t) , to

where H is a nonnegative constant and Q(t) is an integrable function. Then

a(t) <_ R(t) + H t eH(t-s) 8(s)ds to

Proof.

It follows from (4) that

a(t) < S(t) + Hf [R(s) + to t

< 8(t) + H

[,&(s1)

+H

J to

< 6(t) +

H

to

a(s1)ds1]ds J a2

81[,8(s2)

+ HJ a(s3)ds3ds2]dsi to

to

t

to

R(s,)ds1 + to to

... + HnfJto ... + H Th+11

. to

rsn

fto-1Q(sjds1 ... ds

a (sn+1)dsl ...

to

Since ft

el to

to

we have

... Ik a(sk+1)dsk+ldsk ... ds1 to

t

to

(t - sk+l) k a(sk+l)dsk+l k!

,

DIFFUSION PROCESSES

394

...

lim H'°+' `t %__

J tp

8na(sn,

1)dsn+1 ... ds1 = 0

to

Hence

a(t) < R(t) +

Hrt J to

",6(s2)ds2ds1 + ...

R(s)ds +

H'` (t

= R(t) +

to

to

S)k R(s)ds

.

k!

This completes the proof of the lemma. Let us set a(t) = Mxr(t)[e1(t) - 2(t)]2 and G(t) = 0.

Then

MXN(t)[e1(t) - e2(t)]2 = 0 ,

that is, p{e1(t) :?L e2(t)} < {sup I sil(t) I > N}

+ {sup I S2(t) I > N}

.

The probabilities on the right approach 0 because of the continuity (and hence boundedness) with probability 1 of the processes E1(t) and e2(t). This means that e1(t) and E2(t) are stochastically equivalent. Since both processes are continuous with probability This completes the proof

1, we have p{sups I e1(t) - E2(t) I > 0} = 1.

of the uniqueness of the solution to equation (2). Let us now show that equation (2) has a solution. First, let us suppose that M I e(to) I2 < o o. Consider the Banach space B of measurable random functions C(t) that for every t are measurable with respect to the a-algebra 2` t and satisfy the relation sup M I C(t) 12 < °°

with norm = 1/ sup M I W) 12

II fi(t)

Let us define an operator S in the space B by rt

SC(t) = (to) +

Jto

rt

a(s, C(s))ds +

to

a(s, C(s))dw(s)

.

The existence of both integrals follows from the relation I a(s, c(s)) 12 + I U(s, c(s)) ':!g K2(1 + I b(s) 11)

Obviously, SC(t) is measurable with respect to fit. Using the inequality

(a + b + c)2 3(a2 + b2 + C2) and hypothesis b of Theorem 1, we obtain

3.

EXISTENCE AND UNIQUENESS

395

M I SC(t) 12 < 3M I C(to) I2

+ 3(T

t0)MJt K2(1 + I c(s) I2)ds to

+

MK2(1 + I C(s) I2)ds < 3M I C(t0) I2 to

+ 3K2(T - t0)2 + (3K2 + 6K2 II C Ii2)(T - to)

Furthermore,

Thus the operator S maps B into B. SCI(t) 11 < 2[T -

M I SC1(t)

to] rt t0

M[a(s, C1(s))

y ((ff (6(s, US)) - a(s, S2(s))]2ds + 2M\Jt t0 1 - O'(s, b2(s))dw(s))2 YY


t M I US) - S2(s) I2ds J to

< L(t - to) II b1 - b2 II2, if L = 2K2(T - t0 + 1)

.

This inequality shows that the operator S is continuous on B. Furthermore, (1

y M I SnC1(t) - SnYY b2(t) 12 < LJ t M I Sn-1b1(u) - Sn-1C2(u) I2du YY

to

f


<

t

('tl

t0

t0

Ln(t

t to-1M I

I b1

y - b2 I I2

du dtn ... dt,

to

t0)n 11

n!

b

- S2

2

Therefore, for every C(t) in B, II Sn+1C

SnC < Ln(ni

On I

I SC - C I I2

.

The convergence of the series

i 11 S- +1C - SnS I

I

n-1

implies that the process S"C(t) has a limit as n cc. If we let e(t) denote this limit, it follows from the continuity of S that S[S"C(t)] -+ Se(t).

But y

S[Snb(t)] = Sn}1yy S(t) .--3 E(t)

Thus II Si(t) - fi(t) = 0. It then follows from the definition of the norm that e(t) = Se(t) with probability 1 for every t e [t0, T]; that is, e(t) is a solution of equation (2). Let us now prove that equation (2) has a solution in the general case. Let s''(.*0) denote the quantity that is equal to E(t0) when (t0) < N and equal to 0 when I e(t0) I > N. We let N(t) denote

396

DIFFUSION PROCESSES

the solution of the equation rt eN(t) = eN(to) +

to

rt a(s, eN(s))ds +

(5)

6(s, eN(s))dW(s) Jto

Since I eN(to) I < N and M I eN(t0) I2 < o., equation (5) has a solution, and it follows from what was shown above that

sup M I V(t)I2 < t

Let us show that as N--, - the sequence {eN(t)} converges in probability to some process e(t) that is a solution of equation (2). Suppose that N' > N. Let i denote the random variable that is equal to 1 for I e(t0) I < N and equal to 0 for I e(to) I > N. Then [eN(to) - eN'(to)]r7 = 0. The quantity )7 is measurable with respect to the a-algebra to. Starting with the inequality

N(t) - 6 N '(t) I2 < + 2(Jt

[6(s,

2[t [a(s, E (S)) - a(s,

eN'(s))]'/ds]2

L J to

ESN(s)) - 6(s, EEN (s))]r)dw(s))

0

and using condition a and the inequalities involving the integrals (which we have already applied), we see that the constant L is such that t

M I $N(t) - eNl (t) 1272

to

M I EN(S) - EN (S) I2 ,ids

.

Consequently, M I $N(t) - $N'(t) I2'/ = 0,

so that P{I EN(t) - EN'(t) I > 0} < P{I E(to) I > N} .

It follows from the last inequality that EN(t) converges in pro-

bability to some limit E(t) as N-, cc, and that JT (EN(t) - (t))2dt to

converges in probability to 0 as N o o. Using condition a of Theorem 1 and property 3 of Section 2, we obtain the inequalities Jt

a(s, $N(s))ds - :' a(s, E(s))ds

< KY (t - t0)t ($-(s) -(s))2ds , 0

ft P{

6(s, EEN(s))dw(s)

Jto

-

rt to

a(s, EE(s))dw(s) < E} 11

32

<

62

+

EE(S))2ds >

S all

3.

EXISTENCE AND UNIQUENESS

397

from which it follows that we can take the limit under the integral signs in the right-hand member of (5). Thus c(t) is a solution of equation (2). This completes the proof of the theorem. Let us show that under the hypotheses of Theorem

1,

the

solution of equation (2) is a Markov process. To do this, let us prove:

Theorem 2. Let a(t, x) and u(t, x) denote two functions satisfying the hypotheses of Theorem 1 and let et,x(s) denote a process defined for s e [t, T], where t > to, that is a solution of the equation Et,=(s) = x + e8a(u, St,x(u))du + J86(u, et.x(u))dw(u) t

(6)

Then the process E(t), which is a solution of equation (2), is a Markov process whose transition probabilities are given by the relation p(t, x, s, A) = p{E(s) e A I e(t) = x} = P{e,,(s) e Al .

Since e(t) is measurable with respect to z`t and since completely determined by the process w(s) - w(t) for et,x(s) s e [t, T] (independently of fit), it follows that et,x,(s) is independent of e(t) and the events in fit. For s e [t, T], the function e(s) is (by virtue of Theorem 1) the unique solution of the equation Proof. is

s(s) =si(t) + Jt'

a(u,

EE(u))du + J:a'(u, ee(u))dw(u) . t

The process et,ect)(s) is also a solution of this equation. Therefore with probability 1 e(s) = et,,(t)(s). Let us now show that P{e(s) e A I e(t)} = P{e(s) e A I fit}

.

To do this, it will be sufficient to show that for an arbitrary bounded variable C that is measurable with respect to and an arbitrary bounded continuous function X(x),

(7) Let us suppose that we have a function p (x, u) of the MCW(s)) = MCM(x(e(s)) I e(t))

form n

/ p(X, u) _ E ck(x) (k(u) k=1

where *k(u) are independent of

We have

MSk=1 E (Pk( (t))'J'k(U) = Lk=1 [MC Pk(S (t))]M Y' k(u) y

= M k=1 E cPk(e(t))M*k(U) M(E (Pk(e(t)) k=1

/Jk(U)

I

e(t)/I = Ek=1Tk(e(t))M'1 k(u)

(8)

DIFFUSION PROCESSES

398

Since functions of the form (8) are dense we obtain for any function T (x, u) depending only on the future increments w(s) - w(t), s > t M(T(e(t), u) I z`75t) = M(, *SW, u) I e(t)) = M(q (x, u)) I-vo

Then T(e(t), u) = X(e(s))

Let us now take cp(x, u) and we obtain

(9)

g(e(t)) , Consequently,

M(x(e(s)) I

where g(x) = M%(et,,(s)).

P{e(t) e A

Pt,e(t)(s, A)

where Pt,s(s, A) = P{et,x(s) e Al. This completes the proof of the theorem.

Let us show that under certain supplementary assumptions, the process e(t) is a diffusion process with coefficient of diffusion az(t, x) and coefficient of transfer a(t, x). To do this, we prove as a preliminary the auxiliary proposition: Lemma 2. Let et,,(s) denote a solution of equation (6) in which the functions a(t, x) and c(t, x) satisfy the conditions of Theorem 1. Then there exists a constant H such that M I e ,x(s) - x I' < H(s - t)2(1 + I x Proof. Define xN(s) = 1 when sup I et,,(u) - x I < N , t5u<s

and XN(s) = 0 otherwise.

Then

(et,x(s) - x)XN(s) - XN(s)[Ir

yy

EE

sXN(u)a(u, E,x(u))du

+ tBXN(u)6(u, et,x(u))dw(u)] Using the inequality (a + b)' < 8a' + 8b', Cauchy's inequality, and property 7 of Section 2, we obtain et,x(u))du]4

M(et,x(s) - .x)'XN(s) < 8{M[

XN(u)a(u,

+ M[ XN(u)u(u, et,x(u))dw(u)]4} et,y(u))]4dU

< 8(s -

t)3f ttsMXN(u)[a(u,

+ 8.36(s -

t)JM6(u,

et,x(u))'XN(u)du

< 64(s - t)38MXN(u)a(u, x)'du t8

+ 64(s -

eet,x(u)) - a(u, x)]'du

3.

EXISTENCE AND UNIQUENESS

399

+ 8.8.36(s - t) f MXN(u)a(u, x)4du

+ 8.8.36(s - t)MXN(u)[a(u, et,x(u)) - ,(u, x)]4du r

< L(s - t)JgMxN(u) I t,x(u) - x I4 du + R(s - t)2 , t

where L = K4[(T - to)2.64 + 64.38] and

R = L[max I a(t, x)14 + max I a(t, x) Thus if p (s) = M

14]

< h1(1 + I X 14)

x I4XN(s), then

p(s) < L(s -

(t < s)

R(s - t)2

.

It follows from this inequality that p(s)l(s - t) is bounded.

There-

fore, for t < s < t + 1, 'P(s)

s-t

<

tu - t

t) 'P(u) du

+ R(s - t)

< Ls T(u) du+R(s-t). Jt u - t Let us set a(z) = q(t + z)/z. Then a(z) <

Rz

It follows from Lemma 1 that a(z) < R

eLZ - 1

L

Thus MXN(S) I et,z(s) - x 14 < R(s

- t)

eLce-t>

_

L < R1eL($-t)(s - t)2(l + I X 14)

Since the expression on the right is independent of N, and since MXN(s) I et,=(s) - x 14

M I et,x(s) - x

11

as N--+ a, the assertion of the lemma follows from the last relation. Corollary 1. Let fi(t) denote a solution of equation (2). Suppose that a(t, x) and a(t, x) satisfy the conditions of Theorem 1 and that M I (to) I4 < 00 .

Then

sup M I E(t) 14 < 0 .

to5t5T

Proof.

It follows from Lemma 2 that

400

DIFFUSION PROCESSES

M(I E(t) - E(to) I4 I $(to))

< H(T -

t0)2(1 + I e(to) I4)

,

M 1$(t) 1'G 8M I E(to)14 + 8MM(I E(t) - e(to) I4 $(t(,)) I

Corollary 2. Under the hypotheses of the preceding corollary, there exists a constant Hl such that M I S (t) - $(s) I4 < H1(t - s)2 Proof.

For t < s, M b e(t) - E(s) 4 5 H(s - t)2(M I e(t) I' + 1)

Theorem 3. Suppose that the conditions of Theorem 1 are satisfied and that a(t, x) and o(t, x) are continuous with respect to t for t e [to, T]. Then x) and a(t, x) are the coefficients of diffusion and transfer respectively for the process E(t), which is a solution of equation (2).

Proof.

Since the transition probability p(t, x, s, A) for the process

e(t) coincides with the distribution of the variable et,,(s) on the basis of Theorem 2, it follows from Lemma 2 that (y - x)4p(t, x, s, dy) = M I sit,:(s) - x I4 = o(s - t)2

.

Using Remark 1 of Section 1, we see that to prove the theorem it will be sufficient to show that

(y - x)p(t, x, s, dy) = M$,,,(s) - x = a(t, x)(s - t) + o(s - t)

(10)

and

(y - x2)p(t, x, s, dy) = M(61, (s) -

x)2

= a2(t, x)(s - t) + o(s - t)

(1 1)

.

Starting with equation (6), we may write Met.z(s) - x =

Ma(u, t,x(u))du J

= Jta(u, x)du + tM[a(u, $,,z(u)) - a(u, x)]du

But MJt(a(u, et,x(u))

(s -

- a(u, x))du < JtM I a(u, Et.=(u)) - a(u, x) M I Et.x(u) - x

t)3/4[ t

< (s - t3/4H(u [ -

du

I4du]1/4

J

t)2du]114

= O(s - t)3/4+3/4 = o(s - t)

t

It follows from the continuity of a(u, x) that

3.

EXISTENCE AND UNIQUENESS

401

a(t, x)(t - s) - ta(u, x)du = t[a(t, x) - a(u, x)]du = o(s - t) which proves formula (10). Furthermore st.x(u))du + M($t,.(s) - x)2 = M(J9a(u, t

Q(u, fit.=(u))dw(u))2

We note that M[Jta(u, St.x(u))du]2 < (s - t)JtMa2(u, et.z(u))du = o(s - t)

and M(f 8a(u, $t.=(u))dw(u))2 = J'M[t(u, et.z(u))]2du t

t

<

et.=(u)2)du = O(s - t)

Therefore, using Cauchy's inequality, we see that st.=(u))du][Jt6(u, t.s(u))dw(u)]I

M

st,x(u))du)2M(Ja6(u' < lM(Jea(u, t

fit, (u))dw(u))2 = o(s

- t)

Thus

M(st.=(s) - X)2 = M(J,u(u, eet.z(u))dw(u))2 + o(s - t) t

= JtBMo2(u, sts(u))du + o(s

-t

= i2(t, x)(s - t) + [a2(u, X) - Q2(t, x)]du

+

r1M[C2(u, t

et.s(u)) - Q2(u, x)]du + o(s - t)

It follows from the continuity of a(u, x) with respect to u that 5211a2u, x) - ra2(u, x)]du = o(s - t)

.

Furthermore, MJ a[Q2(u, st.x(u)) - Q2(u, x)]du t

JtM[a'(u,

St.=(u)) - a(u, X)]2 du

x TJtM[Q(u, St.=(u)) + a(u, x)]2du

< /O(s - t)(s -

1/2

t)1/2Ot8M

Q(u, fit.=(u)) - Q(u x) I4du)

O(( s - t)1/2+1/4+1/2) = o(s - t)

.

402

DIFFUSION PROCESSES

Thus we have proved formula (11) and with it the theorem. Let us now construct a multidimensional diffusion process fi(t) with transfer vector a(t, x) and diffusion operator B(t, x). We define bk(t, x) = V Xk(t, x)ek(t, x)

where the ek(t, x) are the eigenvectors of the operator B(t, x) and the a,k(t, x) are the eigenvalues corresponding to them. We seek the process e(t) as a solution of the equation "n

de(t) = a(t, e(t))dt + E bk(t, f(t))dwk(t)

,

k=1

(12)

where the wk(t) for k = 1, 2, , in are independent processes of Brownian motion. The functions a(t, x) and bk(t, x) are defined for t e [to, T] and x e R(-), and their ranges are contained in R(m'. Equation (12) is equivalent to the equation

t (t) = NO + t a(s, S(S))ds + J to

tt

1f

k=1

Jt bk(S, S(S))dwk(S) to

(13)

This equation is solved for given E(to), which we shall always assume to be independent of the processes wk(t). Let 2 t denote the minimal u-algebra generated by the quantities E(to) and wk(s) - wk(tp) for k = 1 , 2, , m and s e [to, t]. We shall assume that the solution of equation (13) is a process f(t) such that the integrals on the right side of the equation exist, and such that for every t e [to, T] equation (13) is satisfied with probability 1. Let us formulate the basic properties of the solutions of equation (13). Theorem 4.

Let a(t, x), b1(t, x),

, b,n(t, x)

denote Bore! functions defined for t e [to, T] and x e RI"' with ranges in R(m).

If there exists a K such that In

a(t, x) I2 + E I bk(t, x) 11 < K2(1 + I x

2)

k=1

- a(t, y) I + E I bk(t, x) - bk(t, y) In

a(t, x)

k=1

KIx-Y

for every x and y in K'"), then equation (13) has a solution e(t) that is unique up to stochastic equivalence and continuous with probability 1. This solution e(t) is a Markov process whose transition probabilities p(t, x, s, A)

for t < s are given by the relation p(t, x, s, A) = p{E2,x(s) e Al

4.

DIFFERENTIABILITY OF SOLUTIONS

403

where Lt,x(s) is a solution of the equation

t.x(S) = x + ta(u, e,.(u))du + E bk(u, f,,x(u))dw(u)

(14)

If the functions a(t, x) and bk(t, x) are continuous with respect to t, then the process e(t) is a diffusion process with transfer vector a(t, x) and diffusion operator B(t, x) that satisfies the equation (B(t, x)z, z) _

(bk(t, x), z)2

k=t

The proof follows essentially the proof given for the onedimensional processes treated in Theorems 1 to 3. Remark 1. If et,x(s) is a solution of equation (14), in which the functions a(s, x) and bk(s, x) satisfy the conditions of Theorem

4, there exists a constant H such that for s > t, M I et,x(S) - x 14 < H(s - t)2

.

This assertion is analogous to the one proved in Lemma 1 for a one-dimensional process. Remark 2. If the coefficients a(t, x) and bk(t, x) in equation (12) are independent of t, that is, if the equation is of the form m

de(t) = a(e(t))dt + E bk(t(t))dwk(t) k=1

and a(x) and the bk(x) for k = 1 , 2,

,m

satisfy the hypotheses

of Theorem 4, then the solution fi(t) of the equation is a homogeneous

Markov process; that is, the transition probability p(t, x, t + h, A) is independent of t. To see this, note that P(t, x, t + h, A) coincides with the distribution of Ct,x(h) = et,x(t + h) but according to Theorem

4, rt,1(h) is a solution of the equation ddt,x(h) = a(rt,x(h))dt +E bkTt,x(h))d[wk(t + h) - wk(t)] k=1

with initial condition rt,x(0) = x. Since the common distribution of [wk(t + h) - wk(t)] for k = 1, 2, , m is independent of t, the distribution ',,x(h) is also independent of t.

4.

DIFFERENTIABILITY OF SOLUTIONS OF STOCHASTIC EQUATIONS WITH RESPECT TO INITIAL CONDITIONS

The purpose of this section is to show that the function t,x(s), defined in Section 3 as a solution of equation (14) of Section 3,

DIFFUSION PROCESSES

404

is a differentiable function of x when the coefficients a(t, x) and bk(t, x) are sufficiently smooth. Since et,x(s) is a random function of x, we need to show just what we mean by a derivative of E,,x(s). For our purposes it is convenient to consider mean-square differen-

tiability of random functions. If q(x, u) is a random function depending on the point x in the space R'"`' and x1, , x'" are the coordinates of x, then by acp/axi we mean the random variable for which

lim M QXi (T(xi, ..

Cxi-0

.

xi + Oxi, ... , xu)

- T(xl, .. , xi, ... , x-, u)) -

aq (x, u)

axi

.=0. 2

Just as in the preceding section, we shall give complete proofs only for one-dimensional processes.

Suppose that two functions a(t, x) and a(t, x) are defined and continuous for x e (- a, 00) and t e [t0, T] and suppose Theorem 1.

that they have the continuous bounded partial derivatives a'(t, x), a " (t, x), o'z(t, x), o (t, x)

Then the solution E ,,,(s) of equation (6), Section 3 is twice differentiable with respect to x and the derivatives are mean-square continuous functions of X.

Proof of this theorem rests on: Lemma 1. Suppose that processes C,,(t) where n = 0, 1, solutions of the stochastic equations

C-(t) = T.(t) + Jtto

to

X.(s)C.(s)dw(s)

are

(1 )

We assume that the functions p .(t), *n(t), and xn(t) are for every t measurable with respect to at, and that there exists a K such that with prob0 ability 1, 11, (S) I < K and I X, (s) I < K. If sups M I q,,(t) - q0(t) as n -- - and if *.(t) . -. tro(t) and xn(t) -- x0(t) in probability for every t as n -- o, then sups M I C (t) - Co(t) I2 --- 0 as n -+ a o. Proof.

We note first of all that the existence and uniqueness

of the solution to equation (1) is proved just as in the proof of Theorem 1 of Section 3, and that in the present case, Sup M I C.(t) 12 < °O t

Therefore, we may write

4.

DIFFERENTIABILITY OF SOLUTIONS

405

M Sn(t) - So(t) 12 < 3M I pn(t) - p0(t) 11

+

to

\

(Yfr (s)bn(s) -

b(s)go(s))ds)2

+ - po(t) I2 + 3M(`t *n.(s)KKn(s) - Co(s)]ds

3M (pm.(t) I

+

Jt

Co(s)[*n.(s) - iJro(s)]ds)2 + 3M(f t Xn(s)[Cn(s) Xo(s){di^'(s)12

- b0(s)Jdw(s) +

to

Co(s){Xn(s) ds)2

< an(t) + 6M(t K cn.(s) - Co(s) I Jto

t

+ 6K2toM in(s) - o(s)

gds ,

where to)

an(t) = 3M I cpn(t) - po(t) 12 + 6(t -

-

Ids +

e

MUs)2 I *n(s)

6t tom

Xn(s) - Xo(s) IZbo(s)2ds

Since

sup an(t) < 3 sup M I n(t) -do(t) I2 t

t

+ 6(T - t0 +

1)TMC0(S)2(I

*n(S) - *0(S) IZ

Jto

+ I XL(S) - Xo(S) I2)ds

the sequence {an(t)} converges to 0 uniformly with respect to t. (The integral approaches 0 by virtue of Lebesgue's theorem since the integrand is bounded by 4K2Co(s)2, and approaches 0 in probability.)

If we set H = 6K2(T - to + 1), then M I Cn(t) - Co(t) I2 < sup an(t) + t

M I Cn(s) - Co(s) IIds .

to

Then on the basis of Lemma 1, Section 3, M I C .(t)

- b0(t) 11 < sup an(t)el (T-to) m

t

This inequality proves the lemma. The lemma remains valid if the processes depend on a continuous parameter a and the corresponding limits exist as REMARK 1.

a

0.

DIFFUSION PROCESSES

406

Let us now prove Theorem 1. S x,,x(S) -

Define

OX

Then the process C,,ox(s) is a solution of the equation

C.".(S) = 1 +

t

t'

where a(s, et,x+6x(s)) - a(s, Et,x(s)) x,,x (S)

t,x+,x(S) - Et,x(S) yx,,x(S)

,

_

t,x

(s)) - o(s ,

E t,x

'

(S))

$t,x+11x(5) - et,x(S)

Since a(s, x) and a(s, x) have bounded derivatives with respect to x, there exists a constant K I *x,nx(s) I < K and I /l,x,,x(S) < K. MCx,,x(S)2 < 3 + 3(T -

+

such that with probability

1,

Therefore

K2MCx,,x(u)2du

3K2f 8M I bx,,x(u) I2du < 3 + Hf 8M I bx,,x(u) I2du t

,

t

where H = 3(T - to + 1)K2. that

It follows from Lemma 1, Section 3 Mbyx,,x(S)2 <

3e1(a_t)

and consequently, for some Hl, M t,x+11x(5) - E1,x(s) I2 5 H,(0x)2

(2)

This inequality means that et,+,x(s) - et,,(s) as Ox --. 0. Therefore

0 in probability

*.,,,.(s) --p ax(s, 41,x(5)), Xx,,x(S) - a' (s, $,,.(s))

in probability as Ax

0.

We let cx(s) denote the solution of the

equation

US) = 1 +

ax(s, $t,x(s))Cx(s)ds + Jtcx(s, $,,x(s))gx(s)dw(s)

It follows from Lemma 1 that m I bx,,x(s) - Cx(s) 12 ---. 0 as Ox

(3) 0,

that is, that Sx(S) =

ax

et,x(s)

The mean-square continuity of Cx(s) with respect to x again follows from Lemma 1 since the derivatives ax(s, et,x(s))

and

et,x(s))

4.

DIFFERENTIABILITY OF SOLUTIONS

407

are continuous with respect to x in the sense of convergence in probability, because of the continuity of a'(s, x) and ox(s, x) with respect to x and the mean-square continuity of $,,x(s) with respect to X. To prove the existence of a2et,x(s)/axe, we introduce the processes cx(s)].

/x,Cx(s) _

These processes satisfy the equa-

tions s

mcu

s

77

mc2

s

e

+

t

+

ax(u,st,x(u))-Ix,ox(u)du

80",(u,

Jt

where t az(u, St.x+ox(/u)) - ax(u//, et,x(u))

x;ox(s) -

bx,Ax(u)bx(u)du

86x(u, ;t,x+ox(u)) - o,(u, et,x(u))Yx,ax(u)bx(u)dw(u)

,Px"ex(S) =

ci,x(u)

t

Define

zt (s) = J

t axx

(u, t,x(u))Cx(u)2du

1Py2 (S) _ t axx(u, $,,z(u))bx(u)2dw(u)

Let us show that Jim M I Px';ax(s) AX-0

- Tx,(s) 12 = 0

i = 1, 2

uniformly with respect to t. To do this, we use Lemma 2.

If Cx(s) is a solution of equation (3), then

sup M I Cc(S) 14 < - and s

lim I bx+A (s) - bx(s) I' = 0

Ox-+0

Proof. We note that the solution Cx(s) of equation (3) can be obtained by the method of successive approximations, in which we set Ci01(s) = 1:

C(s) = 1 +

a'x(u, S,,.(u))bzn-"(u)du

t,x(u))bzn I'(u)dw(u) . + ao'(u, t

Then by using the inequalities (a + b + c)4 < 27(a4 + b4 + c4), ax < K, I o ,

K and property 7 of Section 2, we obtain

408

DIFFUSION PROCESSES /(

[ + M1 J'K I Sxn-1'(u) I MClnl(s)4 < 271 \

L

du14

/I

t

+ 36(s -

t

< R1 + R2 f M I Czn-1'(u) I4du

,

t8

where R1 = 27 and R2 = 27K4[(T - t°)3 + 36(T - t°)]. ing that MC11)(s)4 1, we may write ' < R1 + R2 f aR1ds +

Czn)(s)

M

xn 2'(v) I4 du du t

1t

R22

< Rl 1 + R2(s - t) + + R2

s

ul u2

t

t

Remember-

s

t

t) 2

2

+ ... +

R'-'(

1

(s- 1)!

l

u,i,-1

...

M I Ci°j(v) I4du dun_1 ... du1

t

t

< R1ea2ca-0 + Rn

(s - t)n n!

2

This inequality implies boundedness of m CJs) I4. b,,.(s) - Cz+oz(s) =

Furthermore

2.,os(u)Cx(u)dw(u) 5a(u,

+

et,2(u))[Cz(u) - Cz+o2(u)]du

t

+

J,U,.(u, t

,x(u))[c2(u) - C2+ox(u)]dw(u)

where a(s,

'

fit,=+,,(s)) - a(s, eet,z(s))

':,,,(s) - a'(s, t,2(s)) -

t,z+nz(s) - et,2(s)

6(S, t,z+ez(s)) - 6(S, et,=(s))

xz,nz(s) = u'.(s, fit,=(s)) -

,=+AJS) - t,x(S)

Taking the fourth power of both sides of these equations and using Cauchy's inequality and property 7 of Section 2, just as we did above, we see that there exist constants A and B such that M I US) - Cx+oz(s) 4 < A\t8 + I X=,o=(u)

14)

14

I Cx(u) I4du + BJ aM I Uu) - C, e(u) I4du .

Consequently, M I CX(s) - Cx+oz(s)

< A`t M(I

14

14

+ I Xz+os(u) I4) 1 Cx(u)

I4dueB(s-t)

4.

409

DIFFERENTIABILITY OF SOLUTIONS

0, and since Since I rx ex(s) I' + I xx,ox(s) I' _ 0 in probability as Ox the expression on the left does not exceed 32K1, it follows from

Lebesgue's theorem that JS) 'ds = 0 . ,."'.(S) I' + I xx,Ox(s) I') ICSs)

T M(I AX-,0 J t0

This completes the proof of the lemma. We note that from the boundedness of a22 it follows that there exists a constant C such that ax(s, x) - ax(s, y)

x-y


Therefore:

M I x,ex(s) - 1P. (s) Mj = MN' '

2

ax (u, a ,x+nx(u)) - ax(u, Et,x(u)) Sx(u)LSx, x(u) - Sx(u)]du St,x+4x(u) - et,x(u) ( ))

1

+

E1(ht,

< 2(s

C2MCx(u)2[Cx,ox(u) - Cx(u)]2du

- tfi t,t

r a'(u, t,x+Ax(u)) 18MI ('

+

axx(u, t,x(u))]Cx(u)2du

d

t

L

az(u, t,x(u)) t,x+Ax(u) - t,x(u)

axz(u,

The second integral approaches 0 by virtue of Lebesgue's theorem,

since the first factor in the integrand is bounded and approaches 0 in probability as Ox ---> 0, and sup. M I Cx(u) I' < CC. We have the following bound for the first integral: MCx(u)2M I bx,Ox(u) - Sx(u) I2du

t

J&M I Sx,x(u) t

- Sx(u) I'du

,

so that it too approaches 0 in accordance with Lemma 2. This means that lim M x;ox(s)

- x'(S) 20.

Analogously, we can show that lim M I Ax-0

'Px2'ax(s)

- 'px2J(s)

20.

On the basis of Lemma 1, the quantity ?7x,ox(s) converges in the sense of mean square to the process 77x(s), which is a solution of the equation

DIFFUSION PROCESSES

410

/x(s)

t ayx

(u, t,x(u))Cx(u)2du +

t

a (u, t.x(u))Cx(u)2dw(u)

+ 1:a(u, t(u))(u)du +

t

The mean-square continuity of 7x(s) that is, with respect to x follows from the mean-square continuity of the functions q (s) and cps2)(s), and the continuity in probability of a2et,x(s)/axe = r)x(s).

axx(u, et,x(u)) and o," (u, et,x(u)) as functions of x.

For example, for TM(s) we have the inequality M I q ,1)(s) - q&1'(s)

12

< 2(s - t)Mazx(u, et,x(u))2[bx(u) - CY(u)]2du.

+ 2(s -

t)5 M[asz(u, Et,x(u)) - a" (u, et,2(u))]2Cx(u)4du . t

According to Lebesgue's theorem, the second integral approaches

0 as y -- x, since a" (u, E2.x(u)) - axx(u, et.2(u))

is bounded and approaches 0 in probability as y --+ x and sup MSx(u)4 < °O U

For the first integral we have the inequality Ma"(u, t.x(u))2[ySx(u) - yS9(u)]2du

< C2 tM I bx(u) + Cv(u)

8

J8

I bx(u) - CY(u) 4du ,

so that it approaches 0 in accordance with Lemma 2. The meansquare convergence of the function q)2(S) is proved in an analogous manner. This completes the proof of the theorem. In the case of multidimensional processes, we have: Theorem 2. Let E2,x(s) denote a solution of equation (14) of Section 3. S u p p o s e that the functions a(t, x), b,(t, x), , b,(t, x) are defined and continuous for t e [to, T] and x e RI-) and that they have bounded continuous first and second derivatives with respect to the , xm. Then the function et,x(s) as a function of x variables x', x2, is twice mean-square differentiable with respect to x, and the derivatives 2

ax't,x(s)

ax

axit,x(s)

as functions of x are mean-square continuous.

4.

DIFFERENTIABILITY OF SOLUTIONS

411

The proof of this theorem is similar to that of Theorem 1, and for this reason we omit it. REMARK 2. If the hypotheses of Theorem 2 are satisfied and f(x) is a bounded continuous function with continuous bounded is twice derivatives of first and second orders, then u(x) = continuously differentiable with respect to x. Again, we shall prove the validity of this assertion only for the case of one-dimensional processes.

Let us show that

(4)

ux(x) = Mf(et,x(S))Us)

By virtue of the boundedness of [f(x) - f(y)]/(x - y), the convergence of f(tg.z+ilx('S)) - f(t.x(S)) - fz( t,x('S))

to 0 in probability, and the relations lim M I bx,Ax(s) AX-0

MUS), <

- US) Iz = 0 ,

we have f($t,x(s))

-

-f( t,x(S))

bx(S)]]z

$ ,x+dx(S) - t,x(S)

+

2M[.

t,z+d,(s)) - f(et,x(s))

-

fx(et,x(s))]2

Cx(s)2

0

But u(x + Ox) - u(x)

i +Ix_ (

-

f(Et,z+az(s)) -

from which equation (4) follows.

f(et,x(s))) -

f=(et,z(S))bz(S)]z

Analogously, we can show that

uxz = Mf=x(et,x(s))ex(S)Z + Mfx($t,x(sNx(S)

(5)

The continuity of ux and uxz follows from the mean-square continuity of the processes Cjs) and 72,(s). REMARK 3. The processes et,x(s) ,

x

Etx(s)

,

xe

abtx(s)

are stochastically continuous functions of t for fixed s such that s < T, and this continuity is uniform with respect to x on to

DIFFUSION PROCESSES

412

every compact set. Suppose that t < t' < s.

Then for et,,(s) we

have

"Js) - fit-,=(s) = et,z(t') - x + S (a(u, t,Ju)) - a(u, e0, J

(u)))du

+ 9 [u(u, S0,z(u)) - u(u, t,,z(u))]dw(u) . J

Consequently, for some lH, et',x( )j2 G 3M I et,x(t') - x I2

+ HM I E0,z(u) Jt

I2du ,

from which it follows that M I t,z(S) - et-,z(S) 12 < 3M I $t,0(t')

-x

I2ex(s-t-)

.

But M I t,z(t') - x 12 = O(t' - t) uniformly on every compact set (by virtue of Lemma 2, Section 3). Consequently = 0(t' - t) Using the equations aet,z(s)/ax = Cx(s) and x(s)/axe = 7),(s), we can obtain analogous inequalities for these functions. Combining the assertions in Remarks 1 and 2, we obtain: M I t,z(S) - t-,x(S)

12

Theorem 3. Let et,x(s) denote a solution of equation (14) of Section 3 in which the functions a(t, x), b,(t, x), , b,, (t, x) satisfy the conditions of Theorem 2. Let f(x) denote a bounded continuous function defined on R(-' with bounded continuous first and second order derivatives. Then the function u(t, x) = M.f(et,:(s))

defined for tQ < t < s and x c R(-' is continuous and has first and second derivatives that are continuous with respect to the set of variables t, X.

5.

THE METHOD OF DIFFERENTIAL EQUATIONS

In this section we shall derive differential equations that will enable us to determine the distributions of certain functionals of diffusion processes. In so doing, we shall present a new derivation of Kolmogorov's first equation for diffusion processes. Let fi(t) denote a solution of equation (13) of Section 3. Let P(t, x, s, A) denote the transition probabilities for the process e(t). As we have noted above, p(t, x, s, A) coincides with the distribution of the process et,x(s), which is a solution of equation (14) of Section

S.

THE METHOD OF DIFFERENTIAL EQUATIONS

413

Let Mt,x denote the conditional mathematical expectation corresponding to the distribution p(t, x, s, A). To determine the transition probabilities, it is sufficient for us to determine M,,xq[E(s)] for all sufficiently smooth functions q(x). To do this we can use the following theorem (cf. Theorem 3, Section 1): 3.

Theorem 1. Suppose that the hypotheses of Theorem 2 of Section 4 are satisfied for the process e(t). Suppose that the function T(x) is bounded and continuous and that it has bounded and continuous derivatives of the first and second orders. Then the function u(t, x) = Mt,xq[(s)] (t a [to, s)]) has continuous first and second order derivatives with respect to xi, it is differentiable with respect to t, and it satisfies the equation a°(t, x) ax{ u(t, x)

u(t, x) +

at

1E

(1 ) + 2 k,i,j=1 bk(t, x)bk(t, x) axZ u(t, x) = 0 and the condition limt T . u(t, x) = q(x), where ai, bk, and xi are components of the vectors a, bk and x respectively. Proof., The differentiability of the function u(t, x) with respect

to x, and also the continuity and boundedness of the partial derivatives, follow from Theorem 3 of Section 4. We note also that Mt,xMt+ot,E(t+At)(P(f(s)) u(t, x) = = Mt,xu(t + At, 1; (t + At)) . Using formula (10) of Section 2, we may write

u(t + At, f(t + At)) - u(t + At, f(t)) t+nt t

m

L

au (t + At, (s))a'(s, id(s)) axj

+ At, + 2 k,i,j=1 E bi (s, e(s))bk(s, f(s)) d u(t axiaxj +

t+ot " t

(s))

l ds

u (t + At, e(s))bi(s, e(s))dwk(s)

k,i=1 axi

Consequently,

Mt,xu(t + At, (t + At)) - Mt,xu(t + At, e(t)) r

au (t + At, e(s))a`(s, e(s)) axi

+ 2 k,i,j=1 E bi(s,

a2

e(s))bk(s, E(s))

ax°axj

u(t + At, e(s))]ds

DIFFUSION PROCESSES

414 Since

Mt,xu(t + At, fi(t)) = u(t + At, x)

,

we have

u(t, x) - u(t + At, X) = Mt xL au (t + At, (s'))ai(s', e(s')) a2

+

12 k,i,A=1 bk(s', E(s'))bk(s', f(s')) ax'axj u(t + At, (s'))]At

where s' is some number in the interval (t, t + at). Remembering that au/axi and a2u/axioxj are bounded, and that

s' , t as At, 0, using Lebesgue's theorem and taking the limit under the integral sign, we obtain u(t, x) - Qt At, X) Mt x[Z t

ax'

u(t,(t))at(t,(t))

1

+ 2 k,i,9=1 ax'ax' a2 u(t, (t))bk(t, E(t))bk(t, fi(t))] Equation (1) is proved.

The fact that

rp(x) follows

and the continuity of

from the relation Mt,xq tt,x(s)

p{lim (et,x(s) - et,x(t)) = l im et,x(s) - x = 0} = 1

This completes the proof of the theorem. Let us now derive equations that will enable us to determine the distribution of the random variable

I = 'fts, e(s))ds to

where f(s, x) is a sufficiently smooth function and e(s) is a process that satisfies equation (13) of Section 3. We introduce the function v2(t, x) = Mt,x exp {xTf(s, e(s))ds}

(2)

To determine the distribution of the variable I, it will be sufficient for us to determine the function v2(t, x) for t e [to, T] and x c R("') and all imaginary values of X, since v2(to, x) will then give us the conditional characteristic function of the random variable I under the hypothesis f(t0) = x. If we integrate v2(to, x) with respect to the initial distribution, we obtain the unconditional characteristic function of the variable I. Theorem 2. If e(t) satisfies the hypotheses of Theorem 2 of x) for i, j = Section 4, and if the functions fit, x), fi(t, x) and

S.

THE METHOD OF DIFFERENTIAL EQUATIONS

415

1, ,m

are continuous and bounded, then for t e [to, T], the function v2(t, x) satisfies the equation a a'(t, x) - v2(t, x) + Y. ti-1

at

+

a

axi

v2(t, x) a2

1 E bk(t, x)bk(t, x)

2 i,j,k=1

axaxj

v2(t, x)

+ Xf(t, x)v2(t, x) = 0

(3)

and the condition limt I T vz(t, x) = 1. Proof. Validity of the last condition is obvious. The continuity and differentiability of v2(t, x) and the continuity and

boundedness of the derivatives v2(t, x)

axi

and

a2

v2(t, x)

axiaxj

follow from formula (2) and the differentiability of f2,x(s) and f(s, x)

with respect to x, just as in the proof of Theorem 3 of Section 4. From the equation exp (XcT f(s, (s))ds) t

_

- exp (xTf(s, e(s))ds) t /

exp (xTf(u, E(u))du)f(s, e(s))ds

we obtain for t' < t", (xTf(s, e( s))ds) - M2,,x exp (xTfs,

Mtl,x exp

t

t

_ XJt,t Mt,,x.f(S, e(s))MS,e(8) exp (xTf(u,

E( s))ds)

E (u))du)

But Mt-,x exp {xT ;f(S, (s))ds} t

T

= Mt, xMt,-,ect-- exp

t

(S))ds} = Mt,,xv2(t", E(tt t ))

Therefore,

v2(t', x) - Mt,,xv2(t"

e(ttt))

=x

t,

Mt,,xff(S, f(s))v2(S, e(S))

Since

f(s, et,,x(S))v2(S, Lt-,X(s)) - .f(t', x)va(t', x)

0

in probability as s - t' ---> 0, the limit lim

1

t"Mt,,.fls, e(s))v2(s, E(s))ds

tl-tMot" - t' t'

= f(t, x)v2(t, x)

DIFFUSION PROCESSES

416

Therefore the limit

exists.

f(t"))

v,(t', x) - Mt-.xva(t",

Jim

t" - t'

t-V1o t--.t

= lim t,-.t

x)

x) - v2(t [v2(t',

t" - it

+

va(t", x)

- Mt-.xva(t"

(t"))

t" - t'

also exists. But as shown in the proof of Theorem 1, lim Mt-.xyd(t

t,,-t, 10 t,-.t

m (t")) - v2(t , x) = Z a`(t, x)

t"

- t'

i=1

ax

v2(t, x)

+ 2 i,j,k=1 b (t, x)b (t, x) a2y.i(t, x) ax`axi Consequently, the limit v,(t" x) - v,(t', x) = a v,(t, x) Jim 1

t-Vlo t" - t' at tit exists and equation (3) is satisfied. This completes the proof of the theorem. Let us look in greater detail at the case of a one-dimensional process of Brownian motion. If the function f is independent of t, to find the distribution of the random variable I, we can in this case obtain equations of simpler form and with less rigid restrictions. For a process of Brownian motion, the coefficients in equation (2)

of Section 3 are: a = 0, a = 1. It is easy to see that the solution of equation (6) of Section 3 can then be written in the explicit form etjz(s) = x + w(s) - w(t), where w(t) is a process of Brownian motion.

It follows from Theorem 2 that the function v,(t, x) = M exp

J%J

f (x + w(s) - w(t))ds} t

satisfies the equation azv

v2(t x) + 2

(t, axl

x) +

and the condition lima T T v2(s, x) = 1.

xf(x)va(t,

x) = 0

We note that v,(t, x) _

u2(T - t, x), where u,(t, x) = M exp (xtf(x + w(s))ds) . / The function u,(t, x) satisfies the equation U

ua(t, x) + Vf(x)u2(t, x) at ua(t, x) = 2 axQ

and the initial condition limt to u,(t, x) = 1.

(4)

S.

THE METHOD OF DIFFERENTIAL EQUATIONS

417

Equation (4) can be solved by using the Laplace transform with respect to t. We define z,,,,(x) =

Je-1`tu,(t, x)dt .

Here and below, ? denotes a purely imaginary number and p denotes a real nonnegative number. Under these conditions z,,,,(x) is meaningful. Multiplying (4) by e-Pt and integrating with respect

to t from 0 to -, we obtain 1-[7',2(x) - I

-2a

(5)

Xf (x)z',2(x)

22

Equation (5) is valid when the function f and its derivatives are twice

continuously differentiable and bounded. Now let f(x) denote a piecewise-continuous bounded function. Let us choose a sequence of functions fa(x) each of which satisfies equation (5) as applied to z;,;(x) where zana(x) = f e-111M exp Jo

w(s))dsl dt

/

0

In addition, let us suppose that the fa(x) are uniformly bounded

and that they converge to f(x) for each x. One can easily see that the (;(x) are bounded by the number 1/pe and that for every p, x, and x, the sequence jz,1,n)(x)j converges to z,, ,(x) = e-PtM exp (x.tf(x + w(s))ds)dt 0

.

(6)

0

It follows from (5) that the a2zj )(x)lax2 are bounded by the number and that 4 + 2 1 X I C/ p, where C is a constant bounding the Therea2z,,{(x)/axe converges to 2[p - Xf(x)}z,,,,(x) - 2 as n o o. fore the derivative a2z,i,,(x)/axe exists and the sequence {a2zu z(x)/ax2} converges to a2z,,,,(x)/axe (at every point of continuity of the function

f(x)). Consequently at points of continuity of the function f(x), the function z,,,2(x) satisfies equation (5). Thus we have:

Theorem 3. If w(t) is a process of Brownian motion and z,,,,(x) for p > 0 and Re X = 0 is defined by formula (6), where f(x) is a bounded piecewise-continuous function, then z,,2(x) is continuously differ-

entiable, it has a second derivative at all points of continuity of f(x), and it satisfies equation (5). Example.

Let us find the distribution of the quantity

It =

t

sgn w(s)ds 0

DIFFUSION PROCESSES

418

In this case, equation (5) takes the form + 2(X, sgn x

z'',,',2(x)

- ")z,,I(x) = -2

Solving this equation separately for the cases x > 0 and x < 0, we obtain z, ,A(x) =

2Xx) + C2 exp (- 1/2," - 2Xx)

+ C1 exp

1

forx>0, 1

z,,,,(x) =

+ C3 exp (1/2p + 2Xx) + C, exp (- V2p + 2Xx)

for x < 0 . From the assumption that z,(x) is bounded as x--, ± oo, we obtain C, = C, = 0. Furthermore, using the continuity of the functions z,,,,(.x) and az,,,,(x)/ax at the point x = 0, we may write //

1

+C,=

//

+C3,-C2V2pc-2X =C31/2p + 2X

1

From this we get

C=

-1+1/si+ //

3

To determine the distribution of random variable I, it suffices to know z,,,,(0). Substituting the value found for C3, we obtain for _

1

_

1

zE .1 (0)

V

1

x2)-, /2 x2

=

1 : (2n - 1)!! 2n!!

,u n-o

Since n!

tne-'dt =

n=1-1

and

fork odd

0 2

sin' Tdp =

,

(2n - 1)!! 7r for k = 2n , (2n)!!

we have M exp

=

sgn w(s)ds)l =

sin's pdp 2

X2nt2n

(2n)!!

(2n)!

n-o

2

J-r

(2n - 1)!! Xktk

k!

= - (-12 e2t5'n,'dq

nJ

--,2

.

S.

THE METHOD OF DIFFERENTIAL EQUATIONS

419

Consequently, 1

M exp

0t

1 112

sgn w(s)ds) =

/

t

e2s

n'd? = f

eaudF(u)

-Tl2

7L

where 0

-(arc sin x + 1

F(x) =

for x< -l

,

for Ix < 1

,

for

.

7r 1

Ix

>1

Thus pj

REMARK 1.

o

sgn w(s)ds < xI =

I

(

sin t + )

2 Suppose that the function p ,(x) is defined by + sgn x 2

'p, (X)

Then the quantity t

cp+(w(s))ds

-c,+ = 0

represents the time passed on the positive half-axis by the process w(s) during the time t. Using the result obtained, we can find the distribution rt. Specifically, since

rt = 2 + 2

o

sgn w(s)ds

we have p

f t r` < x1 _

= pj-'

t

(arc sin (2x

o

sgn w(s)ds < 2x - 1

- 1) +

for 0< x < 1

Let us use some elementary transformations to obtain the distribu-

tion of the variable rt in a simpler (and more commonly used) form.

If

2 arc cos (1 - 2x) = z then on the one hand,

cos 2z = 1 - 2x ,

2z - sinz z , z =arc sin 1/ x x = 1 - cos 2

and on the other hand,

DIFFUSION PROCESSES

420

arc sin (2x - 1) + 2 arc cos (1 - 2x) = 2. 2 = 2 arc sin V x Thus we have obtained what is known as the "arc sine law." Theorem 4. If zt is the time a process of Brownian motion has spent, up to the instant t, on the half-line x > 0, then

for x<0,

0

p{zt < x} =

2 arc sin ,/ z 1

6.

for 0

x
,

forx>t.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

Let g,(t) and g2(t) denote two continuous functions defined on [0, T] and satisfying the condition g,(t) < g2(t). Let G denote the

region in the tx-plane bounded by these curves and the straight lines t = 0 and t = T. We shall call the process e*(t) a process with absorption on the boundary G if it is fit-measurable and if it has the property: The equation *(t') = g;,(t') imples e*(t) = ge(t) for all t e [t', T]. In this section we shall consider diffusion processes

with absorption on the boundary of the region G, and we shall assume that the functions gti(t) are continuously differentiable. The conditions for absorption on the boundary of G make obvious restrictions on the diffusion coefficients a*(t, x) and a*(t, x) at points dgz(t) = gi(t)dt, we have of the boundary of G: Since

(1 ) Within the region G, the coefficients a*(t, x) and u*(t, x) can a*(t, gi(t)) = gb(t) ,

a*(t, gti(t)) = 0 .

in principle be arbitrary. Suppose that a*(t, x) and u*(t, x) are defined in the region G and satisfy conditions (1). Consider the stochastic differential equation de*(t) = a*(t, $*(t))dt + a*(t, e*(t))dw(t) .

(2)

Let us examine the question of the existence and uniqueness of a solution to this equation, a solution which is a process with absorption on the boundary of G. Theorem 1.

If there exists a number K such that

a*(t, x) - a*(t, x2) I + I u*(t, x1) - u*(t, x2) I < K I x, - x2 I a*(t, x) I+ I a*(t, x) I G K for x1, x2 E (gl(t), g2(t)), and if o* is independent of the process w(t)

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

and satisfies the equation

421

e [g,(O), g2(0)]} = 1, then equation (2)

has a unique solution satisfying the initial condition o = *(O) in the form of a process with absorption on the boundary of G. Proof. Let us first show that equation (2) has a solution that is a process with absorption on the boundary of G. Suppose that

a(t, x) and o'(t, x) are defined for t e [0, T] and x e (- o, W) in such a way that a*(t, x) = a(t, x) and o *(t, x) = o'(t, x) for x e (g,(t), g2(t)), and the functions a(t, x) and 6(t, x) satisfy a Lipschitz condition for all x. Then the equation d$(t) = a(t, $(t))dt + 6(t, e(t))dw(t) (3) with initial condition $(0) = $ has a solution. Let 'r denote the

smallest root of the equation (e(t) gz(t)) = 0 on the interval [0, T]. If the equation has no root in [0, T], we shall take r = T. Let us set $*(t) = fi(t) for t < r but $*(t) = gi(t) if t > r, and fi(r) = gi(r). By construction, this will be a process with absorption

on the boundary of the region G. Let us show that e*(t) is a solution to equation (2). If x(t) = 1 for t < r but x(t) = 0 for

t > r, and if e(r) = gi(r), then

$*(t) - o -

ta*(s,

rt6*(S, *(S))ds

*(S))dH(S)

0

a(s, e(s))ds

-

6(S, $(s))dw(s)]x(t)

+ [e*(t) - 0 - ra*(s, E*(s))ds -

1 r0'*(s, e*(s))dw(s)

0

0

gi(s)ds](1 - x(t)) = 0 .

This is true because if x(t) = 1, then for s < t, a*(s, e*(s)) = a(s, e(s))

,

6*(s, e*(s)) = o'(s, e(s))

*(s) = SO(S)

,

and if x(t) = 0, 6*(s, E*(s))dw(s) _

6(s, e(s))dw(s)

,

a*(s, E(s))ds = oa(s, (s))ds + Jrg;(s)ds and

*(t) - 0 - ra(s, E(s))ds - '6(s, $(s))dw(s) 0

0

$*(r) = gi(t) - gi(r) = tg(s)ds

.

DIFFUSION PROCESSES

422

Let us now prove that a solution of equation (2) that is a process with absorption on the boundary is unique. (As usual, by uniqueness we mean uniqueness up to stochastic equivalence.) Let e* (t) and $ *(t) denote two such solutions. We note that a sufficient condition for them to coincide is that they coincide inside the reigion G, since such solutions reach a given point of the boundary at the same instant and hence coincide. Furthermore, they are processes with absorption on the boundary of G. Let X(t) denote the function that is equal to I if $ (s) G (91(S), 920) for

s E [0, t] and i = 1, 2, and equal to 0 otherwise. Then [ee]T) - $2 (t)]X(t) = X(t)5oX(s)[a*(s, e*(s)) - a*(s, 2 (s))]ds

+

'*(s)) - u*(s, a (s))]dw(s) ,

from which we get (s)))ds]a

M(ee*(t) - $ (t))2X(t) < 2M[ctX(s)(a*(s, a*(s)) - a*(s, $* LJo

+

$*(s)) - a*(s, E2 (s))]dw(s)]2

< LM(ee*(s) - $ (s))2X(s)ds , 0

where L is some constant.

(Here we used the fact that a*(s, x)

and o'*(s, x) satisfy a Lipschitz condition for x e (g1(s), g2(s)).) Just as in the proof of the uniqueness in Theorem I of Section 3, it now follows from the last relation that M(E*(t) - ea (t))ZX(t) = 0;

that is, keeping in mind the continuity of the processes $j(t), we This completes the proof

conclude that P{e1(t) = e2(t), t < z} = 1. of the theorem.

REMARK 2. A solution of equation (2) that is a process with absorption on the boundary of the region G, is a Markov process

whose transition probabilities P*(t, x, s, dy) coincide with the distribu-

tion of the process e*x(s), with absorption on the boundary of G, that is a solution of the equation e*x(s) = x + aa*(u, S*x(u))du

+

5*(u, a*,(u))dw(u) t

(s > t)

.

(4)

The process a*x(s) can be obtained from the process et,x(s), which is a solution of equation (6) of Section 3, if in that equation we substitute those a(t, x) and a(t, x) that were used in the proof

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

423

of Theorem 1. Let z, s denote the smallest root of the equation (e,,z(s) - g,(s)) x (eLrz(s) - g2(s)) = 0 on the interval s e [t, T], and set z,,s = T when this equation has no root. Then *x(s) = st.s(s) for s < z,,x but * z(s) = g;(s) for s z,,x if x(z, x) = g;(z,,x) The first assertion is proved in a manner analogous to that of the proof of Theorem 2, Section 3, and the second assertion follows from Theorem 1. Let us look at certain transformations that enable us to simplify equation (2) in the region G. In addition to the process e*(t), let us consider the process iI*(t) = f(t, e*(t)), where f(t, x) is for each t an increasing twice continuously differentiable function of x for x e [g,(t), g2(t)] and differentiable with respect to t. The process *(t) is also a Markov process since $*(t) is uniquely

This is a process with absorption on the boundary of the region G that is bounded by the curves determined from )7*(t).

NO _ .f(t , ge(t)) ,

g2(t) = .f(t, g2(t))

Using property 8 of Section 2, we see that Y)*(t) satisfies the equation dry*(t)

=

[f'(t, e*(t)) + fs(t, $*(t))a*(t, $*(t)) + 2 .f'x(t,

a*(t))a.*(t, e*(t))2]dt

+ fz(t, $*(t))Q*(t, e*(t))dw(t)

Thus )7*(t) is a solution of a stochastic equation of the form (2): dry*(t) = a*(t, ,*(t))dt + Q*(t, )7*(t))dw(t) , where

a*(t, y) = f'(t, p(t, y)) + fx(t, p(t, y))a*(t, m(t, y)) + 2 fxx(t, P(t, y))6*(t, T(t, y))2 ,

a*(t, y) = fx(t, P(t, y))o*(t, P(t, y))

,

(5)

and the function q(t, y) is the inverse of f(t, x) with respect to x; that is, f(t, p(t, y)) = y, and cp(t, f(t, x)) = x. If we set fz(t, x) = l/Q*(t, x) for x e (g,(t), g2(t)), then a*(t, y) = 1 for y e (g,(t), g2(t)). Suppose that f(t, z) _ Then

*A

O*du

8,(t) u

u)

DIFFUSION PROCESSES

424 82M

gilt)=0,

g2(t)=

du 91(t) a*(t, u)

Let us introduce the process yJ*(t) =

=C(t) 7)*(t)/C(t).

This is also a

Markov process, with absorption on the boundary of the rectangular

region A: {0 < t < T, 0 < x < 1}, and it satisfies the equation

dy)*(t) _ [_c9.() + a*(t, C(t)7)*(t))]dt + d*(t, C(t)i)*(t)) dw(t) .

(6)

C(t)

Now let us define the process C(t) (dw(s)/C(s)). This process is, with probability 1, a continuous process with independent increments and satisfies the relations Mg(t) = 0 ,

DC(t) = f

t

dss)

0 CI(

_ X(t)

Therefore C(t) = w1(X(t)), where w1(t) is a process of Brownian motion. Setting x(t) = s (so that t = X-1(s)) and )7*(X-1(s)) = e*(s) , C'(X (s)) x + a*(X-1(s), C(X-1(s))x) = a*(s, x) C(X-1(s))

and letting x(o,1,(x) denote the characteristic function of the interval (0, 1), we obtain from (6) the following equation for e*(t): dE*(t) = a* (t, e*(t))dt + x(O,1)(e*(t))dw1(t)

(7)

.

Thus an arbitrary problem associated with finding the distribution of any characteristic of the process *(t) that is a solution of equation (2),

can be reduced to finding the distribution of some other

characteristic of the process $*(t) that is a solution of equation (7). The latter is somewhat simpler than equation (2). The transition probabilities of the process e*(t) can easily be obtained in terms of the transition probabilities for the process *(t). Consider' the question of determining the transition probabilities for the process e*(t). From the remarks made above we note that the transition probabilities p*(t, x, s, dy) of the process e*(t) coincide with the distribution of the process et;y(s) constructed as follows: Let $,,,(s) denote a solution of the equation a1(u, et.=(u))du + w1(s) - w(t)

$t,=(s) = x +

,

t

0<x<1. (8) Suppose that a1(u, x) coincides with a* (u, x) for 0 < x < 1.

Suppose

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

425

that this function is bounded and satisfies a Lipschitz condition, and

that zt,x is the smallest root of the equation et,x(s)(l - t,x(s)) = 0 on the interval s e [t, T]. If this equation has no root, let zt x = T. Then e,('z(s) = E,,x(s) for s < z-,,, and ei;s(s) for s > z-,,. Let us show that rt,x is with probability 1 a continuous function of t and x. Since rt,x = min vi'], where zt'x is the smallest root of the equation et,x(s) - i = 0 on [t, T] (r;tix = T if that equation has no root), it follows that to prove the continuity of rt,x it suffices to prove the continuity of vtl% and vi;i. For this we use: Lemma 1. For every h > 0, sup

P{

rt x<s
St,x(s) > 1, 7*zj2 <

T} = P{r,,x < T}

Proof. Let us choose s > 0 and o > 0. Let va denote the smallest natural number such that t,x(ova) > 1 - s. Then p{

sup Sva
ko
et,x(S) > 1, ova < T}

PL <s 1, va = k}

> ko s, va = k} _> E P{ sup ko
ko<s
[w(s) - w(ko)] > s + Ch, v, = k}

where C is a constant bounding a,(t, x).

(Here we used the inequal-

hC + SUpko<s
ity Supko
The process w(s) - w(k3) for s > ko is independent of the event {va = k} since this event is completely determined by the behavior of the process w(u) on the interval [t, ko]. Therefore, on the basis of Corollary 1, Section 5, Chapter VI, P{

sup

ava<s
et,x(S) > 1, ova < T}

>_ E P{ sup ko
ko<s
2/\

2

V 7r

(w(s) - w(ko)) > s + Ch}P{ve = k}

=lJh+C.Jh

exp (- u )dup{ova < T} .

\

2

Since P{r,,,x < ova}

we have

p{

sup le'-s"Io

e(s,) -

I > 6}

426

DIFFUSION PROCESSES

Pf

su p

e".(S) > 1, Z- t(1)

<

r(m))<s
sup

et,x(s) > 1, zt:= > av8, Sv8 < T}

sup

St,x(s) > 1, 31), < T} - P {zt,x < 3vs}'

t8ug<s<8ug+h

>P >_

8vg<s<&vB+h

2

i/ 2ir

,th

P

is,

exp (- uE )du P {8vd < T}

\

2

I>s

sup I St,x(s') '111<6

We also note that 8v8 < zxl) for sufficiently small 3 and hence lim p{hv8 < T} > p{zL', < T} 8-.0

Therefore, taking the limit as a > 0 and remembering that is continuous with probability 1, we obtain sup

Pj

l rL'x<8
>

P

2

et,x(s) > 1, zt;z < T}

J

exp(_)duP

V-2-2r

o h +c h

sup

t,x(s) > 1, Zt,x < T

T}

tx<
>P

sup It(l) t,x t,x <s
t,x(s) > 1, Zt 1,)x < T

>' (//227r o h +c h_ exp \

(-u2

2

)du p {zt;x < T}

by taking the limit first as s --> 0 and then ash assertion of the lemma.

0, we obtain the

Let us now prove:

Theorem 2. The random variable zt,x is, with probability continuous as a function of t and x. Proof.

1,

As already noted, to prove the assertion it will be

sufficient to show that the random variables z,1', and zt°x are continuous with probability 1. Let us show, for example, that z-,11) is continuous. We note that for t, < t2 and x x2 e (0, 1),

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

427

t2

x, - X2 I +

s"'.1(s) - ec2.y2(s)

a1(u, E5,y(u))du e1

s

+

a1(u, $52,y2(u)) du + w(t2) - w(t)

I a1(u, J t2

< I x1 - x2 I + C t1 - t2 I + I w(t2)

+

- w(t)

I L1,y1(u) - ee52,y2(u) I du . t2

Therefore

I $'1"1(s) - $'2,y2(s) I < (I x1 - x2 I + C I

tl - t2

l

+ I w(t2) - w(t1) 1) exp {K(s - t2)}

so that for some L, se1,y1(s) - ec2,y2(s) I

< L(I x1 - x21 + I tl - t21 + I w(t1) - w(t2) 1)

means that

The event zLi'yl > zi2'y2 + $t1,x1(s)

(9)

for S E (Tt21y2' r 2,y2 + ) .

1

Since '11y1('S) - t21y2(s) 1

< L(I x1 - x2 I + I tl - t2 1 + I w(t1) - w(t2) 1)

it follows that

tl - t2 1 + I w(tl) - w(t2) 1)

EL2,y2(S) < 1 + L(I x1 - x2 1 + I for s E (T(2 ?y2, _C"' y2 +

a).

)7a =

We define sup

ct2,y2(S) - 1

TL2, r2<s<_L2' 2+d

-According to Lemma 1, ply), > 0} = 1 for every 3 > 0. As soon as

L(Ix1-x21 +Itl-t2 the event zii'yl >

+ I w(t) - w(t2) I) < 20

,

2 + 3 becomes impossible. Thus ztl'y1 < zi2'y2 +sl,

if L(I x, - X2 I + I t1 - t 2 1 + I w(t1) - w(t2) )_< i

.

(This inequality holds for sufficiently small I x1 - x2 I and

by virtue of the continuity of w(t).) Let us also define

sa = 1 -

sup S<7) i

$'2,y2(S)

2y2-a

Obviously P{Ca > 0} = 1 for all 3 > 0.

L(1x1-x21+It1-t21+

If

w(t) - w(t2) 1) < C1

,

I

t1 - t2

1

DIFFUSION PROCESSES

428 then

sup 8>

tg>x2

_s

a 1,x1(6) <

sup B

et2.x2(s) + 8 = 1

and hence 8. This proves the continuity of v . The continuity of zi°x is proved in an analogous manner, and this completes the proof of the theorem. We shall now prove a theorem that will enable us to determine the transition probabilities of the process i;*(t) that is a solution of (7).

Theorem 3.

Suppose that al (t, x) for x e (0, 1) has uniformly

continuous partial derivatives as* (t, x)/ax and a2a*(t, x)/axe in the region x e (0, 1), t e [0, T]. Suppose also that the function f(x) is twice continuously differentiable and

.f(O) = f(1) = f' (O) _ .f'(1) _ f '(O) = f"(1) = 0 Then the function

u*(t, x) = M(f(S(s)) I E(t) = x) = f(y)p*(t, x, s, dy) where p*(t, x, s, dy) is the transition probability of the process c* (t) in the region 0 < t < s, x e (0, 1), satisfies the differential equation at

u*(t, x) + a*(t, x)

E

au*(t,

-

x) +

2 ax2

u*(t' x) = 0

(10)

and the boundary conditions lim u*(t, x) = f(x) , t-8

lim u*(t, x) = 0 ,

lim u*(t, x) = 0

-

x-+0

x-+1

Proof. Let us first show that the process et,x(s) is, with probability 1, twice differentiable with respect to x, and that the derivatives aet,x(s)lax and a2et,x(s)lax2 with probability 1 are continuous

with respect to the set of variables t and x and bounded by constants independent of chance. We shall assume that a1(t, x) in (8) is chosen in such a way that a1(t, x), aa1(t, x)/ax, and a2a1(t, x)/axe

exist for all x and are bounded and uniformly continuous for t e [0, T], x e (- oo, oo).

(By virtue of the uniform continuity of 2

al

(t' x)'

axa*(t, x) ,

a*(t,

x)

ax2

it is always possible to make such a choice.) We note that continuity of $,,x(s) follows from inequality (9); denote a specifically, I Et,xl(s) - et x2(s) I < L I x1 - x2 I. Let solution of the equation

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

1+

/x,t(s)

Jtaz(u,

429

e,,x(u))rlx,t(u)du .

Then St,x+dx(s) - et,x(s)

- -/x,t(s)

Ax

`$ra(u+ t,x+dx(u)) - a(u, eat,x(u)) t,x+dx(u) - t,x(u) t Ax t,x+dx(u) - st,x(u) as(u, t,x(u))rlx,t(u)Idu

$ra(u+ t.x+dx(U)) - a(u, St,x(u)) - ay(u, t,x(u)) t trx+dx(u) - St,x(u) Set,x-I-dx(u)

x

Ax

x

Ehit,x -

St,-(u) du + f tax(u, a ,x(u)) '

et,x+dx(u) - St,x(u) du . J Ox

]

Since

Ea(u,

ee,+(u)) - a(u, S,(u)) - a(u, S,(u))]

0

t,x+dx(u) - et,x(u)

uniformly with respect to u and x as Ox t,x+dx(u) - bt,x(u) Ax

< L and

0, and since

I a'(u, et,x(u)) I < K ,

we conclude from the inequality st,x+dx(s) - t,x(s)

- 27-,t(s)

Ax

< LJ$ a(u, St.x+dx(u)) - a(u, a ,x(u)) _ az(u, St,x(u)) du St,x+dx(u) - t,x(u) et,x+dx(u) - et,x(u) - 72t.x(u)1du + t Ax it

that _t,x+dx(s) - t.x(s) - 7).,t(s)

Ox

_ < LcBra(u, tt,x+dx(u)) - a(u, t,x(u)) - az(u, bt,x(u))]due'`T t'

bt,zFdz(u) - st,x(u)

follows from the continuity of az(u, et,x(u)) as a function of t and x since equation

that is, t,x(s) - a ,x(s)lax. The continuity of

(11) is meaningful for every sample function and is an ordinary linear integral equation of the Volterra type.

DIFFUSION PROCESSES

430

Analogously we can show that 2

aX2t,x(S)

satisfies the equation Th?2, (s)

_

t

azz(u, t,x(u))77t,x(u)2du

+ 'ax(u, et.x(u)) t

ax

(12)

"/,,x(u)du .

This too is an ordinary Volterra equation and its solution depends continuously on the parameters t and x. It follows from the inequality St,xl(s) - et,z2(s)

L

xl - x2 that aet,x(s)lax < L with probability 1. If C is a constant bounding az(t, u) and a" (t, u), it follows from equation (12) that I

a)7az(s)

< L2C(s - t) + C t

I-

/t,x(u) I du ,

and hence

axe

t,x(s) I < e°TL2CT

with probability 1. Suppose that et;x(s) is defined as indicated above. Let us show that f(et;x(s)) is then, with probability 1, a twice continuously differentiable function of x. If s < rt x, then for sufficiently small 8, we also have s < rt,x, if I x' - x j < a. Therefore et;x,(s) = t,x,(s) for I x' - x I < 8. Hence t.;x,(s) is twice continuously differentiable

with respect to x' in a neighborhood of the point x.

Furthermore, if s > rt x we can easily show that in some sufficiently small neighborhood of the point x', the function t'x(s) is constant with respect to x, so that its derivatives of all orders are equal to 0. In both cases f(et;x(s)) has derivatives with respect to x since f is differentiable. Suppose now that s = rt x. Then

l + B( t,x ttlx(rt,x) -

where 0 < 0 < 1. Since

1

t,x(rt,x)1

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

.=+ox(zt,x) - fit;=(zt,x) I

431

t,x+ox(zt,x) - t,x(Tt,x) I < LAx

;z(zt x)) = 0, because t"(zt,x) is equal to either 0 or 1, it follows that af(et1;x(s))lax = 0 for s = zx,t In an analogous manner we can show that (o2/ax2)f(et;x(s)) = 0 for s = zx,t. Continuity of the derivatives of f(E,"(s)) follows from the relations and

a Pf( (')X(s)) = ff

ox

(S))

=

4x(s) IZ +

Is(S)) ax22 t,x(S)

which are valid for all s. These derivatives are bounded by constants independent of chance by virtue of the boundedness of aet,x(s)/ax and a2et,x(s)/axe. Therefore the function u*(t, x) _

f(Y)p*(t, x, s, dy) =

=(s))

is twice continuously differentiable with respect to x. (We can differentiate under the mathematical expectation sign.) After we have established the existence and continuity of the partial derivatives au*(t, x)lax and a2u*(t, x)lax2, equation (10) can be derived just as equation (1) of Section 5 was derived (cf. proof of Theorem 1, Section 5). From the continuity of e;s(s) with respect to t and the relation ez;'s(s) = x it follows that u*(t, x) f(x) as t s; the boundary conditions u*(t, 0) = u*(t, 1) = 0 follow

from the continuity of Et;i(s) for x = 0 and x = 1 and from the relation ;t;s(s) = x for x = 0 and x = 1. This completes the proof of the theorem. As an example of the application of this theorem, let us determine the probability that the process s(t) = x + -yt + w(t) for t e [0, T] will not get outside the interval [0, 1]. Let e*(t) denote a process that satisfies equation (7) if a*(t, x) _ 7X,0,1,(x), where X(o,,,(x) is the characteristic function of the interval (0, 1).

Then the probability in question is equal to M(Xro,n(s*(T))

M) = x) .

Let f(x) denote an arbitrary function that satisfies the conditions of Theorem 3. Then if u*(t, x) is defined as in Theorem 3 (with

s = T), we have au*(t, x) _

-

7

au*(t, x)

I

a2u*(t, x)

+ 2 axe In view of the homogeneity of the process w(t), we see that at

ax

DIFFUSION PROCESSES

432

u*(t, x) = v(T - t, x), where v(t, x) = Mf(e*(t) I (O) = x) For the function v(t, x) we obtain the equation av(t, x)

a

- 7axv(t'

x)

1

a2v(t, x)

+ 2 ax at with initial condition v(0, x) = f(x) and boundary conditions v(O) v(l) = 0. Solving this problem by Fourier's method, we obtain

v(t, x) = j ake-'x exp

t sin krcx

k27r

(

k=1

2

[72

+2

2]

/

l where ak = 2erxf(x) sin ktrxdx. Taking the limiting case, that is, with f(x) = X(,,,)(x), we obtain the probability that we are seeking. Then

ak -

kr<

y2 + ,22r2

[el(- l)k-1 +

1]

so that

E (0, 1), 0 < t < T}

=

ktc k-1 72 + k27r2

[(_ 1)k-1e-T(x-1) + e-] exp (- [y2 + k2lr2] tl sin k7rx

\

2

//

This formula is convenient to use when t is not very close to 0. Hence when t > 1, to find the desired probability with an accuracy of 10-1, it will be sufficient to take only the first term of the series. When y = 0, we can use another expression for the probability P{0 < w(t) < 1, 0 < t < T}, which we found in Corollary 2, Section 5, Chapter VI:

P{w(t)e(-x, 1 -x),0
T) sin (2k - 1)7rx ,

1 7<

from which we obtain the more general formula P{w(t) E [a a2], 0 < t < T}

= P{w(t) E (

a,

a2

), 0 < t <

T

{ (az a, exp (- (2k - 1)27' 2 T)sin (2k - 1)7r a1 - a2 k=1 (2k - 1)7r 2(a2 - a,)2 //

_i

a2 - a,

az - a,

al)z

1

This formula can be used when T/(a2 - a,)2 is not very small, whereas formula (5), Section 5, Chapter VI is more convenint to use when T/(a2 - a,)2 is small.

Using the differentiability of f(;(;;(s)) with respect to x. we

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

433

can prove the following theorem just as we proved Theorem 2 of Section 5.

Theorem 4. Let f(x) denote a twice continuously differentiable function on the interval [0, 1]. Suppose that f(x) and its first and second order derivatives vanish at the points 0 and 1 and that s*(t) is a solution of equation (7). We define v2(t, x) = M(exp {?t:f(e'(s))ds}

*(t) = x) .

(13)

Then, the function v,(t, x) is continuous for t e [to, T] and x e [0, 1], and it satisfies the equation

a + a(t, x) ax +

axE + x'f(x)v2 = 0

2

(14)

with boundary conditions v,(T, x) = v2(t, 0) = v,(t, 1) = 1.

Let us apply Theorem 4 to the solution of the "second diffusion

problem" (in the terminology of A. Ya. Khinchin, 1936, p. 43). The problem is to determine the probability Px(a, 6) that the diffusion process beginning at a point x will attain the point a before it attains the point 9. Let us assume that a < x < ,8. Consider a diffusion process fi(t) with coefficient of transfer a(x) and coefficient of diffusion 1. Since the diffusion coefficients are independent of t, the process is homogeneous (cf. Remark 2, Section 3). Let us choose an arbitrary point 8, > /3. We let s,(;x(s) denote

a process with absorption on the interval [a, /,] and satisfying the equation ds`i,x(S) = a($e ) x(S))xcn.e, (Si.z(S))ds +

s(S))dw(S)

(15)

with initial condition $i;i(t) = x. Let X(-,M) denote the characteristic function of the interval (a, (3,). It follows from Theorem 4 that the function v,(t, x) = M exp {Tf(e(s))ds}

satisfies the equation I

at

vx(t, x) + a(x) a v,(t, x) +

z

v,(t, x) + ? v,(t, x)f(x) = 0

axe

and the boundary conditions v,(T, x) = v,(t, a) = v,(t, ,8) = I if f is twice continuously differentiable on [a, 6,] and satisfies the conditions

f(a) = f' (a) = f"(a) = .f(,8,) = f' (R,) = f"(/3) = 0. We note that

DIFFUSION PROCESSES

434

v,(t, x) = M exp {tf(E(s))ds}

because of the homogeneity of the process e y (s). Consequently, the function u,(t, x) = M exp {x,tf(E(s))ds}

for t > 0 and x E (a, C3), satisfiesthe equation uz, au, = a(x) au, + 2 axz + Xf(x)u2 ax

1

at

and the boundary conditions u2(0, x) = u2(t, a) = u,(t, i9) =1.

Suppose

now that f > 0 and x < 0. Then the limit u,(x) = lim u,(t, x) = M exp

{0f(e(s))ds}

exists (if we take e = 0). To establish certain properties of the function u,(x), we need: Lemma 2. Let zx denote the instant at which the process eo,z(s) attains the boundary of the interval [a, 6j. This quantity is, with probability 1, finite and for every k > 0, supasxsal Mzx < Proof. Suppose that supasxs,9l I a(x) = ao. Then P{zx < to} > P{ sup rx + oststo L

P{sup

o
w(t) > 0

w(t)>R1-a+toad}=ao>0.

Define

n = max P{zx > nto} x

Then P{-r. > nto} = P{a < eo,x(to) < 6, zx > nto}

= "P{zx > nto I

eo,x(to) = y}P{eo,x(to) E dy}

J

f

"p{-r,, > (n - 1)to}P{ex(to) E dy}

< Xn_1P{a < x(to) < RL} < X.-Al .

This means that X. < a,,X, < V. Since 0 <) , < I - ao < 1, it oo and that zx is finite with probability follows that X1 0 as n 1.

Furthermore, the series nkxn

converges for arbitrary k > 0, and

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

, nkp{zz > n - 1 } <

Mzx

435

nkX -i <

This completes the proof of the lemma. Since

at

ua(t, x) = M exp

f(So's(s))ds)X}(o;=(s))

and f(e("(t)) --- 0 with probability 1 as t

o (because when t > zz,

the quantity a') is equal either to a or to l3 and f(a) = f(/3,) = 0 for x < 0), it follows that lim auz(t, x) = 0 at

Furthermore,

axu'(t,

x) = M exp

(xf(e(s))ds)xt_f(eO,(s))ds

so that the limit lim xua(t, x) -= XM exp (

f

exists since < LMzx

(1)

M

where L is a constant bounding af(E (s))/ax. au2(t, x) = d lim

Obviously

dx ua(x)

ax

t-.m

,

Analogously we can show that lim alu,(t, x) = a, u2(x) t--wo axz

ax2

This means that the function u2(x) satisfies the equation

a(x) u' + 2dx

u,

+ Xf(x)u2 = 0

dxz

(16)

Suppose now that f(x) is positive for x e (e3, (3,) but equal to 0 on [a, ,S]. Then on the interval [a, (3], u2(x) satisfies the equation 2

a(x)

and the condition u2(a) = 1. 1

1im u2(x) = lim M exp

= Mrlim exp

0

ddu2

x +

2

axe

Consider the limit o

(xf(esds)] = Mxo(

f( o;z(s))ds)

DIFFUSION PROCESSES

436

where xo(x) = 1 for x = 0 and xo(x) = 0 for x

0.

The quantity

(s)) ds)

x°\Jo.f(

is equal to 0 if and only if the process co;2(s) reaches the point a before it reaches the point 8, since in the case in which it reaches with probability I (on the ,3, the quantity E ("(s) falls in basis of Lemma 1), so that Jo.f(eo;x(s))ds

is nonzero. relation

Thus the probability in question is determined from the

P.(a, i9) = lim u;(x)

We note that

decreases monotonically as X u.(x) >_ 0, and that u;(a) = 1. If we define u-(x)

C= lim u,(/3),

- 00 ,

that

C>0

we easily see that u(x) = lim u;(x) is a solution of the equation a(x) d dx u(x) +

I

0

2 d ' u(x)

with boundary conditions u(a) = 1 and u(/3) = C. To determine C, we note that u(x) = P,(a,,a) = 1 - P,(9, a) so that u(/3) = 0, since P,(/3, a) = 1. Solving this equation we find 13

B(t)dt

x

N) =

//

(18) (18)

,

B(t)dt

where

B(t) = exp

2a(u)du}

.

RE MARK 2. We have seen that under certain conditions on the smoothness of a(x) and 6(x), we can, by a transformation of the unknown function, reduce a stochastic equation to the case in which a(x) = 1. By means of such a transformation, we can easily show that formula (18) remains valid for arbitrary u(x) if we set

B(t) = exp { f 6a(u) dul

.

(19)

u

Sometimes it is useful to know the mathematicgl expectation

6.

ONE-DIMENSIONAL DIFFUSION PROCESSES WITH ABSORPTION

437

of the instant at which the process goes beyond the boundary of interval [a, 8,]. To determine this quantity, let us differentiate equation (16) with respect to X and then set X = 0. We see that the function I

x

o

= fi(x) = M Jofeolx(s))ds

satisfies the equation z

,p a(x) a-T + 2 dxl ax

+f=0

(20)

and the boundary conditions T(a) = cp(,Q,) = 0. Solving equation (20), we obtain

q(x) = JaB(t)I

- 1rt B(u)) du]dt

where

L B(t)B( u)) du

C= p1B(t)dt

By means of a limiting operation wherein f(x)-->1 for x c (a, e) while remaining bounded, we obtain du t

M7

B(u) (21)

f31B(z)dz

IX

LIMIT THEOREMS FOR RANDOM PROCESSES

Throughout the remainder of this book we shall frequently encounter processes that are obtained from simpler random processes by a limiting operation. These limiting operations yield either sample

functions of the original process (for example, in the solution of stochastic differential equations in Chapter VIII, Section 3, or in the expansion of a Gaussian process in a series of eigenfunctions of the kernel corresponding to the correlation function in Chapter V, Section 2) or simply finite-dimensional distributions of the process (for example, in the construction of stationary Gaussian processes by taking the limit of trigonometric sums, Chapter I, Section 4). In the study of random processes considerable attention is given to methods of finding the distributions of different functionals of a random process, for example t

2f

tl

sup s(t) ,

tlGt StZ

inf e(t)

.

tl5t
Finding the distribution of functionals of random processes is one of the fundamental problems of the theory of random processes,

and it is a method that is used in the solution of many practical problems. Therefore we ask the question: If a process e(t) is obtained from a sequence of processes {e,L(t)} by a particular limiting operation, is it possible to obtain the distributions of the functionals of the process e(t) if we know the distributions of the corresponding

functionals of the processes e,(t)? Of course if we show that the sequence {an(t)} converges uniformly

with probability 1 to a(t), that is,( if P{sup I M0

- E(t) I - 0

1

,

then for every functional f that is continuous in the topology of uniform convergence, f(on(t)) - f(e(t)) with probability 1, and hence 438

1.

WEAK CONVERGENCE OF DISTRIBUTIONS IN A METRIC SPACE

439

the distribution of f(e,,,(t)) converges to the distribution of f(e(t)). However there are two points here that we should consider. First, are not always defined on the same the processes e(t) probability space; second, even when they are defined on the same probability space, the hypothesis of convergence for the distributions of functionals that are continuous in the topology of uniform convergence is too narrow, and restriction to the cases in which it is satisfied considerably decreases the usefulness of the method. In what follows we shall assume that the sequence of processes E (t) at least converges weakly to some process E(t); that is, the sequences of finite-dimensional distributions of e, (t) converge to finite-

dimensional distributions of e(t) (cf. Definition 4, Section 1, Chapter I). These requirements are too weak to imply convergence of the distributions for a sufficiently broad class of functions (for example, for the functionals listed above). Therefore, we must seek supplementary conditions under which the sequence of distributions of a functional (belonging to some class F) of the processes E,, (t) converges

to the distribution of the corresponding functional of the process e(t).

The class F of functionals must be such that f(on(t)) and f(e(t))

are random variables for f c F. Consequently, choice of the class F must depend on the properties of the processes an(t) and e(t). We shall consider the cases in which (1) the processes are continuous with probability 1, (2) they have, with probability 1, no discontinuities of the second kind, and (3) they are measurable and q-integrable for some q. In each of these cases, we shall consider a different class of functions. Limit theorems for random processes are important for the determination of the distributions of functionals of the final limit process from simpler processes by passage to the limit. However,

it is also natural to use such limiting processes to describe the limiting behavior of discrete processes: processes with independent increments (to describe a sequence of sums of independent random variables) and continuous Markov processes (to describe Markov chains with discrete time). In this case we shall examine the procedure of obtaining processes that vary continuously in time by taking the limit, as the distances between the instants at which the changes take place approach 0, of processes for which changes take place only at certain fixed instants of time. For such processes we shall investigate the conditions for weak convergence. As we have already seen, for processes with independent increments and Markov processes we have an analytic apparatus that enables us to find the distributions of certain important classes

LIMIT THEOREMS FOR RANDOM PROCESSES

440

of functionals. Using the known distributions of functionals and applying the limit theorems to random processes, we can obtain limit theorems for sums of independent random variables that are different from the classical limit theorems. 1.

WEAK CONVERGENCE OF DISTRIBUTIONS IN A METRIC SPACE

Suppose that the sample functions of the processes e (t) and e(t) (for t e [a, b]) belong, with probability 1, to some metric space X of functions with metric p ,(x, y) for x, y e X. For example, if en(t) and e(t) are continuous with probability 1, they belong to the space C of continuous functions with metric pc(x, Y) = sup x(t) - Y(t)

For the class of functionals F, for which we are seeking condi-

tions under which the distribution of f(e.(t)) converges to the distribution of f(e(t)), we take the set of functions on X that are continuous with respect to the metric px. For f(i(t)) to be a random variable, it will be sufficient to require that the space X be separable and that for each open sphere S of the space X, the set {u: e(t) e S} be measurable with respect to the original probability space (since in this case the set {u: e(t) e At will also be measurable for every Borel subset A of X). Let us assume that X is a separable space and that the processes Ek(t) and e(t) satisfy the requirement stated above. The measure p (resp. p%) corresponding to the process e(t) (resp. m,(t)) is defined

on the a-algebra 0 of all Borel subsets: p(A) = P{E(t) e Al (resp. p, (A) =

e A})

For every bounded 8-measurable functional f, M f(e(t)) = f(x) p(dx)

(cf. Theorem 3, Section 6, Chapter II). A necessary and sufficient condition for convergence of the sequence of distributions of f(on(t)) to the distribution of f(e(t)) for all continuous functionals f is that lim f(x) pn(dx) _ f(x) p(dx)

(1 )

J

for all continuous bounded functionals f. To see that this is true, note that convergence of the sequence of distributions of f(on(t)) to the distribution of f(E(t)) and the boundedness of f imply con-

1.

WEAK CONVERGENCE OF DISTRIBUTIONS IN A METRIC SPACE

441

vergence of the sequence {Mf( (t))} to M f(e(t)) and hence the validity

of (1). On the other hand, equation (1) implies that for every continuous functional f, the sequence of characteristic functions of the

variables flE(t)) converges to the characteristic function of the variable f(e(t)): lim

Mecat(en(t))

= lim

ei2f(x) ftn(dx) _ e;2f(x) p(dx) = Memaf(e(t))

Definition 1. Suppose that equation (1) is satisfied for all continuous bounded functions fix). We then say that the sequence of the measures U. converges weakly to the measure u and we write

P. - fl. Definition 2. The sequence of measures p. is said to be weakly compact if every subsequence of it contains a weakly convergent subsequence of measures. Theorem 1. Let X denote a complete separable space and let 0 denote a a-algebra of Borel sets. For the sequence of measures 'Un on 58 to be weakly compact, it is necessary and sufficient that a. sup. P.(x) < 00, b.

for every s > 0,

there

exists a compact set K such that

sup . , "n(X `K) < s.

To prove this theorem, we need: Lemma 1.

Let X denote a separable compact space and suppose

that sup, u,,,(X) = H < -. Then the sequence of measures fJ

is

weakly compact.

Proof. Let Cx denote the space of. continuous functions f on X and define I f I I= sup, E x I f(x) I. Then Cx is a complete separable , denote a sequence normed linear space. Let { fk}, for k = 1 , 2, I

that is everywhere-dense in C1. Using the diagonal method, we can choose a subsequence of measures pnk such that for all i the limit kim

exists.

l[fs]

Since

Il[f]-l[f]I
l[f]

=kiln ` f(x) a, k(dx)

LIMIT THEOREMS FOR RANDOM PROCESSES

442

remains valid for all f.

Being the limit of a sequence of linear

nonnegative functionals, 1[f] is itself a linear nonnegative functional

on C. We introduce the set function %(A) defined on 0 by %(A) = inf l[f] f XA

The function MA) has the following properties: 1. %(A) is monotonic, and for every sequence Ak, X(U Ak) k X(Ak), 2. if the distance between the sets Al and A2 is positive, that

is, if p(A1, A2) =

inf E A1,y 6:12

p(x, y) > 0

then x(A, U A2) = M(A) + L(A2) .

To prove (1), we note that if ck(x) > XAk(x), then Ek q (x) XUAk(x)'

To prove relation 2, let us choose arbitrary a > 0 and acontinuous function f such that x(A, U A2) > "[fl- a and f XA, + XA2.

We let TAti denote the function 1

,lti'x O_

if

X G Ai, ,

1 p(x, A;) if 0 < p(x, A,) < s , 0

p(x,A,)>s.

if

Here p(x, y) = infy6A p(x, y). If 2s < p(A A2), then T1 + r2 < 1, 91`A, >_ XA1, and (pAZ x,,2. Therefore ,,(A1 U A2) + a > 1[7'31.fJ + l[7 2f] ? (A1) + '(A2) -

Relation (2) follows from this inequality by virtue of property (1) and the arbitrariness of a > 0. Let s.$o denote the collection of sets for which X(A') = 0, where A' is the boundary of the set A. Obviously 5 3o is an algebra of sets. Let us show that X(A) is an additive set function on V3, Let Al and A2 denote two disjoint members of moo. Let r denote the union of the boundaries of the sets Al and A2. Since X(P) = 0, it follows that for every s > 0, there exists a continuous function g satisfying the condition g > X, such that 1[g] < E. Define

[Ai] n lx' g(x) < 2 }

'

where [A] denotes the closure of A. The closed sets A;,' and A(,'

1.

WEAK CONVERGENCE OF DISTRIBUTIONS IN A METRIC SPACE

443

are disjoint and hence are at a positive distance from each other, since X is compact. Define 2-1

Then X(F,) < 1[2S] < 2s , X(Aa) < x(AzE,) + X(F,) < %(Aa:)) + 2s

Using the monotonicity of . and property (2) we obtain X(A1) + X(A2) < X(AIEN) + ? (AEE>) + 4s

=X(A(0U AZ")+ 4s<MA,UA2)+4s. Taking the limit ass 0 and keeping (1) in mind, we see that 7. is additive on S$o.

Thus the conditions of Theorem 2, Section 7, Chapter II are satisfied for the function x on 0o. Therefore 7 can be extended as a measure p defined on c(00). Let us show that u(Oo) coincides with 58. Note that for every x there exist only countably many radii r such that X(Sr(x)) > 0, where Sr(x) is the sphere of radius r with center at x and boundary ST(x). Therefore Q(uo) contains all spheres and hence all Borel subsets of X. Let us show that for all f e C1,

(2)

1 [f l = f(x)p(dx)

that f is

nonnegative. Let us choose values 0 = . < CN with c,,. > JJf JJ, in such a way that %{x; fix) = ck} = 0. Then if Ak = {x; ck_1 < f(x) < ck}, it follows that Ak e 3o and hence X(Ak) = p(Ak). Let r p k for k = 1 , 2, ,N Suppose

co < C, < c, <

.

denote, a continuous function such that 1[qk] > X(Ak) + s and pk XAk. Since f :!g Ek=1 C,q , we have N

, C Ne +E Ck? (Ak) = Ne + E CkI!"(Ak)

1[fl

k=1

k=1

< Ns + max(Ck - Ck-1) fl(X) + f(x)p(dx)

.

It follows from this inequality that 11f ] < f(x)p(dx)

(3)

for an arbitrary nonnegative function f e C1. Furthermore, 1[1

< f (l - f(x) p(dx) - -L-1 IP ) 11 f1I

.

LIMIT THEOREMS FOR RANDOM PROCESSES

444

Since 1(1) = p(dx) = p(X), we have

- I[f] < -

(4)

f(x)p(dx)

Comparison of (4) with (3) yields (2) for nonnegative functions and

hence for all f(x) e C. Thus for arbitrary f(x) e C1, lcim f(x)li,k(dx) = f(x),u(dx) This completes the proof of the lemma. We can now prove the theorem. Proof of the Sufficiency. Let us choose a sequence {Sm} that approaches 0 as m -. 00 and a sequence {K'm)} of compact sets such that Kim) c K(m+') and sup, p,(X\K'm)) < sm. We define 1jnm)(A) = lJ,(A n K'm)) .

Let us choose a sequence {nk')} such that the sequence of the

measures p") converges weakly to some measure pt'). Let us nk define sequences {nk} such that {nkj)} is a subsequence of {nkj-')l and the sequence converges weakly to some measure p(j). Since nk

p(j) and p(j-1) coincide on K(j-1), it follows that var I P(j) - p(i+P) i <

2sj. Hence the sequence {p(')} converges in variation to some measure p. Let us show that p,kk) converges weakly to p. For every bounded continuous function f, lim f(X),U,kIk) (dx)

k--

< lim k--

L

- f(x),u(dx) J(X)p,kk) (dx) - L m)f(x)p(dx)

+ 11 f 11 (lim p,kk) (X - K(m)) + p (X - Ktm))) !E 211f I Em

This proves the sufficiency of the hypotheses of the theorem. Proof of the Necessity. If {p,} is weakly compact, then 1 I-C,(dx) is a compact set of real numbers and hence the sequence {p,(X)} is bounded.

Let us further suppose that the sequence

{,u,}

is weakly

compact but that condition b is not satisfied. Note that condition b is equivalent to the condition b': For all s > 0 and 8 > 0, there

exists a compact set K such that sup, p,(X\K8) < S, where Ka denotes the set of points x whose distance from K does not exceed 8. That b implies b' is obvious. Conversely, let K") denote a compact set such that sup, p,(X\K,'11)) < s/21. Then the set l,K,(Il)

1.

WEAK CONVERGENCE OF DISTRIBUTIONS IN A METRIC SPACE

445

If b' were not valid then there would exist /an s > 0 and a 3 > 0 such that is a compact set for which condition b is satisfied. sup,, ,u.(X \KS) > s for every compact set K.

Let KM denote a compact set such that p1(X\K(°)) < s. (The existence of such a compact set follows from Theorem 6, Section 7, Chapter II.) Since sup p,,(X\K8°') > s, there exists a number n, such that p,,,1(X\Ka°1) > s; hence, there exists a compact set K"'

>a and K'1' C X\Xb°1 (again on the basis of such that Theorem 6, Section 7, Chapter II). Since sup,, j_t.(X\KS°'\K81') > s, there exist a number n2 and a compact set K(2) C X\K°)\K1) such that ft,2(K(2)) > s. Continuing this process, we choose a sequence of numbers n; and compact sets KW such that u,,,,(K(j) > s and K(j) C X

j-1 Ka") =X \a

j-1UK(i) 8

i=O

i=O

\[

I

Let xi(x) denote a continuous nonnegative function bounded by unity, vanishing on X\Kli2 and equal to 1 on Since the distance between any two compact sets of the sequence Kci) is at least

3, the functions xi(x), for distinct values of i, cannot be nonzero simultaneously. Let us choose from the sequence { ft,,,j} a weakly convergent subsequence { ftk}. Suppose that this subsequence converges

to p. Since the measure It is finite and Ei Xi(x) is bounded, we have xi(x)I-t(dx) _ M1=1

2=1

xi(x)ft(dx) <

and hence

lim j Xj(x)l-t(dx) = 0 P- - 2P On the other hand,

i-P

xi(x)Ik(dx) > p.,,(K`P'') > s

(k = np,)

as soon as k > p; hence for all p,

xi(x) j_t(dx) i=P

= lim k--

xi(x) p'k(dx) > s

.

J

This contradiction proves the necessity of condition b. This completes the proof of the theorem. The completeness of the space X was used only in proving the necessity of the conditions of the theorem. The conditions of the theorem are sufficient for weak compactness of a sequence of measures in an arbitrary metric space. These conditions are also REMARK 1.

LIMIT THEOREMS FOR RANDOM PROCESSES

446

necessary if the space X can be represented as a Borel subset of

some complete separable metric space. With the aid of Theorem 1 we can establish necessary and sufficient conditions for weak convergence of a sequence of measures in the case of a complete space X. Furthermore, these conditions are sufficient for every metric space. Theorem 2. For a sequence of measures pn to converge weakly to some measure ,u, it is necessary and sufficient that the sequence {ttn} be weakly compact and that p ,(A) - p(A) for all A belonging to some algebra 0° such that a(0°) = 0. Proof of the Necessity. Obviously, every convergent sequence is weakly compact. Let us take an arbitrary set A e 3. Let AM

denote its interior (that is, the set of all interior points of A), and let [A] denote its closure. If {pn} converges weakly to p, then by choosing a continuous function f(x) such that f(x) - 1 for x e [A] and p([A]) > f(x)p(dx) - s, we obtain J

p([A]) > f(x)1(dx) - s = lim f(x)1n(dx) - s > lim p. (A) - s . This means that lim pn(A) C p([A])

lim p. (X\,4) < p([X\A]), - lim p,(A) < - (A) . Therefore

)"(All)) < lim p(A) < lim p(A) < ft([A])

.

Thus for all sets A such that p([A]\A(0') = u(A') = 0, where A' is the boundary of the set A, lim j_cn(A) = ,u(A) n-_

Let 0° denote the collection of sets A such that p(A') = 0. Obviously 0, is an algebra of sets. Let us show that u(8°) _ 0. Note that of all the spheres S. with center at a given point x, for only countably many of them do we have ti(S,) > 0. Consequently o(0°) contains all spheres and hence all Borel sets since the algebra of Borel sets is a minimal a-algebra containing all spheres. This completes the proof of the necessity. Proof of the Sufficiency. Let us choose an arbitrary weakly convergent subsequence {p, } of the sequence {p.}. Let f1 denote the limit of this subsequence and let us show that p and P coincide. Suppose that A e 0°. Then as shown in the proof of the necessity,

1.

WEAK CONVERGENCE OF DISTRIBUTIONS IN A METRIC SPACE

u(A(O)) < lim u,tk(A) < lim ",,,k(A) < u([A])

But by hypothesis, lim

a(A).

447 .

Therefore, for all sets in

Q3o,

u(A) < u([A]) .

u(A`°')

(5)

Let {A%} denote an arbitrary monotonic sequence of sets that satisfy inequality (5).

Then since

(U An)c°> = U A"', %

(n An)c°

n An°

[U A.] C U [A.], [n A.] An] = n [A.] [An] , we can see that (5) is also satisfied for the limit of the sequence of sets A,,. Thus the collection of sets 0, that satisfy inequality (5) is a monotone class containing the algebra 0>. This means that this set contains 6( 0) = 0 (cf. Chapter II, Section 1, Theorem 3). Hence inequality (5) is valid for all sets A e 0. Let 02 denote the collection of sets A such that p(A') = 0. Then u(A) = P(A) for all A e ^32 since fl(AM) = ll([A]) = p(A). Obviously, the relation p(A) = AC(A) is also satisfied on the minimal a-algebra c(32) containing 02. As shown in the proof of the necessity of the hypotheses of the theorem, c(82) = 0. Thus the measures u and ,u coincide. We have shown that the sequence is a weakly compact sequence with a unique limit point that is the measure p. From this it follows that the sequence converges weakly to p. This completes the proof of the theorem. Corollary 1. If {p.} - p, then u,, (A) p(A) for every set A such that u(A(1)) = p([A]). The sets A such that u(A(°) = p([A]) are called sets of continuity of the measure p. This means that u, (A) --. p(A) for all sets of

continuity of the measure u if u, -,u. Corollary 2. Let us assume that the measures un correspond to processes e,(t) and that the measure u corresponds to the process e(t). Then the relation Zc ° p as n -f oc implies that the sequence of distributions of { f(on(t))} converges to the distribution of f(e(t)) for all 0-measurable functionals f that are almost everywhere continuous with respect to the measure p. Proof. Let A. denote the set of points of discontinuity of f. Then u(Ao) = 0. Let G. denote the set of those x such that {f(x) < a} and let Ga denote the boundary of the set G.: Ga = [{x; f(x) < a}] n [{x; f(x) >_ a}] . The intersection of the sets G.' and G,'1 for a < a, is contained

LIMIT THEOREMS FOR RANDOM PROCESSES

448

in the intersection of the sets [{x; f(x) < a}] f1 [{x; f(x)>_a1}].

There-

fore the inclusion relation x c G. n Gn1 implies that

lim inf f(y) < a, lim sup f(y) > a1 ; Y-

y_x

that is, G., nGn1EA..

Consequently ji(G., n G,,) = 0.

Hence for an arbitrary sequence

{ak},

p (U Gk) = E f"(G.'k)

From this it follows that the set of numbers a such that ji(Gn) # 0 is no more than countable. Therefore, for all a except possibly countably many values, the set G. is a set of continuity of the measure a, so that ,un(G..) --. ,u(G.) or a} - P{ f(e(t)) < a} as n oc . This completes the proof. REMARK 2. As a rule, in considering random processes we assume that the u-algebra of events of the basic probability space coincides with the minimal u-algebra containing all events of the form {u; fi(t) E C}, where C is a cylindrical set. This means that P{

weak convergence of the sequences of finite-dimensional distributions

of random processes ,n(t) to finite-dimensional distributions of the process e(t) implies convergence of the sequence of measures P. to the measure fa on the algebra 0. of all cylindrical sets of continuity

of the measure a, so that in this case, it

suffices

to verify the

conditions ensuring weak compactness of the measures p. 2.

LIMIT THEOREMS FOR CONTINUOUS PROCESSES

In this section we shall assume that the processes en(t) and e(t) are continuous on the interval [a, b]. Their sample functions belong, with probability 1, to the complete separable metric space C[a, b] of all functions x(t) on [a, b] with respect to the metric p(x, y) = supaStsb I x(t) - y(t) I

We note that in the space C[a, b], the minimal a-algebra of sets U containing all cylindrical sets contains all Borel sets. To see this, if suffices to note that every sphere belongs to U, since {x; sup I x(t) - a(t) I < r} = h {x; I x(tk) - a(tk) I < r}

,

2.

LIMIT THEOREMS FOR CONTINUOUS PROCESSES

449

where a(t) is an arbitrary continuous function and {tk} is an arbitrary sequence everywhere dense on [a, b].

Let H denote a constant and let cob denote a function defined for 3 > 0 that approaches 0 from above as a approaches 0 from above. Let K(H, (o,) denote the set of functions x(t) such that sup I x(t) j <_ H, sup I x(t') - x(t") < w, for all 3 > 0 a5t56

It' - t, , I<8

From a theorem of Arzela (cf. Kolmogorov and Fomin, Heider and Simpson) every compact set in C[a, b] is a closed subset of some set K(H, (o,) and the latter is also compact. We may now use Theorems 1 and 2 of the preceding section. Theorem 1. Suppose that the finite-dimensional distributions of the processes en(t) converge to the finite-dimensional distributions of a process c(t). Then for the sequence of distributions of f(on(t)) to

converge to the distribution of f(e(t)) for all functionals f that are continuous on C[a, b], it is necessary and sufficient that lim sup p{ sup h-.0 It'-t15h

I

en(t') - en(t") > e} = 0

(1 )

for every s > 0.

Proof of the Necessity. Suppose that the conclusion of the theorem is satisfied. Then the sequence of measures ii,,, corresponding to the processes an(t) is weakly compact, so that condition (b) of Theorem 1, Section 1 is satisfied. Therefore, for every ? > 0, there exists a compact set K(H, (o,) such that sup pn(C[a, b]\K(H, (o,)) = sup e K(H, w,)} <)7 . n

n

Then P{

sup

It, - t"I sh

mitt) -

Mt")

I > wh} < P{Sen(t) V K(H, (o8)} < .)

.

If h is sufficiently small, then (oh < s and iim sup P{I Mt') h-+0

n

-

en(t")

I > s}

In view of the arbitrariness of 77 > 0, we obtain (1). Proof of the Sufficiency. The convergence of the sequence of finite-dimensional distributions implies convergence of the sequence of measures t^n to the measure p that corresponds to the process

e(t) on the algebra 0, of all cylindrical sets of continuity of the measure p. But a(580) coincides with the minimal u-algebra containing all cylindrical sets. Hence (as shown earlier in this section) it contains all Borel subsets of the space C[a, b]. Therefore, in view of Theorem 2 of Section 1 it will be sufficient for us to

LIMIT THEOREMS FOR RANDOM PROCESSES

450

prove the weak compactness of the sequence of the measures p,. Let us show that for every rJ > 0 there exists a compact set K(H, o),) such that sup P{en(t) e K(H, (v,)} < 7

Since the sequence of the distributions $n(a) converges to the distribution e(a), there exists an H such that for all n, P{I en(a) I > H} < 2 .

Let us take a sequence {E,} that converges to 0 from above. every z, there exists an hr such that hr < hr_1 and sup p{I

sup

t.

h,

en(t') - en.(t") I > e,} <

I

For

2r+1

Let w, denote a nonnegative nonincreasing function such that hr]. Obviously, ce, 10 as S 10. Furthermore, a), = Er for a e [hr+1.

p{an(t) 0 K(H, (o8)}

P{I $,(a) I > H} + r=1 Z P{. sup

It'-t"I Shr

I $n(t') - En(t") I > E,}

y/7

,,}1 = "/'p

2+

This completes the proof of the theorem. REMARK 1. Instead of condition (1), we may require that lim lim p{ sup En(t') - en(t") I > E} = 0, (2) t'-t"I
h-.0 n-m

I

which is often more convenient to verify. We can use this requirement because it follows from (2) that for every > 0 there

exists a 3 > 0 and an N such that for n > N and h < s, p{

sup t'-t"I
en(t")

I > E} <-1

.

(3)

The continuity of the processes En(t) implies their uniform continuity,

so that for every n, lim p{ h-.0

sup

It'-t" 15h

I

en(t') -

I > E} = 0 .

Therefore, there exists a a such that for h < a relation (3) is satisfied for all n. The next theorem may be more convenient for applications.

Theorem 2. Suppose that the finite-dimensional distributions of the processes en(t) converge to the finite-dimensional distributions of the

process E(t) and that there exist positive numbers a, Q, and H such

2.

LIMIT THEOREMS FOR CONTINUOUS PROCESSES

451

that for all t1, t2 and for all n,

(4)

M I E%(tl) - En(t2) 11 < H I t1 - t2 I1+1

Then for all functionals f that are continuous on C[a, b], the sequence of distributions of f(En(t)) converges to the distribution of f(e(t)). Proof. Obviously, because of the continuity of the processes an(t) we have with probability 1, sup

I

Itl-t21 5h

en(t1) - en(t2) I =

sup p

Itl-t215h

I

en(t1) - en(t2)

t1, t2 E N

where N is the set of all points of the form k/2m belonging to [a, b]. If 2h < 2-k, then sup I en(t') - En(t") I < 2 sup V ,t" C: N

sup (7/2k)<(l/2m)<{(7+1)/2k)

7

<2 since (1/2m) m. Hence

l

-n

2m

- (j12 k) = E1=1 (1/2mr), for k < m1 < m2 < ... < ms <

-±

n(f'Zk) = i=1'[e-(j2k + r=1-') 2-, - en(2k'

2m

\ 2k/

1

n(l2 /

sup m=k+1

n1 2m 1 - n

+

1

r=t 2mr

Now we note that

n(l

sup P{ ¢5(i/2m)<J(it t/2m)J5b

n(Z

< m2a((b

P{

sup t1-k215h

1

1 - n(2m)

2m

a)2m H

>I

- en(2i' )

n(I 2m 1I

m2a E M i

Consequently, if

n(2i-)

2"

m2a

1

L2m3

Em=k.1 (1/m2) < (s/2) where k = 109201h) + 1, then I

en(t1) - e (t2)

m=k+1


P{sup

I > E1l

Sn(i +

E

1)

2-

ti

1

e

(2m)

>

m2

m2a

[1og2(1/h)]+2 2"

-->0

uniformly with respect to n as h , 0. This completes the proof of the theorem.

LIMIT THEOREMS FOR RANDOM PROCESSES

452

It follows from the proof of the theorem that

REMARK 2.

instead of condition (4) we can require that MI

Sn(tl) - e .(t2) 10 < Hp(tl - t2)

where the function q(t) is such that for some ,8 > 0

<0. Furthermore, it is sufficient for us to require the convergence of the sequences of finite-dimensional distributions of the processes en(t) to the finite-dimensional distributions of the processes e(t) for

t in some set that is dense on [a, b]. 3.

CONVERGENCE OF SEQUENCES OF SUMS OF INDEPENDENT RANDOM VARIABLES TO PROCESSES OF BROWNIAN MOTION

Consider a double sequence of random variables $n1, a

2,

, $nk,y

(independent in each sequence) that satisfy the conditions 1.

Meni = 0,

2.

Deni = bni, where E%:1 b,, = 1.

Let us construct a random function en(t), for t e [0, 1]: define Snk

++k

We

e, ' tnk - Eik=1 bni,

,n(t) = Snk +

t - tnk

tn,k+l - tnk

7

[S%,k41 - Snk]

The function en(t) is a random broken line connecting points in the to-plane with coordinates (tnk, Snk) for k = 0, 1, . . , kn. for t e [tnk, tn,k+11+ we also define Sn0 = 0 and tn0 = 0.

In this section we shall study the conditions under which

sequences of finite-dimensional 'istributions of the processes en(t)

and the sequences of the distributions of functionals of these processes converge to the finite-dimensional distributions and the distributions of the corresponding functionals of a process of Brownian motion w(t). Let Fni(x) denote the distribution function of a random variable $ni Then, if for every s > 0, Jim

kn

E \ u2dFni(u) = 0 , fl-,oo i=1 luI>e

(1 )

the random variable ni is said to satisfy Lindeberg's condition. Theorem 1. Suppose that random variables ni satisfy conditions 1 and 2 and Lindeberg's condition. Then, the finite-dimensional dis-

3.

453

CONVERGENCE OF SEQUENCES

tributions of the processes an(t) converge to the finite-dimensional distributions of the process w(t) and the sequence of distributions of f(on(t))

converges to the distribution of f(w(t)) for every functional f that is continuous on C[0, 1]. Proof. Let us first show that the finite-dimensional distributions of the processes en(t) converge to the finite-dimensional distributions of w(t). Let en(t) denote the random process defined by

sen(t) tni
Then for every a > 0, kn

P{I an(t) - n(t) I> a} < P[sup I ni I > a1l < E

_

P!ll eni

I > a}l

a==l

a

f

ul>a

i=1

dFni(u)

1a2 i=1

tt>a

u2dFni(u) , 0.

Therefore, to prove the first assertion of the theorem, it will be sufficient to show that the finite-dimensional distributions of the process an(t) converge to the finite-dimensional distributions of the process w(t). But since w(t) and en(t) are processes with independent increments and w(0) = En(0) = 0, it will be sufficient to prove that the distributions of n(t") - n(t') converge to the distribution also note that w(t") - w(t') for all 0 < t' < t" < 1. MJ L V_
1

ni

that is n(t") - n(t') is the sum of independent random variables for which Lindeberg's conditions (cf. Theorem 5, Section 3, Chapter Consequently n(t") - en(t') is asymptotically 1) are satisfied. normal with mathematical expectation 0 and variance lim

E,

tl5tni
t" - t'

(since I

E

t'5tn t'I

DEn.i kn

< max b,, < s2 + max i

i

f J nI<E

u2dFni(u) < E2 +

u2dFni(u) 2=1

ul>E

and the expression on the right can be made arbitrarily small for sufficiently large n). This proves the convergence of the finite-dimensional distributions.

To prove that the distributions of f(on(t)) converge to the distributions of f(w(t)) for all functionals f that are continuous on C[0, 1], let us show that for arbitrary s > 0,

LIMIT THEOREMS FOR RANDOM PROCESSES

454

lim lim p{ '-,o

h-.0

> e} = 0 ,

sup I n(t') It'-t'IIsh

and let us use Remark 1 of Section 2. < 2 sup

sup

k

< 4 sup k

(2)

Since

sup

en(t) -

sup

en(t) - en(kh)

kh
(kh)

we have sup

p{

en(t') - en(t") I > 6}

I

< k h<1 Plkh< suktljh an(t) - n (kh) I > 41 ppI'

{1l

To find a bound for the probability plkh<

sup t1)h

I

en(t) - n(kh) I > 4

let us use Theorem 2, Section 4, Chapter III. We note that

an(t) - n(kh) < 2

sup kh
sup

f

7-jnk

jnk
where jnk is the greatest of the indices j such that t,j does not exceed kh.

Since j n,k+1

P{

> 8}

E en.j j=jnk

64h

it follows 1that foor sufficiently small h, him P kh<SUktp 1)h

r

1

<

1

an(t) - n(kh) > 4 1

64h

Jim Pj l n(tnjn,ktl + 1) - n(tnjm.k) I > 18} `

62

It follows from the convergence (proved above) of the sequences of finite-dimensional distributions of ,n(t) to the finite-dimensional

distributions of w(t) that lim Pl n(tnjn,ktl + 1) - n(tnj nk) l n-+oo > 1

VT7 h

Consequently

Li>wB)

exp(- u2ldu \ 2h

8}

3.

CONVERGENCE OF SEQUENCES

n--

> c^}

I n(t')

sup

lim P{

455

It

Y

f

1

1

exp

64h 1/2c

kh<1

u2 Idu

\

2/

E2

-

1

1

\

(

1

1/2r1-64hh Jlul>cEash

exp (- u2 Idu

2/

\

s2

Since 1

exp (- u2) du <

`

h J 1..l>W'1

)

\

2

c2 J

nl>W,/hi

u2 exp(- u2)du 2

0

it then follows that equation (2) holds. This completes the proof of the theorem. REMARK 1. It follows from Corollary 2 to Theorem 2, Section 1 that if the conditions of Theorem 1 are satisfied, the distributions of f(on(t)) converge to the distribution of f(w(t)) for every functional

f defined on C[0, 1] and continuous (with respect to the metric of C[0, 1]) almost everywhere (with respect to the measure pw corresponding to a process of Brownian motion on [0, 1]). Let E1, 52, , sn+ . . denote a sequence of independent identically distributed random variables such that ME = 0 and pet = 1. From Theorem 1 we immediately have: Theorem 2. Let en(t) denote a random broken line with vertices {(k/n), (1/1/ n )Sk}, where Sk = e1 +

Then for every functional f defined and continuous on C [0, 1 ] almost everywhere with respect to the measure I.tw, the distributions of f($n(t)) converge to the distribu+ S k.

tion of f(e(t)). Proof. It will be sufficient to show that Lindeberg's condition is satisfied for the variables ink = (1/1/ n )ek. If we let F(x) denote the distribution function of the variable 5k, then the distribution function of the variables Enk is F(1/ _n x) and x2dFnk(x) _

k.=1

xl>[

=

nx2dF(1/ _n x) = J

Corollary. supOSk<-n

x2dF(l/ n x) _

If the conditions of Theorem 2 are satisfied and r, =

Sk, then n

lim p{72n < al/ n

n--

u2dF(u) , 0 .

J Inl>e./n

Ixl>:

}

1/2ar

0

ex p (

\

? du 2

.

LIMIT THEOREMS FOR RANDOM PROCESSES

456

Proof. We have Y),/1/ n = supost<1 en(t), and since supOSt51 x(t)

is a continuous functional on C[0, 1], it follows that lim p( sup gn(t) < a} = P{ sup w(t) < a} n-.oo

05t<1

,

0;5t!5:1

so that it only remains for us to apply Corollary 1 of Section 5, Chapter VI. Analogously, using Corollary 3, Section 5, Chapter VI, we see that

lime{maxISkI
1
a.exp(-(u -2ka)2)du

1 = kE (-1)k 1/2r' -

\

2

JJ

Theorem 3. Suppose that a function p(x) is defined for x E (- oo , oo) is Riemann-integrable over every finite interval and that the random variables ksatisfy \the conditions of Theorem 2. Then cp(/

lim Pj 1 I

'

V-n

Sk J < a} = P{ p(w(t))dt < a}

for all a such that P}J1q(w(t))dt = a} = 0. 0

Proof.

Let us show that the functional f [x(t)] = f cp(x(t))dt o

is continuous law almost-everywhere with respect to the metric of C[0, 1]. Suppose that {xn(t)} converges uniformly to x(t) on [0, 1]. Then cp(x, (t)) --- q(x(t)) for all values of t such that x(t) e A,, where

Alp is the set of points of discontinuity of the function p. Let X, (x) denote the characteristic function of the set A4,. Then the functional f(x(t)) is continuous at the point x(t) c C[0, 1] if for almost all t, we have x(t) V AS , that is, if JOX,(x(s))ds = 0

,

so that (in this case) q(xn(t)) --> p(x(t)) for almost all t and the functions q (x, (t)) are bounded by a single constant since sup,,, I xn(t) I

is finite and q(x) is bounded on every interval. We should note only that

M x,(w(i))dt =

Mxw(w(t))dt = '0 dt

27zt

exp

\

l

2t/du

0,

since A, has Lebesgue measure 0 (by virtue of the Riemannintegrability of the function q). The quantity negative.

Therefore

is non-

3.

CONVERGENCE OF SEQUENCES

457

0} = 0 . 0

If we let A c CEO, 1] denote the set of points of discontinuity of the functional f, then A c {x(t); X (x(s))ds > 0}

.

Hence

f w(A) < P{J xw(w(t))dt 7-, 0} = 0 . If en(t) is the process introduced in Theorem 2, then on the basis of that theorem, limp{J

a} = P{Jo cp(w(t))dt < a}

q(en(t))dt <

if =0.

p{Jo g (w(t))dt = a}

Let q (x) and q), -(x) denote two continuous functions satisfying the inequalities q4 (x) > p(x) > g

(x)

and

- pE [,pF (x)

(x)]dx < e .

continuous function p(x),

For

(en(t))dZ -

l

n k1

k

n Sk)

n(t)) - 1 n

.)(c1-1)ln

kjn

n

k=1

(k-1)/n

sup x-y s 1va

7'

Jdt

n

( n(t)) - 7' l n (-!))

dt

P(x) - ip(Y)

xlstn

where

/n=sup

n 1)- e (n)

SUP k

ek

1/n

I Sk Sn=suk pV n

Therefore

p{ 0

1n k=1

1

n

-

c p{ran < 8} + p{Sn > C}

LIMIT THEOREMS FOR RANDOM PROCESSES

458

where 8 and c are chosen in such a way that Iip-(x) - q5(y) I < s whenever I x - y I < 8 and I x I < C. Since 72, 0 in probability as (this follows from Lindeberg's condition and was established in the proof of Theorem 1), it follows that p{rJ, > 8} ---' 0 as n -> -

n

co

for every 8 > 0. The quantity p{C, > c} can be made arbitrarily small by choosing sufficiently large c for all n (this follows from Corollary 1). Therefore n k-=1

l

P\l/ n Sk) I

-' 0

in probability, so that lim P{

n--

n k=1

(Lsk) < a} = P{J1 rp(w(t))dt < a} n

if P{ f o p(w(t))dt = a} = 0.

Since

Lt

k=1

(p(

n

5k) < a} >

n k=1

> p{1

(5k) < a} /

)J

n

n k=t q

(1/ n Sk) <

a}

oo we find that for every h > 0,

by taking the limit as n

P11 " q(1/ P{\1cE (w(t))dt < a + h} > Tim 0 n-oo n k=1

n

S,) < a}

> n-lim P{L (sk) < a} n k=1 Vn

p (w(t))dt < a -

h}

.

But M

cp(w(t))dt

1PE (w(t))dt J0

J

< <

qE (w(t)) - q (w(t)) I dt 1

(1

dt

s

dt

= 1/270 77

I

`- p (x) - p -,(x)

- V27r J01/ T J

I exp

(- x2)dx 2t

2s

7270 .

This means that the variable distribution of 1 'PE (w(t))dt converges to the distribution of q(w(t))dt as a ----> 0. An analogous assertion

4.

CONVERGENCE OF A SEQUENCE OF MARKOV CHAINS

is valid for pE .

459

Taking the limit as a , 0, we see that for all

h > 0, `p(w(t))dt < a + h} > lim P{

n kE

P(

n

S,,)

< a}

lim p{- q.( -Sk/ < a n

a - h}

P{Jo cp(w(t))dt <

.

Taking the limit as h --+ 0 and remembering that P{ Ju

p(w(t))dt < z}

is continuous at z = a if ,p(w(t))dt = a} = 0, 0

we obtain proof of the theorem. Corollary. Let vn denote the number of positive sums in the , S.. Then for 0 < a < 1, sequence S1, S, lim p{v, < na} = 2 arc sin V -a n m

.

7r

Proof. v.. _ E g+{(1/1/ n )Sk}, where g+(x) = 1 for x > 0 and g+(x) = 0 for x < 0. Therefore on the basis of Theorem 3,

lim P{vn < na} = P{J g+(w(t))dt < a}

for all a such that P{J g+(w(t))dt = a} = 0. Now we use Theorem 4, Section 5, Chapter VIII, and the proof is complete. 4. CONVERGENCE OF A SEQUENCE OF MARKOV CHAINS TO A DIFFUSION PROCESS

Let us consider a double sequence of random

variables

eno+ $nl, , Enk,, that form along in each sequence a Markov chain. We let Pnk(x, A) denote the transition probabilities

P k(Enk, A) = P{En,k+1 E A I nk}

(mod p)

.

Suppose also that 0 = tno < tnl < . . . < tnk,, = 1 is a sequence of partitions of the interval [0, 1]. Let us construct a random broken line E,(t) with vertices at the points (tnk, Enk) In this section we shall study conditions under which the sequences of

LIMIT THEOREMS FOR RANDOM PROCESSES

460

finite-dimensional distributions of en(t) and the sequences of distributions of functionals of e, (t) converge to the corresponding distribu-

tions of the Markov process e(t) representing the solution of a stochastic equation of the type studied in Chapter VIII. We define Otnk = t%,k+1 -tnk ,

an(tnk, x) =

1

0tnk

(y - x)pnk(x, dy)

bn(tnk, x) = on(tnk, x) =

1

Atnk

(y - x)2Pnk(x, dy) - Otnkan(tnk, X)

Theorem 1. Let e(t) denote a solution of the stochastic equation a(s, ee(s))ds +

alt) = Sao +

6(s, $(s))dw(s) ,

where eo is independent of w(t) and where a(s, x) and o'(s, x) are functions that are continuous over the set of variables and satisfy a Lipschitz condition with respect to x:

I a(s, x) - a(s, y) I + I a(s, x) - 6(x, y) < KI x - y For the finite-dimensional distributions of the processes en(t) to converge to finite-dimensional distributions of the process e(t), it is sufficient that the following conditions be satisfied: a.

limn . maxk Atnk =

b.

limn-, Xknl M(I an.(tnk, enk) - a(tnk, nk) I2

c.

for some a > 0

0,

+ 16n(tnk, enk) - 6(tnk, nk)

0,

kn

Jim

E M I n,k+1 - enk n-.eo k=1

Z+d = 0

and kn

Ink) -0

P- Iim SUP E Mli %-,00 k=1 i=k

d. the functions l/on(tnk, x) and an(tnk, x)/a, (tnk, x) are uniformly bounded with respect to n; e. the limiting distribution of the random variable eno coincides with the distribution of the random variable

Proof.

we set g

g ink = Sn,k+1 - nk - anltnk, Snk)Atnkl l6n(tnk, Snk))

1

gg

l

)

Then n,k+l = enk + an(tnk, enk)Atnk + on(tnk, en.k)wn.k

.

Let ink denote the minimal a-algebra with respect to which

4.

CONVERGENCE OF A SEQUENCE OF MARKOV CHAINS

461

, nk are measurable. The quantity mnk is measurable with respect to the a-algebra and the variables S no, n1,

(2)

ink) = 01 M(0) k I Unk) = Atnk Consider the variables /nk defined by the relations M(Wnk I

y'p7

7/n0 = fn01 yy /n,k+1 = /nk + a(tnk, yy )nk)&tnk + Q(tnk, /nk)0)nk

Obviously, the /nk are also We have measurable with respect to the a-algebra

Let us find a bound for M(17nk - enk)2

'/n,k+l - en,k+l = /nk - enk +

[a(tnkk) '7 k)

- a(tnk, $.01

X Otnk + [o(tnk, y)ak) - a(tnk, Cnk)]Unk + Enk r

g

Enk = [a(tnk, Enk) - an(tn.k, Cnk)]Atnk + [Q(tnk, Y)J - 6n(tnk, En.k)]Wnk

Therefore, by using equations (2) and the Lipschitz conditions for a and a and also the inequality 2ab a2 + b2 we obtain y

2

M I /nk

+ 2M('/.k - Cnk) x (a(tnk, /nk) - a(tnk, + 2M I

'/nk y-7

$EEnk))Atnk

Cnk I X I a(tnk, e, ) - an(tnkk, enk) I Ltnkg

+ M

I J k - Cnk Al + 2KOtnk + I tnk)Otnk

+ M I a(tnk, Enk) - an(tnk, ink) I2Atnk

+

2

2M(a(tnkl

a(tnk+ nk) Atnk

f

+ M(6(tnk, /nk) - a(tnk, enk))2 I U nk) + 2M b»k G Myy"/nk - Enk 12(1 + L1tnk) + ank , I

where L = 2K + 1 + 4K2 and ank = M[a(tnk, Enk) - an(tnk, Enk)]' (Ztnk + 2Otnk) 2

+ M[Q(tnk, nk) - 6n(tnk, nk) Atnk Since M /no - ?;no I2 = 0, we have M I EEnl - Y)nl I'

an,,

an0(l + LAtnj + an, S (1 + LO40) + and M Sn3 - 77n3 I2 <[an0 + ant] (1 + LAtnj (1 + LOtn1) + an2 5 [an0 + an, + anz] (1 + LAtnj (1 + LAtnj , M sn2

77n2 I2

`

k-1

M I Enk - Jnk 2 yy77

i=0

k-2

kn-i

i=0

i=o

ani II (1 + LAtni) < e`

ani

It follows from condition b that "Mn-- Ei n0 1 ani = 0. Consequently, the finite-dimensional distributions of the process en(t) converge to the finite-dimensional distributions of the process E(t) if the finite-dimensional distributions of the process rJn(t), where 72n(t) is a random broken line with vertices at the points (Ink, )7nk)

LIMIT THEOREMS FOR RANDOM PROCESSES

462

converge to the finite-dimensional distributions of e(t)-

To continue the proof, we use: Lemma 1. Suppose that w,(t) = tnk
It will be sufficient here to show that for all Xnk such

Proof.

that SUpn,k I X,k I < 00 k1-1

llm rM exp CZ E X.kconk)

\

n-, L

k=0

-

k,,i-1

1\1 ((

M exp CZ E Xnk[w(tn,k+l) - w(tnk)I

1 =0

We note that the function {e", - 1 - iXx + (,2x2 /2)}/I X 11 I x for 0 < 3 < 1 bounded on the entire real line. Therefore < O(X2) I x I2+1

2+a is

.

X2x2)

C

Hence ei2.kmnk MIT k=0

M

11 ei2nkmnk)M[eiAnrmnr I k-1 LL

C) n r I

M( Tf e¢,lnk ink ) MM + 1 +

2

k=0

0)., + O(I 0).r I2+2) t)nr]

M( 110 ei2nkmnk)(1 - ^nr Qtnr) + O(M I ()nk 2+6) 71

'nk Atnk) + 01 E M I wnk 1 1 + 1

1

1

2

kk=0 \

\k-0

It follows from formula (1) for wnk and the boundedness of I /a. and an/o'n that 2+S + I Atnk M I °nk I2+1 < L(M en,k+1 - enk

2+2)

Therefore kn-1

lim E Furthermore k

1

C1 k=0

-

nk Qtnk) 2

-

kl k=0

exp

MI(0)nk2+a=0.

(_

nk Qtnkl 2 J

< O(E Ytnk) = 0(max Atnk) --- 0 . k

Consequently

4.

CONVERGENCE OF A SEQUENCE OF MARKOV CHAINS //

kn-1

lim {M exp ` i

\

m.-.m

k-0

463

k,,b-1 2

Xnk0)nk)

- exp ( - k nk Otnk )} _ 0 \ k-0 2 J

Since

- 2'n'knk At 2

MeilnkCw(tn.k+1)-w(tnk)] = exp

the lemma is proved. REMARK 1. It follows from Lemma 1 that for every continuous function a(t, x) that is uniformly continuous with respect to x in every finite interval, we have M{exp (iXt Eat, a(tnk, x)wnk)

M{exp

nk

t'

a(t, x)dw(t) } J

if In is such that tntre < t'. This is true because just as in the proof of Lemma 1, we can show that M{exp (iX E

t'Stnk
- exp

a(tnk, x)Wnk 19-ln}

J-Y

2

E a(tnk, X)BQtnk -i 0

t'<_tnk
uniformly with respect to x in every finite interval (by virtue of the boundedness of a(t, x) on every finite x-interval). We should note finally that t'St nk
a(tnk, X)2Itnk .-->

a(t, x)2dt t

uniformly with respect to x in every finite interval, since a(t, x) is uniformly continuous with respect to all the variables in every finite x-interval. , xm) are bounded Lemma 2. I f the measurable functions gn(x1, by a single constant, if they converge uniformly on every compact set to a continuous function 80(x1, x2, , x.), and if the distribution functions FZ(x1,x2, then

,xm) converge weakly to a function FO(x1,x2,

lim T n(x1, ... , X,)dFn(Xi, .. , xm)

= J,p0(x1, ... , xm)dF0(x1, ... , Proof.

Since

lim p0(x1, .. , X,)dFn(Xi, ... , n--

= J p0(x1, ... , xm)dF0(x1, . . ., Xm) ,

,xm

LIMIT THEOREMS FOR RANDOM PROCESSES

464

by virtue of the weak convergence of {Fn(x1,

, x.)} to F0(x1,

, X.)

it follows that to prove the lemma it is sufficient to show that go(x1, ... , x.) I dF, (x1, ... , x,n) = 0

lim I gmx1, ... , x,n) J

But for every K > 0 ,

P°-INIdFn< ElxaI

po -

pndFn+

K.

Elx{I>Ii

Ip° -

pnldFn;

z

a

the first integral approaches 0 as n --+ - since {I q - q 1} converges uniformly to 0 for Y, I xi I < K; the second integral can be made arbitrarily small for all n by choosing K sufficiently large (because of the boundedness of 19)o - T. I and the weak convergence of the sequence of the distributions Fn). This completes the proof of the lemma.

for n = 0, 1, , denote m

, Lemma 3. Let {din'}, sequences of random variables. Let tions such that with probability 1, ((n)(X) =

, xk_1) denote func-

x1,

Me:ae1n)

(n)! k

(n)) k-1 =

(n)

M(e'2ekn)

I (n) 7

,

(n) \ k-7/ ,

k = 2 , ... m

Suppose that for all k the functions qy°)(), x1,

, xk_) are continuous , xk_1)} converges to 4)k)(X, x1, . , xk-) for k = 1, 2, , m uniformly as n --> cc on every compact set. Then the joint distributions of the variables ,cm' converge weakly to the and that {(Dkn'(X1, x1,

joint distribution of the variables Rio), ...,o)

Taking k = 1, we see that the distributions of the

Proof.

variables din' converge to the distribution of the variable e11°). Let us suppose that the joint distributions Fkn)1 of the variables

,k-' converge to the joint distribution Fk°1 of the variables

in',

Then

E$°

k-

M exp {i

k

E xj j

j=1

exp eXp J

I

k-1 l

j=1

xjxj }(Dkn)(xk, xl, ..., xk-1)dF (xl, ..., xk-1) 7

I JJ

k-1

((r

--'

}

{i E X jx4J Ck(Xk, x1, ... , j=1

= M exp

(iX1e1°) +

xk-1)dF l(x1, ... , xk_1) ZXkek ))

on the basis of Lemma 2. Therefore, the joint distributions of the variables in), , ek) also converge to the joint distribution of , k ). We prove the lemma by induction on the variables k.

4.

CONVERGENCE OF A SEQUENCE OF MARKOV CHAINS

465

We now return to the proof of the theorem. Let us choose an arbitrary partition of the interval [0, 1]: 0 = zo < T1 < < TN = Define

1.

)7n (TO = yn0

,

n(Tk-1) + rk-15tnr
y

tn(2k)

+

LJ

rk_15L n,
7io (Tk) = 7)0 (Tk-1) +

+

O(tnr, )7n*(Tkk = 1, ..., N

(Tk-1))ds

Jrk-la(s,

Ik

a(s, 7o (Tk-1))dw(s) ,

rk-1

Obviously,

i

'p

I

k = 1, ..., N ,

pp )p70 (T0), ..., /0 (Tk-1)) ='P$,)(a,, /0 (Tk-1))

where k

chk )(X x) = exp iXx + ix

a(s, x)ds -

rk-1

-2

rk

2

rk

6z(s, x)ds}

is a continuous function of x. Let tntk denote the maximum point tn, such that tnr < zk. It follows from the remark to Lemma 1 that when we substitute x for )7*(Z'k M(eiZ7n(

kJ

n (TO),

. .

.,

)7*(Tk-1))

= M{exp (iX2I*(Tk_1) + i) X M[exp (i?

rk-i tnn,
a(tnr, )7n(Tk-1))Atnr) 77

t

rk-15tnr
6(tnr, )/n(Tk-1))(0,,) I (ilk-1J I)7*(TO),

converges to 4)k(0)(X, x) uniformly with respect to x in every finite interval. Therefore, on the basis of Lemma 3 the joint distributions of the variables 77*(zo), , rJ*(zN) converge to the joint distribution of the variables )7o (zo), , i7o (zN). Since Tk) - 'J0(Tk)

frk

S(Tk-1) - 270 (Tk-1) +

+

J rk-1

[a(s, e(s)) - a(s, ')o (Tk-1))]dS

Tk rk-1

[6(S, E(S)) - 6(S, Y)o (Tk-1))]dW(s) ,

by using the Lipschitz condition and some simple transformations we obtain

LIMIT THEOREMS FOR RANDOM PROCESSES

466

M I c(rk) - 170(Tk) I2 = M I e(Tk) - '70 (Tk-1) I2

+ 2M I e(Tk-) - )70(Tk-)

+

\ rk-1

(a(s, ('S))

- a(s,

k r'k-1 rk-1

a(s, c(s)) - a(s, 720 (Tk-1)) I dS

.1

(Tk-1))ds)2

"!0

+ 1 rk (Q(S, c(S)) - 6(S' "/0 (Z*k-1)))dw('S) C M I (Tk-1) - "!0 (Tk-I) I2 E + 2KM S(zk-1) -

rk

I c(S) - '/0 (Tk-1)

/0 (Zk-1) I

'pp

ds

(1r'

2M

rk

r -1([a(s, c(s)) - a(s, "!0 (Tk-1))]2

+ [o'(S, S(S))

Q(S' '/0 (vk-1))]2)ds /0 (Tk-1) I2(1 + H(zk

<M + H Jrk

rk-1

- zk-1))

I2dS

I c(S)

where H = 2K + 8K2. Consequently, using the bounds that we have already applied, we obtain M(e(Tk) - '70 (Zk))2 < He"M I

k-1

\rk rk-1

I c(S)

- c(Tk-)

I2dS

But on the basis of Lemma 2, Section 3, Chapter VIII,

ml$(S) -

c(s2)I2:! CIs,-S2I

,

so that

M(c(Tk) - 72*(7k))2 = O[max (zk - zk-1)] k

(3)

'

By analogous reasoning we can show that g '/*(Zk)]2 = O(max (zk - Zk-)) lim M[y)n(Tk) k

-

We now note that for any real numbers X,, , m' '/1, , Im,

4 ,7m

and

random variables

M exp (ix1c1 + ... + ixmem) - M exp (ix177, + ... + iAm7 m) I M I exp {iX1(c1 - 'pp/1) +

+ iXm(em - /m)} - 1

M x1(, - '71) + . . . + Xm(em - /m) I2 m

m

k=1

k=1

Zk

(5)

M(ek

Let us choose arbitrary numbers s1, assume that Si = zkm..

Then

, sm

in [0, 1].

We may

4.

CONVERGENCE OF A SEQUENCE OF MARKOV CHAINS

467

lim I M exp {i E Xj)7Jzk;)} - M exp {i E %;$(z,)} I n1w

< lim I M exp{i E X;r7n(zk;)} - M exp {i Y, Xj7)*(zk;)}

I

n"W

+ lim I M exp {i Z X;72*(zk;)} - M exp {i E X;)Io (zk;)} I + IBM exp {i E X;7I0*(zk;)} - M exp {i E x;e(zk>)} I

< 0(V m max (zk A:

by virtue of relations (3) to (5) and the convergence of the joint distributions of the variables *(s,) to the joint distributions of the variables 7(s;). Since maxk (zk - zk_1) is arbitrary, the conclusion of the theorem follows. Theorem 2. Under the hypotheses of Theorem 1, for any continuous functional f on C[0, 1], the distributions of f(on(t)) converge to the distribution of f(e(t)). Since the finite-dimensional distributions of the processes

Proof.

$,(t) converge to the finite-dimensional distributions of the process E(t), by virtue of Remark 1 of section 2 we can reduce the proof of this theorem to proof of the equation lim I P{ sup I En(t') - en(t") I > s} = 0 . (6) h-.0

n-.w

It

Following the reasoning used to prove Theorem 2 of Section 3, we see that P{

sup

n(t")

I en(t') kh<1

Pl

sup

l kk5t<(k+I)h

I > 6}

I n(t) - en(kh) I > 61

.

4

Let sk denote the greatest of the indices r such that tnr < kh. Then P lkh 5su(k+1)h I P

n(t) - w(h) I > 4 sup

.k5I159258k+1+1


sup

ek5A5ak+1+t

I

I nj' - n;z I >

nj - nak >

6} 4

sl g1

To find a bound for this probability, we shall need: Lemma 4. If S1, S29 , e,n constitute a Markov chain and if with probability 1, where a < 1, then

LIMIT THEOREMS FOR RANDOM PROCESSES

468

p{skpICkI>2c}<

1

1

ap{I CmI>C}.

Proof. p{sup I Ck I> 2c, I Cm I< c} k

E p{Ie I <2c, i 2c, ICmI
<EP{1C.1<-2c,i2c,1Cm-CkI>C} k=1 M

=EM(p{IC I <2c,i2cICk} k=1 X P{I Cm - Ck I> C I Ck}) m

<_aEMP{IC:I<2c,i2cICk} k=1 aP{sup I ek I > 2c} k

This means that P{sup I Ck I > 2c} < p{I sm I > c} + P{sup I Ck I > 2c, I C. I < c} k

k

< P{I

Cm

I > c} + ap{sup I Ck I > 2c} , k

from which proof of the lemma follows. Let us now return to the proof of the theorem.

Since

p{I P11 nak+l+l -rrnj I > a j Cnj} Cxjll2 I Cnj)

U2 1

`

= Ti M{( El [an(tnr, Cnr)Otnr + on(txr, nr)wJ)2 nj

<

UZ

22

+

an(tnr, Cnr) Atnr /2I en j]

M[( d

II11r

M[(

-j Cn(tnr, Cnr)Wnr 2l Cnj] 7

ML(-El an(L

82

2

\1

nrl tnr /ZI e 1] 1

nr,

-k+1

+

g2

M

+

UZ

M(7=r
Q(tnr, Cnr)M((0nr I nr) I Cnj 6n(tnr,

nr)6n(tnl, a

Mr I\/ E1 an(tnr, \ 1II

1\

1 12+

1

U2 (t." (tnr,

M(0)., I Cnl/ I

I nj ,

n9/

5.

THE SPACE OF FUNCTIONS WITHOUT DISCONTINUITIES

469

since the functions an and an are bounded, and since 8k+1

Otn; < h + 2 max Otn;

*=8k

j

it follows that there exists a constant H1 such that P{1 nsk+1

- nj I >

3 I En;} <

32

(h + 2 max Otn1)

' Consequently, for sufficiently small h and all sufficiently large n, Pl I enak+1+1 - e, I > 161 C 2

Then on the basis of Lemma 4, P

I n(t) - E (kh) I >

lkh< sS_(k+l)h

4

< 2Pl I nsk+1+1 - $nsk I >

16

It follows from the convergence of the finite-dimensional distributions of E,(t) to the finite-dimensional distributions of E(t) that liimm P{I

$n.k+1+1 - %'k I >

16

<=

P{I $((k + 1)h) - $(kh) I >

16

This means that lim P( sup ,8

It'-t'h

I

en(t') - en(t") I > s} P

I

kY1

161

(-)4(kh+h)-(kh) E

kh<1

But for some L we have M (t + h) - E(t) I' < Lh2 on the basis of Corollary 2, Section 3, Chapter VIII, so that lim P{ sup n--oo

It'-t"Ish

I en(t') - En(t") I > a} < 1614Lh 6/

This proves equation (6) and hence the theorem.

5. THE SPACE OF FUNCTIONS WITHOUT DISCONTINUITIES OF THE SECOND KIND

Let D [0, 1] denote the set of real functions x(t) defined on the interval [0, 1] and having right- and left-hand limits at every point. We shall treat two functions that coincide at all points of continuity as the same function. Therefore it is natural to take some sort of standard definition of the values of functions

LIMIT THEOREMS FOR RANDOM PROCESSES

470

x(t) at points of discontinuity. In what follows we shall assume that for all functions in D[0, 1],

x(1)=x(1-0). (1)

x(0)=x(+0),

x(t)=x(t+0),

Study of the space D[0, 1]

is

useful since there are classes of

random processes whose sample functions fail, with probability 1, to have discontinuities of the second kind (for example, processes with independent increments, Markov processes under extremely broad conditions). In order to be able to use the results of Section 1 we need to -define on D [0, 11 a metric in which D [0, 1 ] becomes

a separable metric space enjoying the property that the minimal 6-algebra containing all cylindrical sets coincides with the 6-algebra of Borel subsets of that space. The metric should be sufficiently

"strong" (that is, there should be as few convergent sequences as possible and hence as many continuous functionals in that metric as possible).

The uniform-convergence metric p.u(x, y) = sup I x(t) - y(t) I 05t5,

is not suitable for this since D[0, 1] is not a separable space in that metric. (The set of functions xa(t) = [1 + sgn(t - s)]/2 for 0 < s < 1 has the cardinality of the continuum, but the distance between any two distinct elements of that set is equal to 1.) We introduce into the space D[0, 1] a metric that is somewhat weaker than the uniform-convergence metric. Let A denote the set of all continuous increasing real functions X(t) on [0, 1] such that X(0) = 0 and x(l) = 1 (that is, X is a continuous one-to-one mapping of [0, 1] onto itself). We note that for each ) e A there exists an inverse function ?-1, also in A. If X, and X2 belong to A, then the composite function X1(X2) also belongs to A.

Now for every pair x(t) and y(t) in D[0, 1], we define pD(x, y) =inf [ sup I x(t) - y(e(t)) I + sup t - X(t) ] I

2EA

(2)

t

t

Let us show that p defines a metric on D[0, 11. To do this, we need to show that the function p satisfies the three axioms of a metric: (a) pD(x, y) > 0 with equality holding if and only if x = y; (b) pD(x, y) = p,(y, x); (c) pD(x, z) < pD(x, y) + pD(y, x) for all x(t), y(t), and z(t) in D[0, 1]. Condition (a) is obvious. Condition (b) follows from the relation

pD(y, x) = inf [sup I y(t) - x(X(t)) I + sup I t - M(t) 2EA

t

t

= inf [sup I y(?-1(t)) - x(t) I + sup I X-1(t) - t I ] = pD(x, y) I ICA

t

t

S.

THE SPACE OF FUNCTIONS WITHOUT DISCONTINUITIES

471

Let us look at condition (c), the triangle inequality.

Let x(t),

y(t), and z(t) denote functions belonging to D[0, 1]. For every s > 0 there exist functions X1(t) and X2(t) such that PD(x, y) > sup I X(t) - Y(?1(t)) + sup I t - ?1(t) I - s t t PD(Y' z) > -Sup I Y(t) - z(X2(t)) + sup I t - %2(t) I - s t t

(3)

Then PD(x, z) < sup I x(t) - z(X2(?1(t))) I + sup t - %2(X1(t)) I t t sup I X(t) - Y(X1(t)) I + sup I t - x1(t) t t

I

+ sup I Y(X1(t)) - z(A2(X1(t))) I + sup I ?1(t) - XA1(t)) I t t

t - X1(t) = sup I X(t) - Y(X1(t)) I + sup t t I

I

+ sup I Y(t) - z(X2(t)) I + sup I t - X2(t) I t t

since X1(t) ranges over the interval [0, 1] as t ranges over the interval [0, 1]. From inequalities (3) we obtain PD(x, z) < PD(x, y) + PD(Y, z) + 2s ,

which, by virtue of the arbitrariness of s, implies condition (c). Thus we may take pD as the distance in D[0, 1]. To make further study of the properties of the metric pD, we need some auxiliary propositions. Lemma 1. Let us define for every function x(t) in D[0, 1],

0Jx) =

sup

t-c!9t't5t'15t+C

[min {1 x(t') - x(t) I ; 1 x(t") - x(t) I }]

+ sup Ix(t) - x(0) I + sup

x(t) - x(1)1

1-CSt51

OStSC

Then

lim 0jx) = 0 C-w

Proof. The continuity of x(t) at the points 0 and 1 implies that supo 0.

If we can find an s > 0 such that for arbitrarily small c, sup

min [I x(t') - x(t) 1; x(t") - x(t) 1] > s

then we can also find sequences t' < t < t; such that t' - t" , 0 and

I x(t'n) - x(tm) I > 6

,

x(tn') - x(tn) I > s .

(4)

LIMIT THEOREMS FOR RANDOM PROCESSES

472

By taking subsequences if necessary, we may assume that {tn} converges to some point to in the interval [0, 1]. Then the sequences

{t} and {tn} must also converge to to. Therefore the quantities x(t), x(tn), and x(t') converge to one of the numbers x(to - 0) or x(to) = x(to + 0), so that at least two of these must have the same limit. It is easy to see that the limit of the sequence {x(t,)} must coincide with the common limit of the sequences {x(t' )} and {x(t;; )} when these two sequences have the same limit. This means that at least one of the differences x(t,) - x(tn) or x(t') - x(tn) approaches 0, and this contradicts inequalities (4). This completes the proof

of the lemma. Lemma 2. Let x(t) denote a function in D[0, 1] and let [a, t3] denote a subinterval of [0, 1]. If x(t) has no jumps exceeding s in [a, 8], then the inequality t' - t" I < c for t', t" e [a, ,8] implies I x(t') - x(t") I < 200(x) + s . I

Proof. Let us choose arbitrary 3 e (0, s) and a point z in the interval [t', t"] such that for t e [t', z], x(t') - x(t) I < 0.(x) + 8 and

I x(t') - x(z) I > 0Jx) + 3 . If no such point exists, then I x(t') - x(t") I < A,(x) + S. This means that the assertion of the lemma is satisfied. If a point z does exist then since min [I x(r) - x(t') I; I x(z) - x(t") I] < A,(x)

and

I x(z) - x(t') I > A(x) + 3 , we have x(z)

- x(t") I < 0,(x)

Thus I x(t") - x(t') I

< I x(t") - x(r) I + I x(z) - x(z - 0) 1 + I x(z - 0) - x(t') I

<2A,(x)+3+ Taking the limit as a, 0, we obtain proof of the lemma. Let H,,,,, denote the set of functions x(t) in D[0, 1] that are constant on each of the intervals [(k/n), (k + 1)/n) and that assume values that are multiples of 1/m. Lemma 3. For every function x(t) in D[0, 1] there exists a

5.

THE SPACE OF FUNCTIONS WITHOUT DISCONTINUITIES

473

function x*(t) in H.,n such that PD(x, x*) <

1n + m1 4A 2(x)

.

n

Proof. In each of the intervals [(k/n), {(k + l)/n}) there can exist no more than a single point with a jump exceeding 202/n(x). This is true because if z is one such point, then

x(s)-x(z-0) =min (Ix(s)-x(z-0)I;x(z)-x(z-0) < p n(x)

s e Ln,

for

I

and hence I x(s) - x(s - 0) 1 < 201/,(x) < 202/n(x). Suppose that is a point of the interval [k/n, (k + 1)/n] at which zk I x(zk) - x(zk - 0) > 202/n(x) if such a point exists in that interval. We let X(t) denote a function in A such that X ((k + 1)/n) = zk and t- 1/n < x(t)
function z(t) has discontinuities exceeding 202/n(x) only at points of the form k/n and PD(x, z) < sup I i(t) - x(Mt)) I + sup I t - X(t) I < t

t

1 .

n

Now define

k/

\n

z(n - 11 J n

for for

k+ 1], k
tErk,

n

n t = 1

J

.

\\

Then

PD(z, z*) < sup z(t) - V(t) 15 sup t

t

z(t) - (k

sup

k/n
I

n

Since the jumps of z(t) that exceed 202/n(x) occur only at points

of the form k/n, there are no such jumps in the interval (k + 1)/n).

t(n)

Hence, according to Lemma 2,

- z(t) I < 20n(z) + 2p2(x) for

tc L

Let us find a bound for

n

,

k

1\

n

[k/n,

LIMIT THEOREMS FOR RANDOM PROCESSES

474

sup

-(1/n)
[min {I x(t')

2(t) - z(0) I + + sup 0
t-(1/n)
- x(t)

sup

1-(1/n)
;

2(t) - x(t") }]

x(t) - z(1)

[min {I x(X(t')) - x(X(t)) 1; 1 x(X(t)) - x(X(t")) }]

sup I x(X(t)) - x(1) + sup I x(X(t)) - x(0) + 1-(1/n)
We note that, for t1 < t2 < t1 + (1/n), we have t1

1 <x(t) <X(t2)
n

so that 0 < X(t) - X(t) < 2/n. Therefore (2) < Q 2 (x)

i n

.

n

Hence

p,(-k, z*) < 402(x) n

Finally, let us set x*(t) = (1/m) [mz*(t)], where [x] denotes the integral part of x. Since I x*(t) - 2*(t) I < 1/m, we have

PD(x, x*) < PD(x, x) + PD(x, V) + PD(x*, x) <

I

+ 4A (x) +

n

1 m

.

This completes the proof of the lemma. Corollary.

The space D[0, 1] with the given metric is separable.

This follows from the fact that the countable set Um,n Hm, is everywhere-dense in D[0, 1] in accordance with Lemma 3. Theorem 1. Let L denote a positive constant and let 0)(3) denote a function that is defined, continuous, and monotonic for 3 > 0. Suppose that lim8l0 w(3) = 0. Let K(L, (o) denote the set of functions in D[0, 1] that satisfy the inequalities I x(t) I < L and 0,(x) < w(c). Then K(L, w) is a compact set in the metric pD. Proof. We note first of all that for every s > 0 the set K(L, (o) has a finite s-net consisting of the functions in Hm,n that satisfy

relation I x(t) I < L if m and n are chosen so that 1/n + 1/m + 4w(2/n) < s. The set K(L, w) is closed. It is easy to verify that 0Ax) < Therefore, if pD(xn, z) . -> 0 as n - + oo , where each xn E D[0, 1], the

then for every a > 0,

n

lim OC+b(xn) < w(c + a)

.

5.

THE SPACE OF FUNCTIONS WITHOUT DISCONTINUITIES

475

Because of the continuity of (o, this means that zjz) < o(c). Obviously,

sup x(t) I < supt I Y(t) + PD(x, Y) t so that

-t

sup z(t) I < lim sup I x.(t) I < L .

Consequently, the limit of a sequence belonging to K(L, w) also belongs to K(L, (o). We must now show that every fundamental sequence {x,t(t)} of elements of K(L, (o) converges. (Then, we shall

show that K(L, o) is a complete metric space with finite s-net for every s > 0, which will mean that K(L, m) is compact.) Let {xn(t)} denote a sequence of functions in K(L, w) such that pD(xn, x,,,) ---1 0 as n, m ---p 00 (that is, {xn(t)} is a fundamental sequence). It is sufficient to show that some subsequence {x,k(t)} converges to z(t) since we shall then know by virtue of the inequality PD(xn, x)

PD(xn' xn) + PD(xnk, x)

that z(t) is the limit of the sequence {xn(t)}. Therefore, we may assume that the sequence {xn(t)} is such that pD(xn, xn+1) < 1/21}1. Then there exists a sequence {? (t)} of functions in A such that sup xn(t) - x,41(Xn+1(t)) t and

sup I t - Xn+1(t) I t

_

__<

1

.

2n+1

Let us define p,(t) //,= X1(t) and un(t) = sup I ( ^n(t) - (-^n-1(t)

I < sup Xn(t) - t I <

Since 2n

it follows that {pcn(t)} converges to some nondecreasing continuous

function _t(t) that satisfies the conditions ft(0) = 0 and u(1) = 1. Furthermore, sup I xn(fn(t)) - xn-1(pn-1(t)) I = sup xn(Xn(t)) - xn-1(t) I

2-

Therefore {xn(pn(t))} converges uniformly to some function x*(t) in D[0, 1]. Let us look at the connection between the functions x*(t) and Ic(t). Suppose that (t) is constant on some interval [a, 8]. If x*(a) = x*(8), then x*(t) is also constant on [a,,81. On the other

LIMIT THEOREMS FOR RANDOM PROCESSES

476

hand, if x*(a) # x*(/') there exists a y e [a, iS] such that x*(t) _ x*(a) for t e [a, y) and x*(t) = x*(,S) for t e [y, IS]. This is true because otherwise there would be points t' < t" < t"' belonging to [a, (B] such that x*(t') # x*(t") and x*(t") 'L x*(t"'). Then we would have lim min [I xn(f"n(t')) - xn(f"n(t")) I ; I xn(,un(t")) - xn(, n-I x*(t") - x*(t"') l> 0 = min 11 x*(t') - x*(t") I

n(t,,,))

Il

although 1en(t') < tn(t") < 1an(t"') and the sequences

and {pn(t"')} all converge to u(a). This would contradict the fact that the sequence {xn(t)} belongs to K(L, (o). Let 2(t) denote the function in D[0,1] defined by

(5)

x(t) = x*(a(t)) ,

which is to be satisfied at all points t at which "(s) > u(t) for all s e (t, 1]. Equation (5) defines a unique function 2(t) in D[0, 1]. Let us show that this function 2(t) is the limit of the sequence {xn(t)}. To do this, we construct auxiliary functions cpn(t) in A. , -r, are all points in the interval [0, 1] at Suppose that r , z-,, which 2(t) has jumps exceeding 1/n. We let [a,, 8,] denote the maximum interval on which a(t) assumes the value Zvi (this interval may consist of a single point). Let yi denote a point in the interval [ai, ei] such that x*(t) = X(Ti - 0) for t e [ai, yi) and x*(t) = 2(v) for t e [yi, If ati = yi, then x*(t) assumes the single value 2(v) on [ai, 6j].) Let us choose sn not exceeding 1/n such that DEn(g) < 1/n. Let cpn(t) denote a function satisfying the equation Cpn(yi) = zi and the inequality pn(t) - p(t) < sn. Let us find a bound for I

sup I x*(t) -

I

t

If t does not belong to any of the intervals [ai, Gi], then

x*(t) - x(q,n(t)) = x(f (t))

-

20En(x) +

n

by virtue of Lemma 2 since 2(t) has no jumps exceeding

1/n

between a(t) and pn(t). If t e [ai, Ri), then sup x(Zi - 0) - x(s) < DEn(x) I x*(t) - 2(9m.(t)) 1 < 8 E Sri-En.ii

since 12(21 - 0) - 2(zvi) I > 1/n. In an analogous manner we can show that I x*(t) - 2(cp.(t)) < DEn(2) for t e [yi, 8i]. Consequently, sup I x*(t) - x(pn(t)) I < 2L

(x) +

n

<

3

THE SPACE OF FUNCTIONS WITHOUT DISCONTINUITIES

S.

Let us now find a bound for pD(xn, i).

477

We have

pD(xn, x) < PD(xn(t), x*(fr7"nl(//t//))) + pD(x*(f nl(t)), x(7'n(f'"nl(t)))) + pD(x(t), x(7"n(f'"n1(t))))

< sup I xn(un(t)) - x*(t) I + sup I x*(t) - X(7"n(t)) I t t + sup I t - n(ltn1(t)) I

<

2"

+

n

+ sup I pn(t) - 7'n(t) t

I

<1++1+sn. 2" n

2"

Thus pD(xn, z) ---> 0; that is the sequence {xn} converges to the

function z(t).

This completes the proof of the theorem. Theorem 2. If the finite-dimensional distributions of the processes e ,(t) that have no discontinuities of the second kind converge to the finite-dimensional distributions of the process E(t), and if for every

6>0, lim lim c-+0

6} = 0 ,

n-+_

(6)

then for every functional f defined on D[0, 1] and continuous in the metric PD, the sequence of distributions of f(on(t)) converges to the distribution of f(e(t)). Proof. Using Remark 1 of Section 2, we see that condition (6) implies lim sup p{a(n(t)) > e} = 0 c-.0

n

(7)

Using the sufficiency of Theorem 2 of Section 1 and Theorem 1 of this section, we can see that to prove the theorem, it suffices to show that lim sup p{sup I an(t) I > L} = 0 . (8) But for every function x(t) in D[O, 1] sup I X(t) I < sup 05t51

x\mk

o5kg.

+ a (x)

since for t e [k/m, (k + 1)/m], either I x(t) - x(k/m) < a11,,(x) or I x(t) - x((k + 1)/m I < a11, (x). Therefore, P{sup I an(t) I > L} < pt.;5k;5. sup t

e

m/

> L - 6 } + p{a l (an(t)) > 6}

The random variable SUPQSk<m I een(k/m) I

.

J

is uniformly bounded

LIMIT THEOREMS FOR RANDOM PROCESSES

478

in probability with respect to n.

This follows from the convergence of the finite-dimensional distributions of an(t) to the finite-dimensional

distributions of e(t), which implies convergence of the distributions of supo
This

means that lim sup P{ sup I En(t) I > L} < sup P{0,(en(t)) > s} L- n n t

gym.

Taking the limit as m, 00, we see that equation (8) holds. This completes the proof of the theorem. 6.

CONVERGENCE OF A SEQUENCE OF SUMS OF IDENTICALLY DISTRIBUTED INDEPENDENT RANDOM VARIABLES TO A HOMOGENEOUS PROCESS WITH INDEPENDENT INCREMENTS

Let en e

, nn for each n denote independent, identically distributed random variables. We define Snk -nl + + nk Let us consider the random process an(t) defined by ;n(t) = S,k for t e [(k - 1)/n, k/n) and n(1) = Snn. This process is a process with independent increments and has discontinuities at the points k/n.

2,

In this section we shall study the conditions under which

the finite-dimensional distributions of the processes e, (t) and the distributions of the functionals of these processes that are continuous

in the metric pD converge to finite-dimensional distributions and the distributions of the corresponding functionals of a homogeneous process fi(t) with independent increments. We shall assume that the sample functions of the process E(t) (which we may assume to be stochastically continuous) belong, with probability 1, to D[0, 1]. As we know (cf. Chapter I, Section 4), the characteristic function of a homogeneous process with independent increments can be represented in the form Me'2ect) = exp {t[ix' +

_(ei

-1-

ixu

1 +uU2

udG(u)]

2)_1

} J

(1 )

where G is a monotonic bounded function. Formula (1) completely determines the finite-dimensional distributions of the process e(t). Concerning the sequence of sums Snk, we define ryn = n

x x2dF, (x)

and

Gn(x) = nJ

U 1

+

dFn(u)

u2

where F,(x) is the distribution function of the random variables

ni.

6.

CONVERGENCE OF A SEQUENCE

479

Theorem 1. If there exist a number y and a nondecreasing boundO, then ed function G(x) such that yn y and Gn(x) = G(x) as n the finite-dimensional distributions of the processes en(t) converge to the finite-dimensional distributions of the process 5(t) with characteristic function (1). Proof. Since an(t) is a process with independent increments it is sufficient for us to show that for all t, < t2 in [0, 1], the sequence {En(t2) - en(t)} converges to the distribution E(t2) - s(t,). The characteristic function of the variable n(t2) - en(t1) takes the form Meoj£,i(t2)-en(tl)J = [Meiien1]kn

([Mei)en1]n)kn/n

= Since kn/n --. t, - t, as n

where kn = [nt2] - [nt,].

oc, the theorem

will be proved if we can show that lim (Meilenl\n

ixu 1 1 + u2dG(u)}

= exp (ix^l +

\

l

1 + u2)

u2

(2)

Let us represent the characteristic function of n, in the form Mei1 n1 =

eiz=dFn(x)

1+ J

x dFn(x) + _(ei.tx r1 +x2

= 1 + n [yn

-I-

x )dFn(x) 1 +x2

x x2) 1 + x2 dGn(x)1

w(eiix

.

Then (Meizenl)n = (1 +

x

n [Y. + 1:(ei2x

+2 x2

-1-

1 + x2)

dGn(x)1)n

x

JJ/

cx, and from the hypothesis of the theorem (we But yn --> y as n recall that the function (eizu - I - iXu/(1 +u2)) (I +u2)lu2 is continuous

and equal to - x2/2 at u = 0), it follows that Jei2x _ 1 ao(eilx

ixx ) 1 + x2 dGn(x) 1 +x2 x2 iX,x

1 + x2 )

1 + x2 dG(x) x2

The conclusion of the theorem then follows.

We find that we do not need to impose any additional conditions to ensure convergence of a sequence of distributions of

LIMIT THEOREMS FOR RANDOM PROCESSES

480

functionals that are continuous in the metric pD. Theorem 2. Suppose that the hypotheses of Theorem I are satisfied.

For a functional f that is defined on D[0, 1 ] and continuous in the metric p, the sequence of distributions f(on(t)) converges to the distribution f(e(t)). Proof. Since the convergence of the finite-dimensional distributions of the processes en(t) to the finite-dimensional distributions of the process e(t) was shown in Theorem 1, on the basis of Theorem 2 of Section 5 we see that to prove this theorem it suffices to show

that for every s > 0, lim lim p{0jen(t)) > s} = 0

(3)

.

c-.o, n-,c

To prove this equation we use:

, n are

$2, Lemma 1. Suppose that distributed random variables. Then p{o,isupsnmin

L k=i+1

1k=j+1 1l

ek

]>°I (4)

> 21)2

(p{sup±i

i=1

k

Ek

independent identically

Suppose that i < j < 1, I Ek=i+1 Ck I > E, I j__,k=j+1 Ck I > s. Then either 11k-, Ck I> s/2 or Ek=1 Ck I> s/2, and for all r < j, either I Ek=r+1 Ck I> s/2 or Ek=r+1 k I > s/2. Thus the event Proof.

I

I

ik a

{ sup min i<j
L

E k k=ii-1

k

Ell

+1

, n):

implies one of the events it,. (for r = 1, 2,

1 + ... + k I < 2 k
I 1 + ... + er >

s

;

sup I Er+l + ... + E1 1 > s

2 , i>,

2

Since

P{%,} = P{1e1 + ... + Sk

E k
+ . .. + Cr > 2 1Pfsup l>r

"r+1

;

+ .. . +

>21

,

remembering that the variables i are identically distributed, we obtain

6.

P{

CONVERGENCE OF A SEQUENCE

sup min

k=i+1

a<j
ek

481

$k

k=j+1

]>}


E2 + .

.

1

;

cr+1 + . .. + et I > . er I > -211 PSUp I

P{suple,+...+elI> 26 E

Pf I

1+...+

k

k < r - 1 ; I e1 + . .. +

2

2 >

2

(Pt
This completes the proof of the lemma. Returning now to the proof of the theorem, we note that

Ojx(t)) < sup I x(t) - x(0) I + sup I x(t) - x(l) 1-c5t<1

05t
+ max

sup

0
min [I x(t') - x(t")1; I x(t" - x(t"')

Therefore

PA(En(t)) > 6} I $n(t) - en(O) > ^ ( + P11 sup 11 e%(t) - en(1) P sup 05t5

+ Y, Pt k<71c

I en(t") -

<

min [I e, (tl)

-

en(t")1

en(tll.)11>

2

P{ sup

en(t) - e,(O)

O
+

sup

kc5t'
>1

k<1/c

sup > 64 } + P{1-c5t<1

sup I en(t) - en(kc) I > 4 (P{ kc
S n(t) -n(1) > 4

)2

If n > 1/c, it is easy to see that L (e, (t)) = 0 since en(t) is constant on intervals of the form [i/n, (i + 1)/n]. On the other hand, if n < 11c, then although the number of points of the form i/n in the interval [kc, (k + 3)c] varies with varying k, this number still does not exceed the number of these points in the interval [0, 4c], so that we always have

LIMIT THEOREMS FOR RANDOM PROCESSES

482

P{A(n(t)) > F} <

n(t) - n(O) >

2P{ sup 105 t<4c

4 lIT

1

I (t) - en(O) I > )(P{0 s!9upc t s-

+ (1 + C To find a bound for the probability -I

>} = P j sup N enti >

OSt54c

1k

r

1l

P1 sup I en(t) - En(O) I

4J

4

64

1l

}

k54nc i=1

11

we introduce the variable ni = Sni if I ni I < L and;,; = 0 if I > L, and the variable eni = Cni - eni Then k

P1

sup 15k5N

Sni > i=1

4

sup sup eni I >0 + P 15k5N 1
}< P

k

>4}

i=1

N ,'

i=1

P{I eni I > L}

Pj1skp yL Wni - Meni) > 4 - Li M ni } NDen1

N. P{ni > L} +

(4

- N I Men,

I)2

(Here we used Kolmogorov's inequality, Theorem 1, Section 4, Chapter III. )

If L and -L are points of continuity of the function G, then the following limit relations are valid: lim n P{ I eni I > L} = lim 5

Ixl>L

1 + x2 dGn(x) _ 5 1 + x2 dG(x) x2 IxI>L X2

,

2

lim n Den,

lim n Me21 < lim n M 1 + L en1 = (1 + L2) n-.oo

n-.oo

1

+ e21

-oo

dG(x)

lim n Men1 = lim n LI
= y - lim n5

x x dF,(x) + limn JX -+ x2/ IxI>L 1 + xLn-- IxI1 J

=y-

1 dG(x) + 1

JJ IxI>L X

xdG(x) = 71

.

J IxISL

k>

Therefore, if 4c 17L < a/4, then lim P{1

64 }

kupne

_< 4c xl>L

1 + x2 dG(x) + 4c ( 1 + L2) 5 x2

6

(4

dG(x)

-4cIILI

2

)dFn(x)

6.

CONVERGENCE OF A SEQUENCE

483

This means that for sufficiently small c, lim P{ sup OSt<4c

n\t) - n(O) >

4

Kc ,

were K is some constant such that lim P{i (en(t)) > 6} < C1 + 1 )(Kc)E + 2Kc

Equation (3) follows from this inequality. This completes the proof of the theorem. As examples of corollaries of this general theorem, let us look at a few particular limit theorems. Theorem 3. Let a(t) and b(t) > 0 denote continuous functions and let a denote a real number. If the hypotheses of Theorem 1 are satisfied, then

lim P{a( n) - ab\( l < snk < a\( n l + /

ab(nl k= 1' 2, ... , n

= P{a(t) - ab(t) < e(t) < a(t) + ab(t), 0 < t G 1} for all positive a such that p{a(t) - ab(t) < E(t) < a(t) + ab(t), 0 < t < 1} is continuous at a as a function of a. Proof.

Consider the functional defined on D[O, 1] by

f(x(t)) = sup I x(t) + a(t) 0
b(t)

It is easy to see that this functional is continuous in the metric pD.

Proof of this theorem follows Theorem 2 and the inequality P

la`

n) - ab`

< snk < a\

< P{f(E%(t)) < a + 6} ,

which is valid, if 6 > 0, for all sufficiently large n. Theorem 4. Let g(x) denote a continuous function defined for x e (- oo, ao). Then if the hypotheses of Theorem 1 are satisfied, we have

lim P{ 1 E g(Snk) < a} = %-n k=1

for all a such that

0

g(E(t))dt = a} = 0.

a}

484

LIMIT THEOREMS FOR RANDOM PROCESSES

Proof follows from the fact that the functional f(x(t)) = 0g(x(t))dt

is continuous in the metric pD, Theorem 2, and the equation

1n k=1 L g(S,k) _

0

Let us define a functional y,(x(t)) that is equal to 0 if sup06ts1 x(t) < a and equal to x(r) - a if sup05t,,1 x(t) > a, where z is a point in the interval [0, 1] such that x(t) < a for t < r and x(r) > a. The functional ya(x(t)) is called the time of the first crossing of the function x(t) through the level a. Let us investigate the continuity of ya(x(t)) in the metric pD. One can easily see that for all those x(t) such that suet x(t) < a, the functional ya(x(t)) is a continuous functional. It is also continuous on those functions x(t) such that ya(x(t)) > 0. If x(r) = 0 but sup,.(,,,,,) x(s) > a for every s > 0, then y.(x(t)) is also continuous at the point x(t). Thus, the only possible points of discontinuity of 7a(x(t)) are those functions x(t) for which x(r) = a and sup.. (r, t, x(s) < a for some t > r. We can now formulate: Theorem S. Let a denote a number such that P[ sup x(s) = a} = 0 05e5t

for t e [0, 1]. Then, the distributions of ya(en(t)) converge to the distribution of ya(e(t)).

The proof follows Theorem 2, Corollary 2 of Section 1, and the fact that under the hypotheses of the theorem, ya(x(t)) is continuous almost everywhere with respect to the measure p corresponding to the process E(t) on D[0, 1 ]. 7.

LIMIT THEOREMS FOR FUNCTIONALS OF INTEGRAL FORM

In the preceding sections, by confining ourselves to rather narrow classes of random processes we obtained limit theorems for extremely

broad classes of functionals. A characteristic feature of these theorems was the constructive character of the conditions that were imposed in addition to the condition of convergence of sequences of finite-dimensional distributions. The constructive character of these additional conditions permitted convenient checking of the hypotheses in particular cases. As a rule, these conditions reduced to the requirement of some stochastic equicontinuity of the sequence

7.

LIMIT THEOREMS FOR FUNCTIONALS OF INTEGRAL FORM

485

In this section we shall consider a comparatively narrow class of functionals of integral form: of random processes.

f(x(t))

-

s:

5(si1 S2, ... sk, x(S1), x(SQ), ... I x(`sk))ds1dS2 ... dsk

(1 )

Such functionals can be considered for processes of an extremely general form. For example, if p is a continuous bounded function,

then a sufficient condition for the existence of this functional of the process e(t) is that the process e(t) be measurable. For functionals of the form (1), we can also formulate a limit theorem in terms of some uniform stochastic continuity of the sequence of processes en(t), while we impose almost no conditions at all on the processes c,(t) themselves (that is, on the form of the sample functions en(t)) other than measurability and existence of certain moments.

We first formulate a very simple theorem, from which we can obtain more general statements. Theorem 1. Let e, (t) and e(t) denote measurable processes defined on the interval [0, 1]. If the sequences of finite-dimensional distributions of the processes e, (t) converge to finite-dimensional distributions of the

process an(t), if sup M en(t) I < H < 00 t,n

and if lim lim

sup

h-.0 n-.oo t'-t" I s h

(2)

M I E,(t') - en(t") I = 0 ,

then the sequence of distributions of the random variable

con-

verge to the distribution of the random variable 5e(t)dt. Proof. It follows from condition (2) that for every S > 0, there exists an m such that for all sufficiently large n, M

sup 1m n'=1 n(`m/ - 5e(t)dt 0< It'-t"ISl/m

M

nt')

-

n(ttt)

An analogous inequality holds for e(t), since it follows from equation

(2) and Fatou's lemma (Theorem 2, Section 5, Chapter II) that lim

sup M

%-0 t'-t"Ish

0.

We note that the convergence of the sequences of finite-dimensional distributions of the process en(t) to finite-dimensional distributions

LIMIT THEOREMS FOR RANDOM PROCESSES

486

of the process e(t) implies convergence of the sequence of distributions (1/m) Ek 1 6,(k/m) to the distribution (1/m) E- 1 cn(k/m). fore

There-

lim M exp iX en(t)dt - M exp i? f le(t)dt "-'Co

0

0

< lim M exp {i1e(t)dt} - M exp {ix__ A;=1 ( 0

+ M exp {ix,'e(t)dt} - M exp {iX XIM

+I

\m

m

)JI

(

jt)dt - m

m/

M os(t)dt -

\m/

2I

S.

This completes the proof of the theorem. Theorem 2. Suppose that the sequences of finite-dimensional distributions of the processes en(t) converge to finite-dimensional distributions of the process e(t).

Suppose that for every s > 0

lim lim sup p{jen(t) - e,,,(s) I > s} = 0 , h-,oo n-.oo I t-a 15h

(3)

and that there is a nonnegative function 1/r(x) such that /r(x) T - as I x I --. + 00 and sup, sups M'/r(en(t)) = c < o o. Then for every continuous function (p(t, x) such that lim sup sup I 'At' x) I = 0 N-.oo

t

Ixj>N

*(x)

the sequence of distributions of the variable \(t, ,n(t))dt converges to the distribution of the variable

r1p(t,

0

E(t))dt.

0

Proof. It will be sufficient to show that the hypotheses of Theorem 1 are satisfied for the sequence of processes 7)n(t) = T (t, en(t)).

Convergence of the finite-dimensional distributions of the processes ren(t) to the finite-dimensional distributions of the process 7)(t) = p(t, e(t)) follows from the convergence of the finite-dimensional distributions of the processes en(t) to the finite-dimensional distributions of the process e(t) and the continuity of the function T (t, x). Since I p(t, x) I < K(1 + i/r(x)) for some K, we have sup sup M 1)7.(t) I < K(1 + c) t

Finally, let us show that equation (2) is satisfied for the sequence {ren(t)}. To do this, we define the function gN(x) that is equal to

7.

LIMIT THEOREMS FOR FUNCTIONALS OF INTEGRAL FORM

x for I x I< N and equal to N sgn x for

487

>_ N. We define

Ixl

EN = sup sup I IP(t, X) t xl?N *(X)

Then using the inequality I'P(t, g,(X)) - m c(t, X) I < *(X)

sN ,

we obtain I mq(t1, e (t1))

M

m - q(t2, ag (t)) I

< M 17'(t1,,, $N(Ef(tl))) - q'(t2, gN(t2))) I slmI,(En(t2)) + sNM Y' (ent)) + M I q (t1, gN(en(tl))) - 0111 gN(t2))) I + 2-cc

(4)

Since T(t, x) is a continuous function, it follows that for

every

s > 0 and L > 0, there exists a a > 0 such that 19 (t1, x1) -'(t2, x2) I < s

whenever

Ix1-x2I
It1-t2I

L, Ix2I
Define R = sups, ixlSNI rp(t, x) I. Then for t1 - t2 < 8, nn gN(en(t2))) IIq M I q (t1, gN(en(t))) - 7"(t2,

s + R[P{I en(t1) - en(t2) I > U} + P{I en(t) I > L} + P{I Sn(t2) I > L}] Mi',1f'(L) (En(t)) t s + R P{I en(t1) - en(t2) I > S} + 2 sup

In view of (3) and (4) we obtain lim lim h-,0 n-.

sup M I rJn(t1) - /n(t2) I

Iti-t2l;gh

+ 2sNC +

2Rc[T(L)]-1

The quantities on the right side of this inequality can be made arbitrarily small by the suitable choice of s, N, and L. lim lim h-,0 n-,°

Consequently,

sup M I '7 (tl) - '/n(t2) I = 0

It1-t215h

that is, all the conditions of Theorem 1 are satisfied for the sequence of processes 27n(t). This completes the proof of the theorem.

Theorem 3. Suppose that the sequence of processes en(t) and the function *(x) satisfy the conditions of Theorem 2. Then for every , tk, x1, , xk) satisfying the equation lim sup I q (t1, ... k, 41 x1, ... , Xk) = 0 ,Ixk1'w t1, ,tk E w"I (xi) i=1

continuous function p(t1, 1x11

488

LIMIT THEOREMS FOR RANDOM PROCESSES

the distributions of

...

1

0

1

q(tl, ..., tk, cn(t1), ..., en(tk))dtl ... dtk

0

converge to the distribution of ri

1

0

0

Proof.

7"(tl, ..., tk, (t1), ..., (tk))dt,

dtk .

Define sup

sup

6r,

17' (t1, ... , tk, xl, ... , xk) k

xlISN,...,IxkI N tl,...,tk

i=1

+

sup

yr(xi)

I tP(t1, ... , tk, x1, ... , xk)

sup

x1I5N....,IxklSN t1....,tk

ki/r(N)

It is easy to see that eN --+ 0 as N Let gN(x) denote a continuous periodic function with period 4N that coincides with x for I x I< N and that satisfies the inequality I gN(x) X for I x 1> N. Then I 'P(t1, . . . , tk, x1, ..., xk) p . .. , Op N(xk) q (t1, ... , tk, pN(x1), T

k

ir(Xi)

-'N i=1

Therefore 1

1

0

0

P(tl, ... , tk, e (t1), ... , e.(tk))

M

p - q(ti, . . ., tk, gN(E(t1)), ..., g1p Wtk))dt1 ... dtk G ken,c .

Since on the basis of Fatou's lemma, M-,/r(E(t)) < lim M*(e,(t)) < c, we also have

M

10

...

10

J

(9(ti, ..' , tk, Sg(tl),

.. ', (tk))

- q (t1, . ., tk,

dtk


We now note that the function p(t1, , tk, gN(xl), . . , gN(xk)) can be extended as a continuous periodic function with period 4N with respect to x1, , xk and as a periodic function with period 2 with respect to t1, , tk. Therefore, for every e > 0 there exists a trigonometric polynomial P(t1, , tk, x1, , xk) in the variables t1,

. , tk, xl,

Xk such that I q,, (t1, ... , tk, gN(x1), ... , gN(xk))

P(t1, ... , tk, x1, .. , xk) I < 6 .

dt 01

7. LIMIT THEOREMS FOR FUNCTIONALS OF INTEGRAL FORM

Since

...

urn I M exp fix

P(t1..., tk,

... f

- M exp {ixJ'

489

n(tl), ..., en(tk))dtl ...

tk, E(t1), ..., E(tk))dtl ... dtk1

o

0

< lim m exp

...

{iX1r0

"P(tl, ..., tk, 6X01 ..., Ef(tk))dtl ... dtk) }

J

01p(

...

- Mexp

// / tk, mm(t1), ... , enltk))dtl ... dtl;

tl,

l(l

0

+ M exp

...

1

01'

tk, e(tl), ..., e(tk))dtl ... dtk} 0

- M exp {ix

1

...

1

0

P(tl, ..., tk, (t1), ..., E(tk))dt1 ...

0 1

1

0

0

+ lim m exp %--

lIi

M 0

tke(t1), ... e(tk))dtl ... dtk11rr

0

o

nm

P(t1, ... , tk, en(t1), ... en(tk))dtl ... dtk

P(t1,

- M exp

dtkl

....,1['P(41 tk, en(t1), ..., Ef(tk)) 0

P(t1, . . . , tk, en(t1), En(tk))]dtl ... dtk

+

li_raM

)11

...

n-.-

0

1((

[ 'P(tl, ..., tk, e.(09 ..., en(tk)) 0

E P(t1, ..., tk, e(t1), ..., e(tk))]dtl ... dtk 1

7

+ lim M exp lixJ l ... :P(ti, ... , tk, R(tl), ... , n(tk))dtl ...

...

- M exp {ix

11P(tl,

..., tk, E(t1), ..., e(tk))dtl ... dtkJ

Jo

J0

dtkJ

< 2(s + ks,,c)

+ lim m exp {ix n-0

...

p(t1, ..., tk, en(t1), ..., n(tk))d4 ... dtk} J0

... p(t1, .. , t, e(t1), ... ,

- M exp {ix

... dto

0

to prove the theorem, it is sufficient to show that the distributions of

... 0

1

P(tl,

E

., tk, e (tl), ..., e, (tk))dtl ... dtk

0

converge to the distribution of 1

0

...

1

P(t1, 0

tk, E(tl), ... , e(tk))dtl ... Utk . 7

LIMIT THEOREMS FOR RANDOM PROCESSES

490

But

... P(tl, ... I tk, 4n(t,), ... n(tk))dtl ... dtk 1

0

0

is a sum of products of the form k

fJ J1g,(m;t).f;(n;En(t))dt ;=1 0

where the g;(x) and the f;(x) are trigonometric functions (sin x or cos x) and the m; and n, are real numbers. Therefore the conclusion

of the theorem follows from the fact that the joint distributions of p;(t, $.(t))dt, for j = 1,

an arbitrary finite set of variables

,1 con-

verge to the joint distribution of the variables T;(t, E(t))dt,

for

j = 1,

,1,

so that k

lim M exp i E),; 1q,(t, En(t))dt} n--

i=1

J

0

k

= lim M exp

X;q;(t, $n(t)dt } 0

=1

= M exp ti0;=1 E X, (t, E(t))dt} in accordance with Theorem 2. 8.

APPLICATION OF LIMIT THEOREMS TO STATISTICAL CRITERIA

One of the problems of mathematical statistics is to determine the distribution function of a random variable from the results of observations.

Suppose that the results of some experiment constitute a random variable with unknown continuous distribution function F(x). How

can we approximate the function F(x) if we know n results of independently performed experiments: $11$., . . . , $n? In mathematical statistics we use for this purpose an empirical distribution function Fn (x) defined by F,* (x) =

V .(X)

n

where vn(x) is the number of variables Ek falling in the interval (- oc , x). It follows from Bernoulli's theorem that the sequence {,*(x)} converges in probability to F(x). Thus we can take the function F(x) Of course we must consider the error that arises when we do this. However it is often

8.

APPLICATION OF LIMIT THEOREMS TO STATISTICAL CRITERIA

491

convenient to have an analytic representation for an approximation of F(x). Then we need to answer the question: can a given function

I(x) serve as an approximation of F(x) if we know the results of In both cases it is important to know , fin? an experiment c1, the behavior of the difference between the empirical distribution function and the theoretical distribution function F(x). We shall give some results that have applications in statistics. Theorem 1 (Kolmogorov's criterion). For all a > 0, lim p{1/ n

sup

I F(x) - Fn (x) I < a} _ = (-

n---<x
1)ke-2k2a2

k=-w

For all a > 0, sup (Fn (x) - F(x)) < a} = 1 - e-2a2

-°°<x
Theorem 3 (the (ot-criterion),

For all a > 0,

lim P{n [Fn (x) - F(x)]ZdF(x) <

al

= G(a) ,

where G(a) is the distribution function whose characteristic function is equal to

Ck

r

1

k27C

2

We shall not show how these results can be applied in statistical investigations, but we shall show how they can obtained by means of the limit theorems for random processes. We introduce the process 72.(x) = 1/ n (Fn(x) - F(x)). Lemma 1. The processes i7,(x) converge weakly to a Gaussian process 7)(x) with the properties M)7(x) = 0 and

F(x)[l - F(y)]

for -oo <x
n kE

n) - F(x)]

where s(z) = 0 for z < 0 and s(z) = 1 for z > 0. Since x) = F(x), M[s(ek - x)s(ek - y)] = F(x) for x < y and since the processes s(Ek - x) - F(x) are independent for different values of k, the proof follows from Theorem 4, Section 6, Chapter I. Corollary. Let F-1(t) denote the inverse of the function F(x). We set e, (t) = i7n(F-1(t)) and e(t) = 7)(F-1(t)) for 0 < t < 1. Then the

LIMIT THEOREMS FOR RANDOM PROCESSES

492

sequence of processes e,(t) converges weakly to a Gaussian process e(t) defined for t e [0, 1] with the properties Me(t) = 0 and M(e(t)e(s)) = t(1 - s) for REMARK 1.

t<s.

The process en,(t) can be represented in the form en(t) =

1

,

77 k=1

[E\'/k - t) -t]

where the )7k = F-1(ek) are independent uniformly distributed random

variables on the interval [0, 1]. REMARK 2. The finite-dimensional distributions of the process e(t) coincide with the conditional finite-dimensional distributions of a process of Brownian motion w(t) for 0 < t < 1, with the hypothesis that w(l) = 0. Since the conditional distributions of the process

w(t) with the hypothesis w(l) = 0 are Gaussian distributions, it is sufficient to show that M(w(t) I w(l) = 0) = Me(t), M(w(t)w(s) I w(l) = 0) = Me(t)e(s)

The random variable fi(t) = w(t) - tw(l) is uncorrelated with w(1). Since fi(t) and w(1) have joint Gaussian distributions, the process fi(t) is independent of w(1). Therefore M(e(t) I w(l)) = Me(t) = 0 , M(e(t)e(s) I w(1)) = M(e(t)e(s))

Using the relation w(t) = fi(t) + tw(l) and the preceding formulas, we obtain M(w(t) I w(1)) = tw(l) , M(w(t)w(s) I w(1)) = M(W(t)N(s)) + ts[w(1)]2

= min [t, s] - is + ts[w(s)]2

Setting w(l) = 0, we see the validity of what was said at the beginning of Remark 2. Lemma 2. For every functional f that is continuous on D[0, I], the sequence of distributions of f(e,t(t)) converges to the distribution of fle(t)).

Proof. We note first of all that the separable process e(t)

is

continuous, so that e(t) with probability 1 belongs to D[0, 1]. This is true because for h > 0, the difference (t + h) - e(t) has a Gaussian distribution and M(e(t + h) - e(t)) = 0 ,

M(e(t + h) - e(t))' = (t + h)(1 - t - h)

+ t(l -t)-2t(l -t-h)=h-h2.

8.

APPLICATION OF LIMIT THEOREMS TO STATISTICAL CRITERIA

493

Therefore M I (e(t + h) - e(t)14 = 0(h2) and the process e(t) is continuous by virtue of Theorem 2. Section 5, Chapter IV. The convergence of the finite-dimensional distributions of the processes en(t) to the finite-dimensional distributions of E(t) has been established. On the basis of Theorem 2 of Section 5, it remains only to show that relation 5 of Section (5) is satisfied. Since Oc(x(t)) < sup I x(t') - x(t") I t'-t--is'

the lemma will be proved if we can show that lim lim p{ C-0 n-.-

sup

It'-t" Isc

I en(t') - en(t") I > a} = 0

(1)

for all s > 0. The process en(t) + 1/n t increases monotonically; that is, for t1 < t2 /< t, < t4, -V -n (t,, (- t1) < ((en( \t3) - S En(t2)

en(t4) - en\t1) + V n (t4 - t) Therefore

sup I en(t') - n(t")

It'-t'' I Sc

<

sup

n1

kl-k2I Sc.2m+2

+2Vn

2'

2m

2m

Suppose that the mn are chosen so that V _n /2mn --. 0 as n , oc and 211-->=l. To prove equation (1), it sufficies to/ show that for every 6>0,

limlimp{ c-+0 nom-

sup I kl/2mmn-k2/2 'mnI SC

n(kl)-stn( k2>s0 \ 2-.) \ 2mn

We note that sup I

(kl/2m,n_k 2/2mn) I5c

n\2 n/ - n\22n/

<2

mn

sup

n\-n (2r)

(With regard to the last inequality, see the proof of Theorem 2, Section 2.) Let us choose a < 1 such that 2a4 > 1. Then where m(c) is the smallest integer such that c2-(c) > 1.

sup

p{

<

'mn

mn

p sup

< AM

1

2r

1

2r-1

i mn

- S. \2mu

e-(2.%)

lI (k1/2'mn-k2/2mn) I sc

nl

l

2r

>

1 - n\2r/ >

) - n (2r)

2

ar- n(c)

1-a}

> 2(1 - a) } 4

1

s

sar-'nW

- S42ir/

2r-1

n{ Z 2r

s}

24(1 - a)4 s4a4(r-m(c))

LIMIT THEOREMS FOR RANDOM PROCESSES

494

that fall in the interval

Let fin denote the number of variables [t, t + h]. Then P{,un = k} = Cnhk(1 -

h)n-k

and

- hl

en(t + h) - en(t) = 11-n (u" \n

/

Calculations show (cf. Gnedenko, 1963.) That h

M(een(t + h) - n(t))4 < 3h2 +

< 3h2 + Zhnn

n

This means that for h > 1/2"`n we have h) - en(t))4 < 4h2

This inequality and the preceding one show that P{

Sup m n) I <=c I (k)/2mn-k2/2 mn

n(2 n)

24(1 - a)4

- n(2 n> I >

4

2r

r-,, (c) c4a4(r-m(c))

<

26(1 - a)4


1

1

)

t

r-0 (2a4)r

64

E 'Lm(c) ,

where

L = 26(1 - a)4 e

E4

m

r-

1

(2a4)r

This completes the proof of the lemma. Let us now proceed to prove Theorems 1 to 3. We note that the functionals Px(t)) = sup I x(t) I 05t51 f2(x(t)) = sup x(t) , OSt51

.f3(x(t)) =

J0

[x(t) ] 2dt

are continuous on D[0, 1]. On the other hand, 1/ n sup

-<x< I Fn (x)

n

- F(x) = fi(rs(t))

sup (Fn (x) - F(x)) =

n(F.(x) - F(x))2dF(x) = nl[FF(F-1(t)) 0

- t]2dt =

Therefore, on the basis of Lemma 2, it is sufficient to show that

a} _

k=- (-

(2)

8.

APPLICATION OF LIMIT THEOREMS TO STATISTICAL CRITERIA

495

p{f2(e(t)) < a} = 1 - e`2

(3)

3cect>> = C 1 -

(4)

Mei

21c, 1-1 k27C 2

k-1

To prove formula (2), we use Theorem 2, Section 5, Chapter VI, from which we easily find the joint distribution of sup05t<1 I w(t) I and w(1): For I x I < a,

p{ sup I w(t) < a, w(l) < x} O
J

[exp { -

(u + 4ka)2}

- exp { - 2 (u + 4ka + 2a)2}]du

.

From this formula we find the conditional distribution of sup0
under the hypothesis that w(l) = 0. This conditional distribution coincides with the right-hand member of equation (2). Analogously, by using Theorem 1, Section 5, Chapter VI, we obtain formula (3).

To prove formula (4), we note that on the basis of formula (18), Section 2, Chapter V, E(t) = 1/ 2

ek

sin ktrt k7r

where the ek are independent normally distributed random variables such that Wk = 0 and DEk = 1. Therefore Cl 1

M

e

k 2( t) dt _ k1 k22r2

exp{iaJ e2(t)dt}

=

- 2ia exp -2x2 k27r2 k2tr2 k=1 -

dx

11

This proves formula (4).

:

2tr

_ k1

k2 7r2 k27r2

2ia

Bibliographic Notes

These notes contain selected references to the literature on the questions discussed in each chapter. Their purpose is not to give a complete biblio-

graphy or to clarify the history of the basic ideas in the theory of random processes. In many cases we refer not to the original works, since they are not always accessible, but instead to later texts and monographs that contain a bibliography on questions we are studying. CHAPTER 1.

§ 1. A systematic study of the questions of the theory of random processes was begun in the works by Slutskiy (1928) and Kolmogorov (1933 b), (1938 a).

A significant role in the construction of the theory of random

processes was played by the works of Kolmogorov. § 2.

The concept of a stationary process

in

the broad sense was

introduced by Khinchin (1937).

§3. The multidimensional generalization of the central limit theorem was first obtained by Bernstein (1946). § 4. Oscillations with random amplitudes and phases have been studied in many works. See, for example, the appendix by Yu. A. Krutkov in the books by Einstein and Smoluchowski, and Bogolyubov. § S, The spectral representation of the correlation function of a process that is stationary in the broad sense was obtained by Khinchin (1938). Its multidimensional generalization was produced by Cramer. The representation of a structural function of processes with increments that are stationary in the broad sense was found by Kolmogorov (19.40 b). The formula for the correlation function of a homogeneous isotropic random field is contained in the work by Schonberg.

CHAPTER 11.

In this chapter the exposition of the theory of measure and the integral was geared to applications in probability theory and the theory of random processes. A more detailed exposition can be found in the books by Saks, Halmos, Kolmogorov and Fomin, and Heider and Simpson. 497

498

BIBLIOGRAPHIC NOTES

CHAPTER III.

§1. The set-theoretic axiomatization of probability theory that is generally accepted at the present time was proposed by Kolmogorov in 1929 and was expounded in his monograph (1933 b). § 2,

The fundamental theorem (Theorem 3) of this section is from

Kolmogorov (1933 b) (in which it is proved for families of random variables).

The more general approach to the question has been proposed by Bochner (1947). § 3.

In connection with the material of this section see Kolmogorov

(1933 b), Halmos and Loeve (1963). In Halmos one can find a theorem on

the existence of a probability space that is a representation of an infinite sequence of independent elements with values in an arbitrary space X. § 4. The basic ideas and results of this section are from Kolmogorov and Khinchin and Kolmogorov (1929 a). Series of independent random variables are studied in greater detail in Doob, and Loeve (1963). § 5. The appearance of ergodic theory is connected with the problems of providing a foundation for statistical mechanics. See Khinchin (1949). The ergodic theorems of von Neumann and Birkhoff served as the starting point of an intensive development of ergodic theory. Khinchin freed the proof of Birkhoff's ergodic theorem from unnecessary assumptions and made possible its application to probability theory problem. A simple proof of the Birkhoff-Khinchin theorem has been presented by Kolmogorov (1938 b). Hopf and Jacobs give good surveys of ergodic theory.

§ 6.

The theory of conditional probabilities and conditional mathematical

expectations was constructed in the monograph by Kolmogorov (1933 b). See also Doob, and Loeve (1963). CHAPTER IV.

§ 1. The problem of constructing a random process that is stochastically equivalent to a given process whose sample functions satisfy specified conditions of regularity was first examined by Ye. Ye. Slutskiy and Kolmogorov (cf. Slutskiy's article [1949]). In the subsequent development

of the problems which arose here and the different variations of the axiomatic definition of a random function, many essential results belong to Doob. References to the original works are found in his monograph.

§2 and 3. The basic results of these sections are from Doob. §4. Theorem I was proved by Chentsov and Theorem 2 was proved by Kinney (for Markov processes). The fact that stochastically continuous processes with independent increments have no discontinuities of the second kind was shown by Levy (1934).

BIBLIOGRAPHIC NOTES

§ 5.

499

Theorem 2 belongs to Kolmogorov and was first published in the

article by Slutskiy (1949). Theorem 4 was proved independently by Dynkin, and Kinney (1952) (for Markov processes). CHAPTER V.

§ 1. An introduction to the theory of Hilbert spaces can be found in Kolmogorov and Fomin, and Heider and Simpson. A more complete exposition can be found in Akhiyezer and Glazman.

§2. Cf. Slutskiy (1923) and Loeve (1948), (1963). §3. The theory of stochastic integrals was proposed by Cramer. Kolmogorov first showed the connection between stochastic integrals and the theory of spectral representations of random functions with the theory of Hilbert spaces (1940 a), (1940b), (1941). §4. Theorem 1 belongs to Karhunen (1947), Theorem 2 to Cramer, and Theorem 3 to Kolmogorov (1940 b). § 5.

By means of the theory of filters, one can easily obtain the

spectral decomposition of stationary processes (cf. Blanc-Lapierre and Fortet).

A more general theory of linear transformations of random processes can be constructed with the aid of the theory of generalized random processes as proposed by I. M. Gel'fand and K. Ito (cf. Gel'fand and Vilenkin, and Ito (1954)). § 6.

The basic results for stationary sequences were obtained by

Those for processes with continuous time were obtained by Harmer, and Karhunen (1950). Kolmogorov (1941).

§ 7. The general formulation of the problem of linear prediction (for stationary sequences), its connection with the geometry of a Hilbert space, and its reduction to a problem in the theory of functions belongs to Kolmogorov (1941). N. Wiener developed methods for the actual solution of problems of linear prediction and filtration for processes with continuous time. A method by A. M. Yaglom with a large number of examples can be found in his survey article.

§8. The theorem on the decomposition of a stationary sequence and the concept of determined and undetermined processes belongs to Wold. The general solution of the problem of the prognosis of a stationary sequence from the entire past was obtained by Kolmogorov (1941) and for processes with continuous time, by Krein (1944), (1954). The problem of prediction

of a vector-valued stationary sequence was studied by Rozanov (1958). Details of the prediction of processes with continuous time can be found in Doob, Grenander and Szego, and Rozanov (1966).

500

BIBLIOGRAPHIC NOTES

CHAPTER V1.

The decomposition of a process into a continuous and a jump component was used by Levy (1934) in the derivation of the characteristic function of the process. The idea was presented by Ito (1942). § 1. Measures constructed from the jumps of a process were introduced by Ito (1942) and (1960).

§2. Theorem 2 belongs to Khinchin (1936), chapter 1, §2. § 3.

The representation of the discontinuous part of a process by

means of stochastic integrals belongs to Ito (1942) and (1960). The general formula for the characteristic function of a process was found by Levy (1934). § 5.

A process of Brownian motion was first considered by Bachelier.

Rigorous construction of this process and study of the properties of its sample functions were carried out by Wiener (1923). Theorem I is from Bachelier, and Theorem 2 follows from the general results of Petrovskiy. § 6. The growth of homogeneous processes with independent increments was studied by Khinchin (1939) and Gnedenko (1943), (1948). The law of

the iterated logarithm for a process of Brownian motion was obtained by Khinchin (1936), Chapter V. CHAPTER VII.

§ 1.

Markov first studied sequences of associated random variables for

which the "future" does not depend on the "past." Kolmogorov's article (1938) served as the basis of the general theory of Markov processes. §2. Theorem 2 is attributed to Kolmogorov (1951). A more detailed exposition of the questions studied in this section and bibliographic references

can be found in the article by Reuter and the monograph by Chung. § 3. Integro-differential equations of jump processes were introduced in Kolmogorov's article (1938). Theorems on the existence of solutions to these equations were considered by Feller (1936), (1940).

§ 4, Numerous examples of Markov processes in natural science can be found in Feller (1966) and Bartlett. § 5.

Branching processes with discrete time were first studied in the

article by Watson and Galton. The general definition of a branching process was given in the article by Kolmogorov and Dmitriyev. The survey article by Sevast'yanov was used for the present section. § 6. The definition of a Markov process was analyzed by Doob. For more general definitions and properties of Markov processes see Dynkin

(1961). § 7.

The structure of sample functions of a homogeneous Markov

BIBLIOGRAPHIC NOTES

501

jump process and also the construction of a process from the function q(t, x, A) were studied by Doob (who gives references to earlier works). A similar investigation of processes with countably many states, including all possible pathological cases, was made by Chung, and a summary of his results are contained in his book. The property proved in Theorem 3 of

a process was called the "condition of strict Markovity" by Dynkin. Dynkin (1961) studied general, strictly Markov processes. For the strict Markovity of homogeneous processes with countably many states, see the monograph by Chung. Theorem 5 and 6 were proved in the article by Reuter. CHAPTER VIII.

A probabilistic treatment of the phenomenon of diffusion was made by Khinchin (1936), Chapter III. Stochastic differential equations for random processes were studied by Bernstein (1934), Gikhman (1950), (1951), and Ito (1946), (1950). Here, we have used the basic terminology and notation of Ito. § 1. Continuous Markov processes and equations for transition probabilities were considered by Kolmogorov (1938). Solutions of Kolmogorov's equations and their generalizations were studied by Feller (1936), (1954, a, b), and Dynkin (1964).

§2.

The basic results of this section are from Ito (1944), (1951).

§3. Equations in this form were studied by Ito (1946), (1950), who proved the theorem on existence and uniqueness and showed that the solution is a Markov process. § 4. Differentiability of the solutions of stochastic equations from initial conditions was shown by Gikhman (1951). § 5, The idea of deriving Kolmogorov's equations from initial conditions by using the differentiability of the solution of a stochastic equation is from Gikhman (1951). The derivation of equations for the distribution of an

additive functional from a process of Brownian motion belongs to Kac (1949), (1951), and in the general case to Dynkin (1955 b). Theorem 4 was proved by Levy (1939). § 6. The application of differential equations to random observations in a bounded region was proposed by Petrovskiy. Diffusion processes in

bounded regions were studied by Khinchin (1936), Chapters III and IV. A general study of various boundary conditions for homogeneous diffusion processes was made by Feller (1954 a). The distributions of certain functionals connected with the time of reaching the boundaries was studied by Khas'minskiy.

BIBLIOGRAPHIC NOTES

502 CHAPTER IX.

The limit theorems for probabilities of events that depend on the entire

trajectory of a process (the probabilities that the process remains in a curvilinear strip) were first studied by Kolmogorov (1931), (1933 a), Petrovskiy, and Khinchin (1934). The first general limit theorem for arbitrary functionals

that are continuous in the metric of C was obtained by Donsker (1951) (in the case of convergence of sums of identically distributed independent random variables to a process of Brownian motion). § 1. Questions of weak convergence of measures in a metric space were studied by Prokhorov (1953), (1956).

§2. The general limit theorem for continuous processes was obtained by Prokhorov (1953), (1956). § 3.

The limit theorems for different particular cases were considered

by Kolmogorov (1931), (1933 a), and Kac and Erdos (1946 a), (1946 b).

The

general theorem belongs to Prokhorov (1953), (1956). A particular case (Theorem 2) was obtained earlier by Donsker (1951). § 4, Specific cases of convergence to diffusion processes were studied by Bernstein (1934), (1946), Khinchin (1936), and Gikhman (1953), (1954).

The theorems for C-continuous functionals were studied by Maruyama, Prokhorov (1954), and Skorokhod (1965), (1963). § 5. The convergence studied in this section was introduced by Skorokhod (1956). An interesting limit theorem for processes without dis-

continuities of the second kind was obtained by Chentsov. § 6. The conditions for convergence of the distributions that we considered in this section follow from the limit theorems for sums of independent random variables. See, for example, Gnedenko and Kolmogorov.

The limit theorem for the probability that a sequence of sums will be contained in a curvilinear band was obtained by Gikhman (1958). General theorems for functionals were studied by Skorokhod (1957) and Prokhorov (1956). § 7.

Smirnov.

With regard to the criteria mentioned, see the survey article by The method given here of proving the criteria of Kolmogorov

and Smirnov belongs to Donsker (1952).

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Index of Symbols

VxX for all xeX, 9 x X there exists an x e X, 0 the empty set A c B B includes A (the event A implies the event B)

union of sets (sum of events) intersection of sets (coincidence of events) (1 A\B the difference between the sets A and B A the complement of the set A {x; Al the set of elements x satisfying the relation A p{A} the probability of the event A M$ the mathematical expectation of the random variable U

C

DC

h I

the variance of the random variable C

a (h 1 a) h approaches a from below (from above) (x, y) the scalar product of vectors x and y for i = j and equal 8;; the Kronecker delta (equal to 1

to 0 for i# j)

511

INDEX

Absolute continuity, 76 Absolute variation, 46 Absorption, process with, 420 Additive set functions, 42 Adjoint, 178 Admissible filter, 212 Admissible functions, 207 Algebra of sets, 41 "Almost-everywhere," 55 Autoregression, process of, 240

Baire function, 53 Banach space, 49 Basis, 177 Bessel's inequality, 177 Birth and death processes, 324 Borel-Cantelli theorem, 112 Borel function, 53 Borel sets, 42 Bounded in probability, 154 Bounded operator, 178 Branching processes, 327 Brownian motion, process of, 21, 282

Caratheodory measurable, 81 Cartesian product, 93 Cauchy condition, 102 Central moment functions, 7 Chapman-Kolmogorov equation, 298

Characteristic functions, of joint distributions, 6 of sets, 52 Charges, 46 Chebyshev's inequality, 103 Coefficient, of correlation, 8 of diffusion, 372 of transfer, 209, 372 Compactness, weak, 441 Compatibility conditions, 2 Complement, orthogonal, 176 Completion, 89 Condition, Lindeberg's, 452 Conditional distribution, 141

Conditional expectation, 134 Conditional probability, 134 regular family of, 139 Continuity, absolute, 76 stochastic, 154 Continuous random process, 169 Convergence, 45 almost certain, 102 almost uniform, 58 in mean, 74 in measure, 56 in probability, 56, 102 in variation, 49 weak, 5, 440 with probability one, 102 Converse equation of Kolmogorov, 373 Convolution, 209 Correlation, coefficient of, 8 Correlation function, 5, 8

joint, io Correlation matrix, ix Covariance functions, 5 Cylindrical sets, io8

Decomposable class, 86 Decomposition, spectral, 203 Delta function, 208 Derivative, mean-square, 183 Determined process, 243, 252 Differential, stochastic, 387 Differential equations, stochastic, 391 Diffusion, coefficient of, 372 Diffusion operator, 376 Diffusion process, 370, 420 Discrete time, 4 Distribution, conditional, 140 Yule-Furry, 322 Distribution function, xox joint, 1

Eigenfunction, 210 Eigenvalue, 210 Elementary measure, 86 513

INDEX

514

Energy, of a random process, 22 Equivalence, stochastic, 145 Equivalent functions, 55 Equivalent random functions, 222 Ergodic stationary sequence, 130 Ergodic theorems, 123 Error, mean-square, 227 Expectation, conditional, 134 mathematical, 61, 102 Extensions of measures, 8o

Jensen's inequality, 103 Joint correlation function, to Joint distribution function, 1, lox Jump process, 312, 347

Kolmogorov, converse equation of, 373 first system of differential equations, 309, 315

general o or i law, 114 second system of differential equations, Fatou's theorem, 69 Filter, admissible, 212 physically realizable, 216 Filtering, of stationary processes, 226 First attainment, instant of, 283 First crossing, instant of, 283 Fokker-Planck equation, 374 Fourier transform, 178 Frequency characteristic, 209, 212 Fubini's theorem, 97 Function of regular growth, 289 Fundamental sequence of functions, 89

Gaussian processes, 13 Generalized function, tog

Hilbert random function, 181 Hilbert spaces, 174 Homogeneous random processes, 273 Homogeneous systems, 207 Hull, linear, 177

Impulse transfer function, 208 Increments, independent, 255 Independence, of events, 111 of random variables, 115 Independent increments, processes with, 255

Input function, 207 Integrable function, 65 Integral, 61, 65 Lebesgue-Stieltjes, 93 stochastic, 19o Ito's, 381 Invariant event, 130 Isometric mapping, 179 Isomorphic mapping, 179 Isotropic random field, 37 Iterated logarithm, law of, 294 Ito's stochastic integral, 381

310, 314 three-series theorem, 121 Kolmogorov-Smirnov theorems, 491

L°, 12 L(?), 5 L1 spaces, 68 Laplace transform, 367 Laplace-Stieltjes transform, 360 Law of iterated logarithm, 294 Lebesgue outer measure, 85 Lebesgue-Stieltjes integral, 93 Lebesgue's theorem, 69 Limit(s), 44 inferior, 44 mean-square, 181 superior, 44 theorems on, 438 Lindeberg's condition, 452 Linear growth, process of, 322 Linear hull, 177 Linear transformations, 174, 207 Lower function, 289

Mappings, 76 isometric, 179 isomorphic, 179 measure-preserving, 124 Markov processes, 344 birth and death, 324 branching, 327 definition of, 343 homogeneous in time, 301 improper, 303 jump, 297 basic properties of, 347 of pure growth, 320 Markov property, 345 Markov systems, 298 Mathematical expectation, 61, 102 conditional, 134 Matrix, correlation, 11 structural, 12

INDEX Mean power, 2z Mean-square derivative, 183 Mean-square error, 227 Mean-square limit, 181 Mean value, 8 Measurable, Caratheodory, 81

Measurable functions, 50 Measurable space, 43 Measure(s), 41

absolute continuity of, 76 completion of, 89 convergence of, 441 elementary, 86 extensions of, 8o outer, 8x sequence of, 441 singular, 77 space with, 43 spectral, 204 stochastic, 190, 204 subordinate, 203 theory of, 40 Measure-preserving mapping, 124

(mod /L), 56

Moment functions, 5 Monotonic class of sets, 45

Negative variation, 46 Norm, 178

515

Radon-Nikodym theorem, 78 Random field, i isotropic, 37 Random function(s), 2, 144 continuous, 169 equivalent, 222 Gaussian, 13 Hilbert, 181 integral representations of, 200 measurable, 156 separable, 150

subordinate, 222

Random process(es), x continuous, 169 limit theorems for, 448 energy of, 22 homogeneous, 273 limit theorems for, 438 linear transformations of, 174, 207 mean power of, 22 Random sequence, 4 Random signals, 319 Random variable(s), lox independent, 115 independent of the future, 356 separable, 149 Random vector, 104 Realizability, physical, 211 Rectangle, 94 Reduction, 179 Regular family of conditional probabilities,

'/

139

Operator, bounded self-adjoint, 178 diffusion, 376 Order, 5 Orthant, 104 Orthogonal complement, 176 Orthogonal vectors, 175 Orthonormal system, 177 Oscillations with random parameters, 22 Outer measure, 81 Lebesgue, 85 Output function, 207

Parseval's equality, 179 Physical realizability, 210, 216 Piecewise-constant process, 274 Positive variation, 46 Prediction, of stationary processes, 226, 241 Probability, bounded in, 159 conditional, 134 convergence in, 102 transition, 297 Probability space, loo construction of, 104 Projection, 176 Pure growth, processes of, 320

Regular growth, function of, 289 Regular process, 243, 252 Representation, 107 Response, 207

a--algebra, 42 complete, 55

Sample function, x Scalar product, 174 Self-adjoint operator, 178 Separability, set of, 151 Separable random function, 150 Separable random variable, 149 Sequence, invariant, 130 of sets, convergent, 45 transitive, 130 uncorrelated, 217 Set function, additive, 42 Sets, algebra of, 41 of continuity, 447 of separability, 151 Simple function, 52 Singular measure, 77 Singular process, 243, 252 Space, measurable, 43 Spectral decomposition, 203

INDEX

516 Spectral function, 25 Spectral representation, 27, 29 Spectrum, 23

Transition probability, 297 Transitive sequence, 130

Stationarily connected processes, 10

Stationary increments, process with, 4 Stationary processes, 4

filtering of, 226

in the broad sense, to prediction of, 226, 241 Stationary sequence, ergodic, 130 Step function, 378 Stochastic continuity, 154 Stochastic differential, 387 Stochastic differential equations, 391 Stochastic equivalence, 145 Stochastic integrals, 190 Stochastic measures, 19o subordinate, 2o2 Stochastic spectral measure, 203 Structural function, 12 Structural matrix, 12 Subordinate random function, 222 Subspace, 176 Symmetric process, 289

Uncorrelated sequence, 217 Undetermined process, 243, 252 Unitary operator, 178 Upper function, 289

Variation, absolute, 46 convergence in, 49 negative, 46 of a function, 279 positive, 46 Vector, length of, 175 orthogonal, 175 random, 104

Weak compactness, 441 Weak convergence, 5 of measures, 440 Trace,

29

Transfer, coefficient of, 210, 372 Transform, Fourier, 178 Laplace, 367 Laplace-Stieltjes, 360

Wiener process, 21

Yule-Furry distribution, 322