Fourier Analysis in Probability Theory (Probability and mathematical statistics)

Fourier Analysis in Probability Theory

~robability

and Mathematical Statistics

A Series of Monographs and Textbooks

Editors

Z. W. Birnbaum

E. Lukacs

University of Washington Seattle, Washington

Catholic University Washington, D.C.

1. Thomas Ferguson. Mathematical Statistics: A Decision Theoretic Approach. 1967 2. Howard Tucker. A Graduate Course in Probability. 1967 3. K. R. Parthasarathy. Probability Measures on Metric Spaces. 1967 4. P. Revesz. The Laws of Large Numbers. 1968 5. H. P. McKean, Jr. Stochastic Integrals. 1969 6. B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev. Mathematical Methods of Reliability Theory. 1969 7. Demetrios A. Kappos. Probability Algebras and Stochastic Spaces. 1969,1970 8. Ivan N. Pesin. Classical and Modern Integration Theories. 1970 9. S. Vajda. Probabilistic Programming. 1972 10. Sheldon M. Ross. Introduction to Probability Models. 1972 11. Robert B. Ash. Real Analysis and Probability. 1972 12. V. V. Fedorov. Theory of Optimal Experiments. 1972 13. K. V. Mardia. Statistics of Directional Data. 1972 14. H. Dym and H. P. McKean. Fourier Series and Integrals. 1972 15. Tatsuo Kawata. Fourier Analysis in Probability Theory. 1972

In Preparation Fritz Oberhettinger. Fourier Transforms of Distributions and Their Inverses: A Collection of Tables L. E. Maistrov. Probability Theory: A Historical Sketch

FOURIER ANALYSIS IN PROBABILITY THEORY TATSUO KAWATA Department of Engineering Science Faculty of Engineering Keio University, Yokohama, Japan

ACADEMIC PRESS New York and London

1972

CoPYRIGHT 0 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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INC.

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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWI

LIBRARY OF CONGRESS CATALOG CARD NUMBER:

AMS(MOS) 1970 Subject Classification: 60-01 PRINTED IN THE UNITED STATES OF AMERICA

72-84279

Contents

PREFACE.

I

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xi

Introduction

1.1 1.2 1.3 1.4 1.5 1.6

1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16

Measurable Space; Probability Space . Measurable Functions; Random Variables. Product Space . . . . . . . Integrals The Fubini-Tonelli Theorem ·Integrals on R 1 Functions of Bounded Variation Signed Measure; Decomposition Theorems The Lebesgue Integral on R 1 Inequalities . . . Convex Functions . . . . . Analytic Functions . . . . . Jensen's and Carleman's Theorems . Analytic Continuation . . . . . . . Maximum Modulus Theorem and Theorems of Phragmen-Lindelof Inner Product Space . . . . . v

1 6 7 8 12 13 16 18

20 24

27 28 31 32 35 37

vi

CONTENTS

II Fourier Series and Fourier Transforms 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

The Riemann-Lebesgue Lemma Fourier Series . . . . . . . . . . . . . . . . . The Fourier Transform of a Function in Ll(-oo, oo) Ma~tude of Fourier Coefficients; the Continuity Modulus More about the Magnitude of Fourier Coefficients Some Elementary Lemmas Continuity and Magnitude of Fourier Transforms Operations on Fourier Series . . . . . . Operations on Fourier Transforms . . . . Completeness of Trigonometric Functions Unicity Theorem for Fourier Transforms . Fourier Series and Fourier Transform of Convolutions. Notes . . . . . . . . . . . . . . . . . . . . . . .

43 49 52 53

56 59 62 64 67

69 73 74 78

III Fourier-Stieltjes Coefficients, Fourier-Stieltjes Transforms and Characteristic Functions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Monotone Functions and Distribution Functions Fourier-Stieltjes Series . . . . . . . . . . . . Average of Fourier-Stieltjes Coefficients Unicity Theorem for Fourier-Stieltjes Coefficients Fourier-Stieltjes Transform and Characteristic Function Periodic Characteristic Functions . . . . . . . . . . . Some Inequality Relations for Characteristic Functions . Average of a Characteristic Function Convolution of Nondecreasing Functions . . . . . The Fourier-Stieltjes Transform of a Convolution and the Bernoulli Convolution Notes . . . . . . . . . . . . . .

81 84

86 88 89 93 95 102

104 108 111

IV Convergence and Summability Theorems 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Convergence of Fourier Series . . . . . . . . . Convergence of Fourier-Stieltjes Series . . . . . Fourier's Integral Theorems; Inversion Formulas for Fourier Transforms Inversion Formula for Fourier-Stieltjes Transforms Summability . . . . . . . . . . . . (C,1)-Summability for Fourier Series . Abel-Summability for Fourier Series .

113 121 123 1:l8 131 135

140

vii

CONTENTS

4.8 4.9 4.10 4.11

V

Summability Theorems for Fourier Transforms . . . Determination of the Absolutely Continuous Component ing Function . . . . . . . . . . . . . . . . . . Fourier Series and Approximate Fourier Series of a Transform . . . Some Examples, Using Fourier Transforms . Notes . . . . .

. . . . . . . . of a Nondecreas. . . . . . . . Fourier-Stieltjes

145 151 154 158 164

General Convergence Theorems

5.1 Nature of the Problems . . . . . . . . 5.2 Some General Convergence Theorems I . 5.3 Some General Convergence Theorems II 5.4 General Convergence Theorems for the Stieltjes Integral 5.5 Wiener's Formula . . . . . . . . . . . . . . . . . . 5.6 Applications of General Convergence Theorems to the Estimates of a Distribution Function Notes

VI

£2-Theory of Fourier Series and Fourier Transforms

6.1 6.2 6.3 6.4 6.5 6.6

Fourier Series in an Inner Product Space . . . . Fourier Transform of a Function in L 2(-oo, oo). The Class H 2 of Analytic Functions A Theorem of Szego and Smirnov . The Class .p• of Analytic Functions A Theorem of Paley and Wiener Notes

166 167 174 178 181 186 192

194 201 210 214 218 228 231

VII Laplace and Mellin Transforms 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

The Laplace Transform . . . . . . . . . The Convergence Abscissa . . . . . . . . Analyticity of a Laplace-Stieltjes Transform Inversion Formulas for Laplace Transforms The Laplace Transform of a Convolution . Operations of Laplace Transforms and Some Examples The Bilat~ral Laplace-Stieltjes Transform Mellin-Stieltjes Transforms The Mellin Transform Notes

232 238 242 246 252 259 265 269 273 276

Vlll

CONTENTS

VIII More Theorems on Fourier and Laplace Transforms 8.1 A Theorem of Hardy . . . . . . . . . . . . . . . . . . . 8.2 A Theorem of Paley and Wiener on Exponential Entire Functions 8.3 Theorems of Ingham and Levinson . . . . 8.4 Singularities of Laplace Transforms . . . . 8.5 Abelian Theorems for Laplace Transforms . 8.6 Tauberian Theorems 8.7 Multiple Fourier Series and Transforms . 8.8 Nondecreasing Functions and Distribution Functions in Rm 8.9 The Multiple Fourier-Stieltjes Transform Notes . . . . . . . . . . . . . . . .

278 284 288 298 301 304 314 323 326 328

IX Convergence of Distribution Functions and Characteristic Functions 9.1 9.2 9.3 9.4 9.5 9.6 9.7

Helly Theorems and Convergence of Nondecreasing Functions Convergence of Distribution Functions with Bounded Spectra Convergence of Distribution Functions . . . . . . . . . . . Continuous Distribution Functions: A General Integral Transform of a Characteristic Function . . . . . . . . . . . . . . : A Basic Theorem on Analytic Characteristic Functions . . . Continuity Theorems on Intervals and Uniqueness Theorems The Compact Set of Characteristic Functions Notes . . . . . . . . . . . . . . . . . . . .

330 339 341 353 356 357 362 365

X Some Properties of Characteristic Functions 10.1 Characteristic Properties of Fourier Coefficients . . . 10.2 Basic Theorems on Characterization of a Characteristic Function 103. Characteristic Properties of Characteristic Functions 10.4 Functions of the Wiener Class 10.5 Some Sufficient Criteria for Characteristic Functions 10.6 More Criteria for Characteristic Functions Notes . . . . . . . . . . . . . . . . . . . . .

366 372 376 383 385 392 397

XI Distribution Functions and Their Characteristic Functions 11.1 11.2 11.3

Moments, Basic Properties . . . . . . . . . . . . . . 400 Smoothness of a Characteristic Function and the Existence of Moments 408 More about Smoothness of Characteristic Functions and Existence of Moments. . . . . . . . . • . . . . . . . . . • . . . . . . . 419

CONTENTS

11.4 Absolute Moments 11.5 Boundedness of the Spectra of Distribution Functions 11.6 Integrable Characteristic Functions . . . . . . . . 11.7 Analyticity of Distribution Functions . . . . . . . Mean Concentration Function of a Distribution Function 11.8 11.9 Some Properties of Analytic Characteristic Functions 11.10 Characteristic Functions Analytic in the Half-Plane 11.11 Entire Characteristic Functions I . 11.12 Entire Characteristic Functions II Notes . . . . . . . . . . .

IX

428 431 437 439 445 452 456 460 466 473

XII Convergence of Series of Independent Random Variables 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12

Convergence of a Sequence of Random Variables The Borel Theorem . . . The Zero-One Law . . . . The Equivalence Theorem . The Three Series Theorem Sufficient Conditions for the Convergence of a Series Convergence Criteria and the Typical Function . . Rademacher and Steinhaus Functions . . . . . . Convergence of Products of Characteristic Functions Unconditional Convergence . Absolute Convergence Essential Convergence Notes . . . . . . .

475 486 488 492 496 501 504 507 513 519 526 530 534

XIII Properties of Sums of Independent Random Variables; Convergence of Series in the Mean Continuity and Discontinuity Properties of the Sum of a Series Integrability of the Sum of a Series . . . . . . . . . . . . . Magnitude of the Characteristic Functions of the Sums of Series Distribution Functions of the Sums of Rademacher Series; Characteristic Functions of Singular Distributions . . . Further Theorems on Rademacher Series 13.5 13.6 .Sums of Independent Random Variables . 13.7 Convergent Systems . . . . . . . . . . 13.8 Integrability of Sums of Series; Strong and Weak Convergences of Series 13.9 Vanishing of the Sum of a Series 13.10 Summability of Series Notes . . . . . . . . . . . . 13.1 13.2 13.3 13.4

536 543 546 556 565 573 583 586 590 596 602

X

XIV 14.1 14.2 14.3 14.4

CONTENTS

Some Special Series of Random Variables Fourier Series with Rademacher Coefficients Random Fourier Series . . . . . . . . . Random Power Series, Convergence . . . Convergence of Random Power Series with Identically and Independently Distributed Random Coefficients . . . . . . . . . Analytic Continuation of Random Power Series . . Fourier Series with Orthogonal Random Coefficients Notes . . . . . . . . . . . . . . . . . . . .

625 628 634 640

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

642

INDEX • •

661

14.5 14.6

• •

• • • • •

• •

• • •

• • • •

• • • •

• • •

• • •

• • • •

•

605 612 619

Preface

The methods and results of Fourier analysis have been effectively utilized in the analytic theory of probability. Moreover, simple analogs of some results in Fourier analysis have actually given rise to many significant results in probability theory. However, one often hears the complaint that in seeking pertinent results from Fourier analysis which are needed in the study of probability, the standard texts give a presentation that is, in most cases, too detailed to be useful. The author's primary purpose, therefore, was to present useful results from the theories of Fourier series, Fourier transforms, Laplace transforms, and other related topics, in a fashion that will enable the student easily to find the results and proofs he desires before he proceeds to more detailed investigations. To further this purpose, particular attention has been given to clarification of the interactions and analogies among these theories. Chapters 1-8, present the elements of classical Fourier analysis, in the context of their applications to probability theory. This is done in a comprehensive but elementary fashion. Chapters 9-14 are devoted to basic results from the theory of characteristic functions of probability distributors, the convergence of distribution functions in terms of characteristic functions, and series of xi

xu

PREFACE

independent random variables. It is hoped that this presentation will help the reader better to understand the workings of Fourier analysis. It should also serve to exhibit some detailed classical results from these fields. The use of Fourier analysis has spread into almost all parts of probability theory, and it would be impossible to discuss all of these areas within the scope of this book. For this reason, certain limitations have been imposed. Some important and recent results such as the almost everywhere convergence of the Fourier series of a function of LP( -:n, :n), p > 1 could not be included; even LP and HP theories with 1 < p, p =I= 2 were not given since it was felt that to be too ambitious in scope would jeopardize the use of the book. One particular omission, contained in the original plan of the book, is especially regretted by the author. That is the strong analogy between the theory of Fourier series with gaps, or more general series with properties similar to gap conditions, and the theory of series of independent random variables. I am very grateful to Professor E. Lukacs for having given me the opportunity to write this book and for his unfailing encouragement and stimulation during my nine years as a professor at the "Catholic U niversity of America. I should like to extend my appreciation also to Dr. G. R. Andersen, Mr. N. Monsour, and Dr. B. McDonagh who read the manuscript, helped me in many respects and corrected the mistakes. My thanks are also given to Mrs. J. Lawrence for typing the most part of the manuscript, Mrs. J. Schafer for her help in many respects, and the Catholic University of America and the National Science Foundation (GP-6175) for financial support.

I Introduction

In this chapter we summarize the basic notions and results from real and complex function theories and probability theory, that will be used in this book.

1.1. Measurable Space; Probability Space Let D be a set of elements (space) and let d be a class of subsets of D. Disregarding the trivial case, we assume that D is nonempty. Suppose that d contains the empty set 0 and is closed with respect to complements and with respect to countable unions; that is, (i) if E e .N, then the complement E0 of Eisa set of d; (ii) if {En, n = 1, 2; ... } is a countable sequence of sets of .N, then

Then d is called a a-field.

I.

2

INTRODUCTION

.N contains the space D. Let C be any class of sets of D. Then there is the smallest u-field .N which contains C, that is, for any u-field M with M ::::::> C, J;l/' c M holds, and .N is called the minimal u-field over C. If S is the class of all subsets of D, then S is a u-field containing C and dis given by the intersection of all u-fields containing C. Consider a fixed u-field .N. A couple (D, d) is called a measurable space. Any set of D which belongs to dis called a measurable set. A nonnegative set function p(E), defined for all sets of .N that satisfy (1)

for any sequence Ei of disjoint measurable sets, is called a measure on a measurable space (D, p); p(E) could be +oo. We suppose, disregarding the extreme case, that for some E e .N, p(E) < oo. The property (1) is also referred to as the countable additivity of a set function p(E). From (1) we may derive the basic properties of the measure: (i) p(0) = 0. (ii) If E 10 E 1 e .N, and E 1 c E 2 , then

p(E1) p(E1)

-

~

(2)

p(E2 ),

p(E1) = p(E2

-

(3)

E1)·

(4)

(iii) If {En, n = 1, 2, ... } is a sequence of sets of .N (not necessarily disjoint), then (5)

(iv) Suppose that lim En= E,

n-+oo

Ene.N'.

n = 1, 2, ... ,

that is,

E =lim sup En= lim inf En, n-+oo n-+oo where lim sup En =

n-+oo

lim inf En =

n-+oo

00

00

n-1

k-n

00

00

n-1

k-n

0 U E, U 0

Et.

Then E e .N and lim p.(En) = p.(E).

(6)

n-+oo

A triplet (Q,.N, p.) is a measure space (Q,.N) together with a measure p. defined on d. If p.(Q) < oo, then p. is called a finite measure. In this case, p.(E) < oo for all E e d. If there is a sequence En, n = 1, 2, ... , in ..N such that

and p.(En) < oo, n = 1, 2, ... , then p. is called a a-finite measure. In probability theory, we deal with a measure space (Q, .N, P), where Pis a measure with P(Q) = 1. This triplet is called the probability space and the measure Pis called the probability or the probability measure. Let 91 be a class, containing the empty set 0, which is closed with respect to complements and with respect to finite unions, that is, Ec e 91 with E e 91 and for any finite sequence {En , n = 1, 2, ... , m} of sets of91,

Then 91 is called a field. A measure m(E) on a field 91 is, by definition, a nonnegative set function defined on 9J such that (i) (ii) that

m(0) = 0, if {En, n = 1, 2, ... } is a sequence of disjoint sets of 91 such

then (7)

where m(E) may be +oo. A finite or a-finite measure on 9J is similarly defined as in the case of measure on a a-field. We have the following extension theorem.

Theorem 1.1.1. Let m be a measure on a field .9J in D. Then there exists a measure p. defined on the minimal a-field ..N over 9J such that p.(E)

=

m(E)

for any

E e 91.

I.

INTRODUCTION

p is finite or a-finite according as m is finite or a-finite. Such extension

is unique if m is a-finite. (See, for example, Royden [1], p. 219.) A measure space ( D, d, p) is called complete if d contains all subsets of sets whose measure is zero; that is, if E e d and p.(E) = 0, then any F c: E is contained ins/. Of course p.(F) = 0. In this case, pis called a complete measure.

Theorem 1.1.2. Let (D,.N', p.) be any measure space. Then there always exists a complete measure space (D,..r¥'0 , p. 0 ) such that (i) d c: d 0 , (ii) for any E Ed, p.(E) = p 0 (E). Actually, .s/0 is defined from .1¥' by adding to .1¥' all the sets of the form E u F, where E e d, F is any subset of a set of d of measure zero, and p.0 is defined to be p 0 (E u F) = p(E), p 0 (E) = p(E) for any E e d and any F a subset of a set of d of measure zero. Now we shall discuss the special case in which D is the Euclidean space. A set of points x = (x 1 , x 2 , ••• , Xn) in the n-dimensional space Rn, such that ai <xi< bi, i = 1, 2, ... , n, where ai, bit i = 1, 2, ... , n are given numbers, is called an open interval in Rn. The unique minimal a-field over the class consisting of all open intervals is called the Borel a-field or simply the Borel field. Sets of the Borel field are called Borel sets or Borel measurable sets. Consider the class C of all sets obtained by the finite number of set operations (complements, unions, intersections) on open intervals in Rn. Obviously, Cis a field (not a a-field). It is also obvious that the minimal a-field over C is identical with the Borel field.~. Let F(x 1 , x 2 , ••• , xn) be a finite-valued function defined over Rn. The nth difference of F is defined by L1!" L1!~i 1

••

·L1~ 1 F(xl,. .. ' Xn-l' Xn) = F(xl+hl,. .. ' Xn-l+hn-1> xn+hn) -F(x 1 , x 2 +h 2 , ••• , Xn+hn)- · · · -F(x1+h 1 , •• • , Xn-l+hn-l, Xn)

+···+··· + (-1)nF(xl>x

2 , ...

,xn);

that is,

F(x 1 , • •• , xi- I, xi + hi, xi+l, . .. , Xn) - F(x 1 ,. • • , xi•· . . , Xn), • L1~'F = L1!"(L1~i'( · · · L1~'F) · • ·).

..1:; F(x1 ,. •• , Xn) = L1!" Ll!~i 1

• •

(8)

1.1.

5

MEASURABLE SPACE; PROBABILITY SPACE

Suppose that

F(x 1, x 2 , ••• , xn) is nondecreasing for each xi, i = 1, 2, ... , n; Ll!" Ll~i'· · ·Lf~'F(x 1 , •• • , Xn) ~ 0 for h1 ,. •• , hn ~ 0.

(i) (ii)

Now define the set function C/>(E) on C in such a way that

C/>(E) = F(a1 + h1 - 0, ... , an-1 + hn-1 - 0, an + hn - 0) - F(a 1 + 0, a 2 + h 2 - 0, ... , an + hn - 0) - · · · - F(a 1 + h 1 - 0, ... , an_ 1 + hn_ 1 - 0, an + 0) + •·· + (-1)nF(a 1 + 0, a2 + 0, ... , an+ 0)

(9)

for intervals E: ai < xi < ai + hi, i = 1, 2, ... , n, and extend C/>(E) for all sets of C to be additive in a natural way. We then easily show that C/>(E) so defined on C is a measure on the field C. Then, in view of Theorem 1.1.1, we may define a measure cJ> on the Borel field .5YJ. cJ> is called a Borel measure. Since F is finite-valued for any point in Rn , the measure cp is a-finite; and, if lim

lim :1:1~-xl, ..•

,zn.-+-oo

are finite and hence F(x 1, ... , xn) is bounded, the measure cp is a finite measure. Let (Rn, 2', C/>0 ) be the completion of (Rn, .5?1, C/>) (the complete measure space). The measure C/>0 is called the Lebesgue-Stieltjes measure. A set of.:? is called a Lebesgue measurable set and .:? the class of Lebesgue measurable sets. When the completed measure C/>0 is specialized in such a way that C/>(E) = h1h2 • • ·hn for E ={a;< xi< ai + h;, i = 1, 2, ... , n}, it is called the Lebesgue measure on Rn. Throughout this book the Lebesgue measure of a set E will be denoted by m(E) or mE. If, besides, the conditions (i) and (ii) of F(x 1, x 2 ,. •• , Xn), F(x 1, x 2 , ... , xn) satisfy the conditions that (iii) limzr+-oo F(x 1 , x 2 ,. •• , Xn) = 0 for each i, (iv) limz,-+oo, ... ,zn-+oo F(x1 , x 2 , ••• , Xn) = 1, and (v) F(x 1, . .. , xn) is left-continuous for each xi, then F(x 1 , x 2 , ••• , xn) is called a distribution function. Condition (v) is emphasized in probability theory. However in analytical considerations, (v) is sometimes ignored, since the measure cJ> gen-

6

I.

erated by F is not affected by whether (v) Section 9.1.)

IS

INTRODUCTION

assumed or not. (See

1.2. Measurable Functions; Random Variables Let a measurable space (.Q,d, p,) be given. In the following, the letter w is used to denote an element of .Q. A real-valued function /( w) is supposed to be defined on .Q. The following definitions and properties of functions are mostly true for extended real-valued functions that may assume the values ±oo; but for simplicity, we assume throughout that a function is finite valued for any we .Q. A function f( w) is called a measurable Junction if, for every real number x, the set of {w :f( w) < x} is a set of d. [ {w :f( w) < x} denotes the set of w for which /( w) < x holds.] It is noted that the con clition {w :f( w) < x} E d for every x is equivalent to each of the conditions: {w:.f(w) ~ x} Ed for {w :f( w) > x} E ~c/ for {w:J(w) ~ x} E ~c/ for {w :f( w) E A} E d for

each each each each

x, x, x, Borel set A on R 1 •

A measurable functionf(w), defined on a probability space {.Q,d, P}, is called a random variable. The probability measure of a set on which a random variable/( w) assumes values of a Borel set A is called the probability that f(w) assumes values of A and is denoted by P(f(w) E A). Similar notations are used for more general propositions. If f(w) and g(w) are measurable and c is a constant, then f(w) c, g(w) are measurable, and if f(w)fg(w) is cf(w), f(w)g(w), and f(w) finite, then it is measurable. A functionf(x 10 ••• , Xn) which is measurable on (Rn,.9.9, p,), where~~ is the Borel field on Rn , is called a Borel measurable function or a Borel function. If f(x) is a Borel function on R 1 and g(w) is measurable function on (.Q,d, p,), thenf[g(w)] is measurable on (.Q,d, p,). If a proposition A holds except on a set of measure zero of a measurable space (.Q, d, p, ), then A is said to hold almost everywhere or almost everywhere (p, ), and it is sometimes described as: A holds a. e. or a. e. (p, ). If p, is a complete measure and/( w) is a measurable function, then a function g( w), which is equal to J( w) almost everywhere (p, ), IS also measurable.

+

+

1.3.

7

PRODUCT SPACE

So far we have talked about measurable functions that assume values in R 1 (the real-valued measurable functions). Similar definitions are applied to vector-valued functions or multivariate functions on (Q, d, p. ). A finite functionf(w) = [/1 (w),j2 (w), ... ,fm(w)] defined on (Q,d, p.) is measurable if the set of w such that [Ji w ), ... , fm ( w)] E A, A being any Borel set in Rn, is measurable; that is, if d, then the multivariate function f( w) is called measurable. This definition is equivalent to saying that {w:f(w) ~ x} E J:Y'for each x, where x = (x 1 , ••• , xm) andf(w) ~ x means fi( w) < xi, i = 1, 2, ... , m. The above properties of real-valued measurable functions continue to hold for vector-valued measurable functions. The m-variate random variables on a probability space are defined similarly. Let/1 (w),j2 (w), .. . ,fm(w) be m real-valued random variables. If, for any set x 1 , ••• , Xm of m real values, m

P[j1 (w) < xl>j2 (w) <

X 2 , •••

,Jm(w)

< Xm] =

Il P[Ji(w) <xi]

(1)

i-1

holds, then / 1 ( w ), ... ,Jm ( w) are called independent, and {fi( w ), i = 1, 2, ... , m} is a set (a sequence) of independent random variables. Let f( w) be a random variable. Then the function of x in R 1 ,

F(x)

= P[J(w) < x]

(2)

defines a distribution function on R 1 , that is, a nonnegative, nondccreasing, left-continuous function with F(- oo) = 0, F( + oo) = 1 and F(x) is called the distribution function off( w ). For m-variate random variables/( w) = [/1 ( w ), ... ./m( w )], the function on Rm

is the distribution Junction on Rm off( w ).

1.3. Product Space Let (Q 1 , ~cY'1 ), (Q 2 , ~o/2 ), ••• , (Q,, ~cY'n) be an n-measurable space, ..Ni being a a-field in Qi, i = 1, 2, ... , n. The set of elements ( w1 , w2 , ••• , wn) is called a product set of A 1 , A 2 , ••• , A,, where w; E Ai (Ai is a subset of Qi, i = 1, 2, ... , n) and is denoted by A 1 X A 2 X··· X An. D 1 X D 2 X • • • X Qn is called the product space of D 1 , D 2 , ••• , Qn.

I.

8

INTRODUCTION

If D, = R 1 , i = 1, 2, ... , n, then D 1 X • • · X Dn = R 1 X ···X R 1 is the n-dimensional space Rn. Consider the class of all product sets A 1 X A 2 X • • • X An such that A i e ..Ni, i = 1, 2, ... , n. The minimal a-field over this class is called the product a-field of ..N1 , ..N2 , ••• , ..Nn and is denoted by

d=dt X dz X··· X dn. In particular, when D, = R 1 , ~ is the Borel field !JfJ, i = 1, 2, ... , n, then..N is no more than the Borel field in Rn. Note that if..N = Ji"1 X··· X ..Nn and 1 ~ k < n, then ..N = (..N1 X ..N2 X • · • X .N'k) X (.N'k+l X··· X dn)· Now consider n-measurable spaces (D 1 , .N'1 , ,u 1 ), (D 2 , ..N2 , ,u 2 ), ••• , (Dn, dn, ,Un)· Define the measure of the product set A 1 X A 2 X • • · X An for any Ai e ~. i = 1, 2, ... , n, by

It is easily seen that the class of sets obtained by a finite number of set operations on product sets of A 1 , A 2 , ••• , An, Ai E ..Ni, (i = 1, 2, ... , n) constitutes a field and on this field the measure is defined generated by (1 ), and hence by Theorem 1.1.1, we can define the measure of any set of ..N=..N1 X .N'2 X··· X ..Nn. This measure ,u on (D,..N), (D = D 1 X··· X Dn), is called the product measure and (D, d, ,u) is called the product measure space. If each .Ui is a probability measure, then (D, d, ,u) is called the product probability space.

1.4. Integrals Let a measurable space (D,d, ,u) be given. Throughout this section,

,u is supposed to be complete. A set function IE(w) defined on D 1 , which assumes 1 on a set E and 0 on Ec, is called the indicator or the characteristic Junction of the set E. A real-valued function defined on D that assumes a finite number of real values is called a simple Junction. A simple function J( w) can be described as n

J(w) =

L x;!E (w), 1

i-1

where {Ei, i = 1, 2, ... , n} is a finite set of disjoint sets of D. A simple function is measurable if and only if the sets Ei are all measurable. The

1.4.

9

INTEGRALS

above representation of a simple function f( c.o) is not unique when the xi are not necessarily distinct. In order to define the Lebesgue integral of a measurable function on D, we begin with the case in which p. is a finite measure (although the direct definition without this restriction is possible, which might be more elegant; for instance, see Royden [1 ], p. 196 ). Suppose that {) is a finite measure space. Let n

f(c.o) =

L xJE,(c.o) i=l

(1)

be a given measurable simple function. (i) The integral of f(c.o) over D is defined by (2)

The representation (1) of a simple function f(c.o) is not unique, but the value on the right-hand side of (2) is uniquely determined and it serves as the definition of J0 f( c.o) dp.. The value of the integral is always finite since p.(Ei) < oo. (ii) Letf(c.o) ~ 0 be a nonnegative measurable function. It is shown that there is a nondecreasing sequence of nonnegative simple functions fn( c.o) which converges to f( c.o) for every c.o. Using the definition (2) for the integral of fn( c.o ), we define the integral off( c.o) by

f

f(c.o) dp. =lim

D

n-+oo

J fn(c.o) dp..

(3)

D

The limit on the right always exists as a finite or an infinite value, and we may show that the limit is determined independently of the choice of Un( c.o)} of simple functions. The integral J0 f( c.o) dp. defined above is nonnegative. If the right-hand side of (3) is finite, then f( c.o) is said to be integrable or it is said that the integral off( c.o) exists. (iii) Let f(c.o) be any measurable function on (D,st', p.). Let f+(c.o) and j-( c.o) be the nonnegative and negative parts off( c.o ), respectively; that is,

f+(c.o) = { {(c.o)

on on

{c.o:f(c.o) ~ 0}, {c.o:f(c.o) < 0};

j-(w) = { -;;J(c.o)

on on

{c.o :f( c.o) < 0} , {c.o:f(c.o) ~ 0}.

I.

10

INTRODUCTION

We then havef(co) = j+(w)- j-(w). By (ii), Jnf+(w) df.J and J0 j-(w) df.J are defined. When one of these integrals is finite, we define the integral ofj(w) by (4)

This may be +oo or -oo. If both fnf+(w) df.J and fnJ-(w) df.J are finite, then f(w) is said to be integrable. This definition implies that if f( w) is integrable, then If( w) I is also integrable. The integral of a measurable function f( w) on a measurable set E is defined by (5)

Iff( w) is integrable, then it is integrable on any measurable set, namely, JE f( w) df.J is finite. We easily see that any bounded measurable function is integrable, and that ifj(w) andg(w) are measurable such thatf(w) = g(w) a.e., then the integrability of one of them implies the integrability of the other and the two integrals are of the same value. Next we define the integral over a a-finite measure space. . Let f( w) ~ 0 be measurable. Then we may show that there is a nondecreasing sequence Un(w ), n = 1, 2, ... } of simple functions, eachfn(w) vanishing outside a set of finite measure such that fn( w) converges to f(w) at every w of D. We define the Lebesgue integral of f(w) by

f

D

f(w) df.J = lim n-+oo

J

fn(w) df.J.

(6)

D

Again we may show that the value of the right-hand side of (6) is determined independently of the choice of simple functions fn( w) with properties described above. For a general measurable functionf(w), we define

as before, where at least one of the right-hand integrals is finite. If both the integrals are finite, then f( w) is called integrable. The integral f E f( w) df.J is defined as before, where E is any measurable set. The following elementary properties hold for integrals on a a-finite measure space (D,d, f.J). (i)

If f(w) and g(w) are integrable functions, then cd(w)

+ c~(w)

1.4.

11

INTEGRALS

is integrable and

where c1 and c2 are constants and E is any measurable set. (ii) If E 1 and E 2 are disjoint measurable sets, then

(iii)

Ifj(w)

~g(w)

a.e., then

f

D

(iv)

j( W) dp,

~

f

D

g( W) dp,.

With f(w), lf(w) I is integrable.

The following convergence theorems are frequently used. Suppose throughout, as before, that p, is a a-finite measure.

Theorem 1.4.1 (Fatou's Lemma). If Un(w), n = 1, 2, ... } is a sequence of nonnegative measurable functions and fn( w) converges to a measurable function a.e. on a measurable set E, then (7)

Theorem 1.4.2 (Dominated Convergence Theorem). Let Un(w) n = 1, 2, ... } be a sequence of measurable functions such that lfn(w) I ~g(w),

(8)

where g( w) is integrable on a measurable set E and such that fn( w) converges almost everywhere on E to a measurable function f( w ). Then

(9) In particular, we have:

Theorem 1.4.3 (Bounded Convergence Theorem). Suppose that p, is a finite measure. If a sequence {fn(w), n = 1, 2, ... } converges a.e. on a measurable set E to a measurable function f( w) and fn( w) is uniformly

I.

12

INTRODUCTION

bounded on E:

1/n(w} I~ K, K being a constant, then (9) holds.

Theorem 1.4.4 (Monotone Convergence Theorem). Let {/ w ), n = 1, 2, ... } be a nondecreasing sequence of nonnegative measurable functions converging a.e. on a measurable set E to a measurable function f(w). Then (9) holds. 11 (

The integrals in (9) in this case could be infinite. However, the theorem says that if either side of (9) is finite, then the other hand side exists as a finite value and the equality holds. A consequence of Theorem 1.4.4 is:

Corollary 1.4.1. Let {/ w ), n = 1, 2, ... } be a sequence of nonnegative measurable functions on a measurable set E. Then 11 (

(10)

1.5. The Fubini-Tonelli Theorem Suppose two measurable spaces of a-finite measure, (D1 , .N'1 , p 1 } and (D 21 .N'2 1 !Ja)· Let/( w1 , wa) be a function defined on the product space (D 1 X Da,.N'1 X .N'2 , p 1 X !Ja)· p 1 , p 2 , and p 1 X !Ja are assumed to be complete. The function over D a, for each w1 defined by

is called the cross section of/( w1 , w2}; that is, where/( w1 , wa) is thought of as a function of w2 for each fixed w1 , it is called the cross section. /OJ.( w1} is similarly defined.

Theorem 1.5.1 (Fubini's Theorem). Let /( w 1 , w 2} be an integrable function on the product space D = D 1 X D 1 of two measurable spaces of a-finite measure. Then (i) the cross section /OJ 1(w 2 } is an integrable function on D 2 for almost all w1 (with respect to p 1 }; (ii) f 0 J(w 1 , w2 ) dp 1 is an integrable function on D 1 ;

1.6.

INTEGRALS ON

(iii)

Ja f( w

1,

R1

w 2)

13

dp = =

Ja, [ Ja. f( w

1,

w 2 ) dp 2 ]

dp 1

fu Ja, f( W1, w2) dp1] dp2, 1

(1)

[

where p = 1-'1 X 1-'2.

Theorem 1.5.2 (Tonelli's Theorem). Let/( w 1 , w 2 ) be a nonnegative measurable function on the product spaceD= D 1 X D 2 of two measurable spaces of 0'-finite measure. Then (i) the cross section fw,(w 2 ) is a measurable function on D 2 for almost all w 1 [,u 1 ]; (ii) the set E in D 1 , in which Ja./( w 1 , w 2 ) dp 2 is finite, is measurable and the integral Ja./( w 1 , w 2 ) dp 2 is a measurable function, finite on E; (iii)

fa, [ fa. f(

w 1 , w 2) dp 2 ]

dp 1 =

fa. [ fa, /( W1,

w 2) dp1]

dp 2 (2)

where p = p 1 X p 2 , in the sense that if one of the first two integrals in (2) exists as a finite number, then the other two in (2) exist as finite numbers and they are all equal. These theorems are useful, particularly in the case where, in a repeated integral, we want to verify whether the interchange of the order of integration could be permitted. Condition (iii) of Theorem 1.5.2 can also be stated in a clearer way: (iv) If Ja./( w 1, w 2) dp 2 is finite a.e. [,u 1] and Ja, [Ja./( w 1, w 2) dp 1 is finite, then Ja, f( w 1 , w 2 ) dp 1 is finite a. e. [,u 2] and Ja 1 [Ja, f( w 1 , w 2 ) dp 1] dp 2 is finite and both the integrals are equal to Ja f( w 1 , w 2 ) dp. X dp 2]

In this book we use the notation (3)

1.6. Integrals on R, Let (R 1 , !?, p) be given. !? is the completion of the Borel field on R 1 , the class of Lebesgue measurable sets. The measure on it is p, the 0'finite measure. Let g(x) be a measurable function (Lebesgue measurable

I.

14

INTRODUCTION

function). The integral

f

Rt

g(x) dp.,

when it exists, is called the Lebesgue-Stieltjes integral of g(x) (with respect top.). Write (J)(x) = p.(-oo, x), the measure of a set (- oo, x ). The integral is written by

f

Rt

g(x) d(J)(x)

Suppose thatg(x) is a Borel function andf(w) is a measurable function defined on a measurable space (.Q,~ P). Write F(x) = P(-oo, x). If g(f(w)) is integrable on (.Q,~ P), then we have

fu g[f(w)] d'JI

=

J:g(x) dF(x)

(1)

in the sense that the right-hand side of (1) exists and the equality holds. We also see that if the right-hand integral of (1) exists, thengff(w)] is integrable on .Q. Since (R 1 , !?, p.) is a-finite, we can see that

f

oo

g(x) d(J)(x) =

lim A-+-oo,B-+oo

-oo

f

g(x) d(J)(x),

(2)

IA,BJ

where g(x) is integrable, and (J)(x) = p.(-oo, x) as before. We use the notations:

f f f f f


la,bl

(a,bl

la,bl

{a}

=

JIHI g(x) d(J)(x),

g(x) dp. =

Jb-HI g(x) d(J)(x),

g(x) dp. =

Jb-HI g(x) d(J)(x),

g(x) dp. =

JIHI g(x) d(J)(x),

g(x) dp.

a-HI

a-o

a-HI

a-o

g(x) dp. = Ja-HJ g(x) d(J)(x), a-o

where {a} is the set consisting of a single point a.

(3)

1.6.

INTEGRALS ON

R1

15

We also use the notational convention: For -oo
J:

g(x) df/>(x)

= g(a)[f/>(a + 0)- f/>(a)]

+

+ g(b)[f/>(b)- f/>(b- 0)].

J::

g(x) df/>(x)

(4)

We have

f bg(x) df/>(x) = a

-g(a)[f/>(a)- f/>(a- 0)]

+ 0)- f/>(b)]. The notation would be clear for J~, J:_

+ fb.W g(x) df/>(x) a--o

-g(b)[f/>(b

0•

The integral on the left-hand side of (4) is the integral of g(x) over the space (.Q, S?, p.), where .Q = [a, b]. With the notation of (4), we have

J:

g(x) df/>(x)

=

J:

g(x) df/>(x)

+

J:

g(x) df/>(x),

for a

< c < b. (5)

Note that, as was mentioned in Section 1.1, for any nondecreasing function f/>(x) given in advance, we may introduce the complete measure p.. Hence (2)-(4) hold for any given nondecreasing function f/>(x) in so far as they exist. If g(x) is continuous on [a, b], -oo
J:

g(x) df/>(x),

with a finite nondecreasing function f/>(x), is identical with the RiemannStieltjes integral defined by the following: Let a = x0 < x 1 < · · · < X 11 = b be any partition of [a, b] and form n-1

un =

L ga,)[f/>(xi+I)- fl>(xi)],

(6)

i-o

where Ei is any point in [xi, xi+ 1 ], i = 0, 1, ... , n- 1. Let n go to infinity in such a way that maxo;:;;i;:;;n- 1 (xi+ 1 - xi)--+ 0. Then lim n-+oo

U = Jb g(x) df/>(x). 11

(7)

a

REMARK. Even if g(x) is not continuous and f/>(x) is not necessarily nondecreasing, the Riemann-Stieltjes integral

J:

g(x) df/>(x)

I.

16

INTRODUCTION

is defined by (6) and (7) in so far as the limit of an, in the described fashion, exists. If (2) exists and g(x) is continuous, we have (8)

with the Riemann-Stieltjes integral

J:

g(x) dl/>(x).

Even when

exists, the Lebesgue-Stieltjes integral

does not necessarily exist. For this limit we describe

where J! g(x) dl/>(x) is the Lebesgue-Stieltjes integral for any pair (A, B),

B>A.

1.7. Functions of Bounded Variation Let a finite function f(x) be defined on a closed interval [a, b] on Ru a< b. For any partition of [a, b], a= x 0 <xi<···< Xn = b, form n

s = L I/(xi)- f(xi-1> I·

(1)

i-I

If Sis bounded by some constant, independent of the choice of partitions, thenf(x) is called a function of bounded variation. The least upper bound of Sin (1) with regard to the choice of the partition of [c, b] is called the total variation of f(x) over [a, b]. This definition is also applied to a function on RI = (- oo, oo ). A function f( x) is said to be of bounded variation on R I if S in (1) is bounded irrespectively of choices of division points - oo < x 0 < xi < · · ·

1.7.

17

FUNCTIONS OF BOUNDED VARIATION

< Xn < oo. The total variation is the least upper bound of S for all division points (x 0 , x 1 , • •• , xn)· If f(x) is of bounded variation on [a, b], then for any c, a< c
Theorem 1.7.1. Every function of bounded variation is the difference of two nonnegative bounded nondecreasing functions. This theorem is true also for a function of bounded variation over (- oo, oo ). Another well-known theorem is the following theorem of Lebesgue.

Theorem 1.7.2. Every function of bounded variation has a finite derivative a.e. [m], m being the Lebesgue measure. An application of this theorem is the following:

Theorem 1.7.3 (Fubini). Let {fn(x), n = 1, 2, ... } be a sequence of nondecreasing (or nonincreasing) functions in [a, b ]. Suppose that 00

Lfn(x) = s(x) n-1

is a convergent series for every x in [a, b], and s'(x) exists a.e. [m]. Then 00

I:/,/(x)

n-1

(wherefn'(x) exists a.e.) converges a.e. [m] to s'(x). A function of bounded variation has at most a countable number of discontinuities. Discontinuities of a function of bounded variation are of the first kind.

Theorem 17.4 (Lebesgue Decomposition Theorem). Any finite nondecreasing function f(x) on R 1 can be decomposed as f(x) = /d(x)

+ /ac(x) + / (x), 8

(2)

18

I.

INTRODUCTION

where fd(x) is purely discontinuous, that is, constant except on at most a countable set of discontinuities;fac(x) is absolutely continuous in every finite interval; andf.(x) is a continuous singular function, that is, a continuous function whose derivative is zero a.e. [m] and which is not a constant. For a probability distribution function F(x), this theorem can be stated as:

Theorem 1.7.5. A distribution function F(x) can be described as (3) where a 1 , a 2 , and a 3 are nonnegative numbers such that

and Fd(x), Fac(x), and F.(x) are, respectively, a purely discontinuous distribution function, an absolutely continuous distribution function, and a singular distribution function.

1.8. Signed Measure; Decomposition Theorems In this section, the notion of a function of bounded variation and the theorems in 1. 7 will be generalized to multidimensional functions. It is convenient to talk about set functions instead of point functions. We note that to give a nondecreasing function is equivalent to the assignment of a measure [Section 1.1, Eq. (9)] and that the total variation of a function of bounded variation is an additive interval function. We shall discuss this on a more general measure space. Let (D,.N") be a given measure space, where dis a a-field as usual. Any finite countably additive set function 11(E), 11(E) being finite for any E and defined overJlf, is called a signed measure. As in the case of the ordinary measure (nonnegative, countably additive set functions on d), 11(E) is continuous. In order to have the analogs of the Jordan and Lebesgue decomposition theorems, notions of absolute continuity, singular measure, and derivatives of set functions are introduced. Two finite measures f.t 1 and f.t 2 on (D, d) are said to be mutually singular if there is a pair (E, F) of measurable sets with E n F = 0, E u F = D such that f.t 1 (E) = f.t 2 (F) = 0. Jordan's decomposition theorem for a signed measure is

1.8.

19

SIGNED MEASURE; DECOMPOSITION THEOREMS

Theorem 1.8.1. Let v be a signed measure on the measurable space (D, d). Then there are two mutually singular measures v+ and v- on (D,d) such that

v = v+- v-. Moreover there is only one such pair of mutually singular measures. Let v be the signed measure on (R 1 , 93'), ,~ being the Borel field. If f( x) is defined by v(- oo, x ), then we find that f( x) is of bounded variation over (-oo, oo). (vis a finite set function.) Withf1 (x) = v+(-oo, x), f 2(x) = v-( -oo, x),f1(x) andf2 (x) are nonnegative, nondecreasing functions such that / 1 ( - oo) = / 2 ( - oo) = 0, and

f(x) = /1(x)- f2(x).

(1)

This decomposition is unique with / 1( - oo) = / 2 ( - oo) = 0. Conversely, let a functionf(x) of bounded variation be given over R 1 • From Theorem 1.7.1, we have the Jordan decomposition (1);/1 and/2 generate the finite measures p, 1 and p, 2 on (R 1 ,.~) and v(E) = p, 1 (E) - p, 2 (E) gives a signed measure. Let p, and v be two finite measures on (D, ..N'). The measure v is said to be absolutely continuous with respect to the measure p, if v(E) = 0 for each measurable set E for which p,(E) = 0, and this relation is described as v ~ p, in symbols. It is shown that in this case there is a nonnegative integrable function f( w) on D such that

v(E)

=

tJ(w) dp,.

(2)

The function f( w) is uniquely determined in the sense that if g( w) is any other measurable function with the same property as f( w) in (2 ), then f( w) = g( w) a.e. [.u]. · The above results still hold if p, and v are a-finite. (2) is true for every E with p,(E) < oo. The function f( w) in (2) is called the Radon-Nikodym derivative of v with respect top, and is denoted by dvfdp,. A change of variable in the integral calculus corresponds to the following result: Let p, and v be two a-finite measures on d such that v ~ p,. For a function f( w ), integrable with respect to v, we have

(3)

20 If three a-finite measures

11,

~-'•

and A are given, and

I.

INTRODUCTION

11

< 1-' < A, then (4)

The Lebesgue decomposition theorem has the following form:

Theorem 1.8.2. Let (.Q,d, !-') be a measurable space with a-finite measure 1-'· Any a-finite measure defined on ~ can be represented by 11

=

11ac

+

(5)

11s •

where 1180 is a measure which is absolutely continuous with respect to 1-' and 118 is a measure which is singular with respect top. The decomposition of 11 as in (5) is unique. The clause stating that 118 is singular with respect to 1-' is no more than saying that 118 and 1-' are mutually singular. If 1-' is the Lebesgue measure on .Q = R 1 , the theorem is seen to be equivalent to Theorem 1.7.4, in which 118 corresponds to the measure generated by /d(x) + / 1 (x) in Section 1.7, Eq. (2). For the proofs of the results in this section, refer to Royden [1 ], for example.

1.9. The Lebesgue Integral on R, Some basic properties of the integrals on (R 1 , .s:?, m) (Lebesgue integrals) that will be used in this book are given. The Lebesgue measure is m as usual. We use the term "almost everywhere" [or almost all (every)] in this section, for "almost everywhere [m]" [almost all (every)]. Let f(x) be an integrable function on (a, b). A point x, a < x < b, at which lim h-1JA I f(x A->-o

o

+ t)- f(x) I dt =

0

(1)

holds, is called a Lebesgue point. A continuity point of f(x) is a Lebesgue point.

Theorem 1.9.1. If f(x) is integrable on (a, b), then almost all points in (a, b) are Lebesgue points. The indefinite integral F(x) = J:J(t) dt of an integrable functionf(x)

1.9.

THE LEBESGUE INTEGRAL ON

R1

21

has the derivative f(x) almost everywhere or more precisely at Lebesgue points of f(x). Let E be a measurable set of positive measure and let x be a given point. If (2) lim m[E 11 (x- h, x + h)]f2h 11-+0+

exists and is equal to one, the point x is called a point of density of the set E. A consequence of Theorem 1.9.1 is the following:

Theorem 1.9.2. Almost every point of a set of positive measure is a point of density. This is so because, letting f(x) be the indicator of E,

(2h)- 1 f

~11

z-11

f(u) du

converges almost everywhere to f(x), which is equivalent to the fact that the limit exists in (2) and is equal to one. We mention another theorem that is very useful.

Theorem 1.9.3. If f(x) is integrable on ( -oo, oo), then

~i~ J~oo If(x

+ h)- f(x) I dx =

0.

A basic problem in integral calculus is: Is the integral of the derivative of a function the given function ? More exactly, does

f(x) = f(a)

+ J:f'(t) dt

(3)

hold when f(x) has the derivative f'(x)? A classical result is:

Theorem 1.9.4. If f'(x) exists everywhere in [a, b] and is integrable on (a, b), then (3) holds for every a ~ x ~b. The weaker condition thatf'(x) exists almost efJerywhere is not sufficient for (3). What we have is: If f(x) is a continuous nondecreasing function in [a, b], then f'(x) (which exists almost everywhere) is integrable on (a, b) and

J:f'(t) dt ~f(x)- f(a).

(4)

I.

22

INTRODUCTION

This is a consequence of the Lebesgue decomposition theorem (Theorem 1.7.4) if it is noted that a function is absolutely continuous if and only if it is an indefinite integral. We also give a theorem requiring (3). For this purpose, we introduce four derivatives of Dini (see Saks [1]). Letf(x) be a function over a neighborhood of x. Define

D+f(x)

=

lim sup t1 11 (x)fh,

D+f(x) = lim inf t1 11 (x)fh,

A-+o+

D-J(x) = lim sup t1 11 (x)fh,

-+o+

D_J(x)

=

A-+o-

lim inf t1 11 (x)fh, A-+o-

+

where t1 11 (x) = f(x h) - f(x). If D+f(x) = D+f(x), the common value is the right derivative, and if D-J(x) = D_f(x), the common value is the left derivative. When the four derivatives are identical, it is the ordinary derivative.

Theorem 1.9.5. Suppose that for a given function F(x) on a closed interval, one of four derivatives of F(x) is finite everywhere. The necessary and sufficient condition that the derivative be integrable in the interval, is that F(x) be of bounded variation on the interval. In this case, F(x) is the indefinite integral of the derivative. [That is, F(x) is absolutely continuous and (4) holds.] For this theorem and further theorems on this matter, see Hobson [1]. We remark that Dini's four derivatives are not independent of each other; that is, these four derivatives cannot take any values independently of each other. (See Saks [1], p. 271.) A frequent use of integration by parts will be made. If f(x) and g(x) are integrable on (a, b), then

I:

F(x)g(x) dx = F(x)G(x)l:-

I:

G(x)J(x) dx,

(5)

where F(x) and G(x) are indefinite integrals ofj(x) andg(x), respectively. For the Lebesgue-Stieltjes integral, we have: If F(x) and G(x) are two functions of bounded variation over [a, b], then

I:

F(x) dG(x) = F(x)G(x)l:-

I:

G(x) dF(x).

(6)

Note that the notational convention of Section 1.6, Eq. (4) has been used here. For (6), see Saks [1], p. 102.

1.9.

23

THE LEBESGUE INTEGRAL ON R1

The discrete analog of the formula for integration by parts is called summation by parts or Abel's transformation. Let L:,o an and L:,o bn be two series and let

Then, for 1 ;:;;;; m < n, (7)

The mean value theorems for integrals will often be used. The simplest form is given by the first mean value theorem:

If f(x) is continuous on [a, b] and ¢(x) is a nonnegative integrable function over (a, b), then there is a number ~such that a ;:;;;; ~ ;:;;;; b and I:J(x)lf>(x) dx

= Ja)

I:

(8)

lf>(x) dx.

There are two forms of the second mean value theorem. The Weierstrass form is: Ifj(x) is bounded and monotone over [a, b] and ¢(x) is integrable over (a, b), then for any A and B such that ~f(a

+ 0),

B

~f(b-

0),

A ;:;;;;j(a

+ 0),

B

~f(b-

0),

A or

according asf(x) is nonincreasing or nondecreasing,

I:J(x)lf>(x) dx =A

J: lf>(x) dx + B I: lf>(x) dx

(9)

for some~ in [a, b]. The number ~ may depend not only on f and 4> but also on A and B. A special case of (8) is

J:!(x)lf>(x) dx = f(a) I: lf>(x) dx

+ f(b)

I:

lf>(x) dx.

(10)

The ~ in this formula may be different from the ~ in (9). Actually we may show (9) from (10). The Bonnet form of the second mean value theorem is:

I.

24

INTRODUCTION

If f(x) is a nonnegative bounded, nondecreasing function on [a, b] and 4J(x) is integrable over [a, b], then, for any number B such that B ~f(b- 0),

= B

J:J(x)4J(x) dx

J:

(11)

4J(x) dx

for some a ~ ~ ~ b. For a nonnegative bounded, nonincreasing f(x), we have J:J(x)cf>(x) dx

= A

~f(a

for any number A such that A

J:

(12)

cf>(x) dx

+ 0) for some a ~ ~ ~b.

In particular, we may take A= f(a), B = f(b), in each of (11) and (12). For the proof, see, for instance, Hobson [1].

1.10. Inequalities We begin with the following elementary remarks. Let a = (a 1 , a 2 , ••• , an) be a set of nonnegative numbers. Write M,(a)

=

Ln a{ )1/r,

( n- 1

ro:/=0,

k-1

and max a= max(a 1 , a 2 , ••• , an), min a= min(a 1 , a 2 , ••• , an)· If r < 0, all aft are assumed to be strictly positive. We know that min a

~

M,(a)

~

max a

for

r> 0

(1)

and that lim M,(a) = max a, r-+oo

(2)

lim M,(a) = min a. r-+-oo

The continuous analogs are the following: Let f(x) be continuous on a ~ x ~ b [ -oo
= {(b- a)- 1 j~a [f(x)]' dx }1h,

ro:j=O.

For a negative r,f(x) is supposed to be strictly positive over [a, b]. Then min f(x) a~z~b

~

M,(f)

~

max f(x), a~z;:;iib

r

> 0,

(3)

1.1 0.

25

INEQUALITIES

and

lim M,(f)

= max J(x), a:;;.x~b

r-+oo

(4)

lim M,(f) = min J(x). a;;;z;;;b

r-+-oo

Let (D, ~ p.) be a measure space with a-finite complete measure p.. The class of all measurable functions/( w) such that J0 I/( w) IP dp. < oo, p > 0, is denoted by LP(dp.) or LP(D) or simply LP unless confusion is expected. We list some fundamental inequality relations. If p and q are nonnegative numbers such that 1, and if f(w) e LP and g(w) e LP, then f(w)g(w) e £1,

HOLDER INEQUALITY.

(1/p) and

+ (1/q) =

f

0

I f(w)g(w) I dp. ~

[J I/(w) IP dt.t ] Jo I g(w) lq dp. lip[

]1/q

0

.

(5)

The inequality holds, if and only if, for some constant c,

lf(w) IP = c I g(w)

lq

a. e.

[l.t].

A special case is the following discrete case : If p and q are nonnegative numbers such that (1/p) for any {ak}, {bk}, k = 1, 2, ... such that

+ (1/q) =

1, then

00

L I ak IP < k-1

oo,

then

and (6)

The equality holds if and only if I ak IP = c I bk independent of k. MINKOWSKI

f(w)

[ fo

INEQUALITY.

+ g(w) e LP and I /(

w> + g( w > I p d,_,.] 1/p

If p ;;:;; 0 and j(w)

~

[

fo

1/(

lq for some constant E

LP, g(w)

w> 1p dp.] 11 p +

E

c

LP, then

( fo 1 g( w>1p dp.] for p ;?; 1,

11

p

(7)

26

fo

I. INTRODUCTION

lf(w)+g(w)IPd,u ~t lf(w)IPd,u+

fo lg(w)IPd,u, for

Q

[1/(w)l + lg(w)I]Pd,urp;;;; [

p ~ 1.

(8)

I/(w) I = c I g( w) I a.e. for some con-

The equalities hold if and only if stant c. We also have

{f

~

0

f

Q

I f(w)IP d,u rp + [ for

f

Q

I g(w)IP d,u

0 ~ p ~ 1.

rp. (9)

Discrete analogs are the following: If 00

L I bk I < p

P ;;;; o,

oo,

k=1

then for p ;;;; 1, 00

L I ak + bk IP k-1

00

~

(10)

00

L I ak IP + k=1 L I bk IP k-1

for

0 ~p ~ 1

(11)

Moreover

[~1 (lakl + lbki)Prp;;;; (~1 1 aklprp + c~1 ~ bklprp for JENSEN'S INEQUALITY.

0

~

If 0 < p < q, then for ak > 0, k

p

~

1.

(12)

= 1, 2, ... , n, (13)

for all integers n. The equality does not occur if n ;;;; 2. Note that the integral analog of ( 13) does not hold. Jensen's inequality involving a convex function will be given in Section 1.11. The following inequality of Hardy (see Zygmund [1, 5]) is also important.

Theorem 1.10.1. Let p > 1, p - 1 > q, and f(x) ;;;; 0 for x ;;;; 0, be integrable on every finite interval. Define g(x) = x- 1 J:J(u) du

for

x

> 0.

(14)

1.11.

27

CONVEX FUNCTIONS

If [J(x)]Px'l E £1(0, oo), then [g(x)]Px'l E £1(0, oo) and

J~ [g(x)]Px'l dx ~ [pf(p- q- 1)]P J~ [J(x)]Px'l dx.

(15)

Another useful inequality is due to Hardy-Littlewood and is known as the Hardy-Littlewood maximal theorem (Hardy and Littlewood [4], Hardy et al. [1 ]).

Theorem 1.10.2. Ifj(x) e LP(a, b), p > 1, -oo
O(x,f) =sup (x- E)-1 a~e~b

e"'z

Jze If(u) I du,

a< x

<: b,

we see that O(x,f) e LP(a, b) and

J:

[O(x,f)]P dx

~ 2[pf(p-1)]PJ: 1/(x) IPdx.

(16)

This ceases to hold for p = 1.

1.11. Convex Functions A functionf(x) defined over I= [a, b] or (a, b) is called convex on I if

f[(x

+ y)/2]

~ Hf(x)

+ f(y)]

for

x,y e I.

(1)

The values of a and b are allowed to be infinite. It can be shown that ifj(x) is continuous, then (1) implies the inequality

where x 1 , x 2 , ••• , Xn e I, ak > 0, k = 1, 2, ... , nand :L~- 1 ak = 1. If, in (1 ), the opposite inequality sign holds, then the function f(x) is called concave on I. If f(x) has a finite second derivative /"(x) in (a, b), then a necessary and sufficient condition that f(x) should be convex is that f"(x) ~ 0 in (a, b). Basic properties of convex functions are included in:

Theorem 1.11.1. Suppose that f(x) is convex on I= (a, b).

I.

28

INTRODUCTION

(i) Ifj(x) is bounded above in some interval interior to I, no mattre how small, then f(x) is continuous in /. (ii) A continuous convex function has a left derivative and a right derivative everywhere in /. Both derivatives are nondecreasing functions in /, and the right derivative is not less than the left derivative. (iii) A continvous convex function is of bounded variation over every closed interval in /. (iv) If f(x) is continuous convex on [a, b] and D+f(x) denotes the right derivative, then

f(x) (v)

=

f(a)

+

J:

D+f(u) du.

(3)

A measurable convex function is continuous.

For (i)-(iii), see Hardy et al. [1] and for (iv), Zygmund [1], p. 69. Property (v) is due to Blumberg [1] and Sierpinski [1]. Also see Ostrowski [1] and Korosonel'skii and Buticka [1]. The following inequality of Jensen is of frequent use: Suppose that g(x) is defined on (a, b) and a ~ g(x) ~ p, where a, b, a, and p may be finite or infinite, thatf(x) is a continuous convex function on (a, p), and that q(x) e V(a, b) and nonnegative. Then jENSEN's INEQUALITY.

f

(

J>(x)q(x) dx) < J:J[g(x)]q(x) dx b = b '

(~~~

(~~~

(4)

provided that the right-hand side exists and is finite. [If the right-hand side exists and is finite, then the left-hand side also exists and (4) holds.] (See Hardy et al. [1 ]. )

1.12. Analytic Functions A set S in the complex plane, is called arcwise connected if any pair of points z 1 and z 2 of Scan be joined by a polygonal line, all the points of which lie in S. In complex function theory, a domain means an open arcwise-connected set. Let S be the closure of a set S. Then S n Sc is the boundary of S. Let z = z(t) be a complex-valued continuous function for a ~ t ~ p (-oo
1.12.

29

ANALYTIC FUNCTIONS

If z(t) is such that z(t 1 ) of=- z(t 2 ) for any pair t 1 , t 2 , t 1 of=- t 2 , then the curve is called a simple arc or Jordan arc. It is a simple closed curve if t 1 < t 2 , z(t 1 ) = z(t 2 ) implies t 1 = a, t 2 = {J. A line segment joining z 1 and z 2 has length I z 2 - z 1 I· The length of a polygonal line, consisting of segments joining z 1 and z 2 , z 2 and z 3 , ••• , Zn-l and Zn, is I z 1 - Z 2 I I z2 - z3 I I Zn-l - Zn I· Take a simple arc C: z = z(t), a ~ t ~ {J. Cis said to be rectifiable if

+

+ ··· +

n-1

I 1z(tk) -

z(tk+ 1 )

k=l

1

is bounded for any partition of [a, {J] into a = t 0 < t 1 < · · · < tn = {J. For this to be so, it is necessary and sufficient that the real and imaginary parts of the function z(t) be of bounded variation over [a, {J]. The length of C is, by definition, n-1

sup

L I z(tk)- z(tk+ I = 1)

l(C),

k=O

the supremum being taken over all partitions of [a, {J]. The length l(C) is denoted by f~ I dz(t) 1. If [a, {J] breaks up into a finite number of subintervals [a 0 , a 1], [a 1 , a 2], ... , [am_1 , am], a 0 =a, am= {J, and z(t) is differentiable and z'(t) is continuous in each of them, then l(C) =

mf k=O

f"'k+l I z'(t) I dt. "k

Now let f(z) be a complex-valued function of a complex variable z, which is continuous and single-valued in a domain D. If lim h- 1 [f(z 0 +h) -f(z0 )],

z 0 ED,

(1)

h-+o

exists as a finite number and is independent of how the complex increment h tends to zero, thenf(z) is said to be differentiable at z 0 • The limit (1) is the derivative of f(z) at z = z 0 • If f(z) is differentiable everywhere in D, then f(z) is called regular (analytic, holomorphic) in D. If f(z) is regular in a neighborhood of z 0 , then it is said to be regular (analytic, holomorphic) at z 0 • Let C be a rectifiable curve in a domain C: z = z(t), a ~ t ~ {J, z(t) being a continuous function whose real and imaginary parts are of bounded variation over [a, {J]. The curve C is assumed to be oriented by the

I.

30

INTRODUCTION

parametrization; that is, z(t 1 ) precedes z(t 2 ) on C when and only when tl < tz. Let f(z) be continuous on C. By definition

J 0 f(z) dz = J:J[z(t)] dz(t). A fundamental theorem in the theory of analytic functions is:

Theorem 1.12.1. Letf(z) be analytic in a domain D. Let C be a simple closed rectifiable curve inside D. Then (2)

J 0 f(z) dz = 0.

If C is a simple closed curve, then the complement of C is the union of exactly two mutually exclusive domains and every point of C is a boundary point of each domain (Jordan curve theorem). The domain that does not include infinity is called the interior of the curve C and is denoted by ci. Let C be z = z(t), a ~ t ~ p, a simple closed rectifiable curve. It is said to have positive (negative) orientation, if for a point z 0 lying within the interior of C, arg(z(t)- z 0 ] increases (decreases) by m when t increases from a to p. The Cauchy integral theorem is:

Theorem 1.12.2. If C is a simple closed rectifiable curve of positive orientation, ifj(z) is analytic in Ci and continuous on Ci (closure of Ci), and if Z E Ci, then

J

f(z) = (mi)-1 c [!(~)/(~- z)] d~.

(3)

A function f(z), analytic in a domain, has derivatives of all orders in the domain and

m= 1,2, ... , C being the curve in the theorem. Ifj(z) is analytic in a circle I z- a I < r, r panded in a Taylor series, or 00

f(z) = where

c" =

f<"'(a)fnl.

L Cn(z- a)",

n-o

(4)

> 0, thenf(z) can be ex-

I z-a!


1.13.

31

JENSEN'S AND CARLEMAN'S THEOREMS

If f(z) is analytic in the annulus 0

< r 1 < I z - a I < rro then

00

f(z) =

L C,(z- a)", n--oo

where Cis IE- a I= r, r 1 < r < r 2 • Analyticity is defined also for functions of a real variable. Let f(x) be a function over a < x < b. It is called analytic at x 0 (a < x 0 < b) if there is a function ifJ(z) of a complex variable z, which is analytic in some circle C with center z = x0 and ifJ(x) = f(x) on the subinterval of (a, b), contained in the circle C. This definition is the same as saying that f(z) is analytic at x 0 if and only if f(x) is differentiable infinitely many times and the Taylor series of f(x) about x 0 is convergent in some neighborhood of x0 • This is simple to show since the Taylor series of f(x) about x0 , with the variable z replacing x, gives the analytic function ifJ(z). If f(x) is analytic at every point in (a, b), then it is called analytic in (a, b).

1.13. Jensen's and Carleman's Theorems An important and useful theorem is Jensen's, and its analog in a halfplane is known as Carleman's theorem.

Theorem 1.13.1 (Jensen). Let f(z) be analytic in I z I < R. Suppose that f(O) # 0 and let r 1 ;;::;; r 2 ;;::;; • • • be the moduli of zeros of f(z) in I z I < R. Then for rn ~ r ~ rn+1, log[r" l/(0) l/r 1r 2 • • ·rn] = (2:n)- 1

I:

loglf(rei6 ) I dO,

(1)

where zeros are counted as many times as their multiplicities. Equation (1) is also called Jensen's formula. If n(x) is the number of zeros of f(z) in I z I ~ x, then (1) is written in the form

I:

[n(x)fx] dx = (2:n)- 1

J:,. loglf(rei

6)

I dO-logl/(0) I

for r < R. (2)

An immediate consequence of (1) or (2) is

(Z:n)- 1 providedf(O) # 0.

I:

loglf(rei6 ) I dO ~ logl/(0)

I,

(3)

I.

32

INTRODUCTION

In the Jensen theorem above, the domain is a circle. We may have the analog of the Jensen-type theorem in a half-plane (given by Carleman).

Theorem 1.13.2. Let z = x Let the zeros ofj(z) in 1 ~I z Then

+ iy and I~

R, 0

let f(z) be analytic in y ~ 0. arg z ~ :n be r 1ei81 , ••• , rnei8".

~

T) 1I" loglf(Re18 ) I sin (J dO Ln(1 - Rk 2 sin Ok = -R Tk :7t

k-1

0

+ 21:n IR ( x•1

-

1

1 ) loglf(x)f(-x) I dx R•

+ 0(1),

where 0(1) denotes a function of R which is bounded as R - oo. A trivial consequence is 0 < 12:n

IR (1 -1 ) log f(x)f(-x) I x R 2

1

+ :n~

2

J: logl f(Rei

8)

I sin (J d(J + A,

where A is a constant independent of R. For the proofs of the above results, see Titchmarsh [3], pp. 125 and 130.

1.14. Analytic Continuation Let D 1 and D 2 be two domains with common points. Ifj1 (z) andf2 (z) are analytic in D 1 and D 2 , respectively, and / 1 (z) = / 2 (z) in D 1 n D 2 , then / 1 (z) and / 2 (z) are called analytic continuations of each other. In fact, if l/J(z) is defined by / 1 (z) in D 1 and by f 2 (z) in D 2 , then l/J(z) is analytic in D = D 1 n D 2 •

a

\Y

FIGURE

1.1.

Suppose that a domain D, with a simple rectifiable boundary, is divided into two domains D 1 and D 2 by a segment s so that s is the common boundary of D 1 and D 2 (see Fig. 1.1).

1.14.

33

ANALYTIC CONTINUATION

If/1 (z) and/2 (z) are analytic in D 1 and D 2 , respectively, and continuous on their boundaries (including s) and if/1(z) = f 2 (z) on s, then there is a function 4>(z) which is analytic in D and is such that 4>(z) = / 1 (z) in D 1 and on s, and 4>(z) = / 2 (z) in D 2 and on s. In other words, 4>(z) is an analytic continuation of / 1 (z) and / 2 (z). This is well known (Titchmarsh [3], p. 157). A little more general form is:

Theorem 1.14.1. Suppose that / 1 (z) and / 2 (z) are analytic in D 1 and D 2 , respectively, and are continuous on their respective boundaries except on s, and thatf1(z) and/2 (z) are bounded on D 1 and D 2 , respectively. If / 1 (z) and fz(z) converge to common values as z E D 1 , z-+ ~ and z E D 2 , z-+ ~. respectively, for almost all ~on s, then there is a function 4>(z) analytic in D (including s) such that 4>(z) = / 1 (z) on D 1 , 4>(z) = / 2 (z) on D 2 • The meaning of "almost all ~ on s" will be quite clear. The proof of the theorem is carried out in the following way, which is a standard method. Without loss of generality, the segment s may be supposed to be an interval (a, b) on the real axis. Let D 1 and D 2 lie in the upper and the lower half-plane, respectively, with respective boundaries cl u sand C 2 us so that C 1 and C 2 are the respective boundaries of D 1 and D 2 in the upper and lower half-planes (see Fig. 1.2). Proof.

c,

o, FIGURE

1.2.

0

a

b

02 C2

Define f(z) = / 1 (z) for z on C 1 and f(z) = / 2 (z) for z on C 2 and construct

4>(z)

=

(Zni)-1

fc

[f( C)/( C- z)] dC,

(1)

for z E D, where C = C 1 U C 2 is of positive orientation. 4>(z) is welldefined and analytic in D. Let P 1 = a 1 iy, Q 1 = {3 1 iy (y > 0) be

+

+

I.

34

INTRODUCTION

the points on C 1 and let P 1 and Q1 approach a and b, respectively, along cl as y - 0+. Take any point z e D 1 and choose y so small that 0 < y < Im z. Then / 1

(z)

= (2.ni)-1

J

Ct'

+ (2.ni)-

1

[f( C)/( C- z)] dC

J::

[f(u

+ iy)J(u + iy- z)] du,

(2)

where C 1 ' is the path from Q 1 to P 1 along C1 • The function f(u + iy) converges boundedly for almost all u, a < u < b, as y -+ 0+ to a limit which we denote by / 1 (u). We see therefore that the second integral in (2) converges to (2.ni)- 1

J:

(2.ni)- 1

J

[f1(u)J(u- z)] du

and the first to Ct

[A( C)/( C- z)] dC.

Hence /1(z) = (2.ni)-1 J [f( C)/( C- z)] dC Ct

+ (2.ni)-1 Jb [fi(u)J(u- z)] du G

for

zeD.

(3)

Moreover, if z e D 2 , the right-hand side of (2) is zero and then we have 0 = (2.ni)-1

JCt [f( C)/( C- z)] dC + (2.ni)- Jb [f1(u)J(u- z)] du 1

G

for Similarly we have, noting that f(u

+ iy)-+ /

1

z e D2 •

(4)

(u) also for y-+ 0,

/2(z) = (2.ni)-1 J [f( C)/( C- z)] dC- (2.ni)-1 Jb [f1(u)J(u- z)] du C1 G for z e D 20 (5) 0 = (2.ni)-1 J [f( C)/( C- z)] dC- (2.ni)- 1 C1

Jb [fi(u)J(u- z)] du G

for

z e Dl"

(6)

Adding (3) and (6 ), we get

J

/1(z) = (2.ni)-1 c [f( C)/( C- z)] dC

for z e D 1 •

(7)

1.15.

35

MAXIMUM MODULUS AND PHRAGMEN-LINDELOF THEOREMS

From (4) and (5) we get / 2

(z) = (lm)-1

J [f( C)/( C- z)] dC

(8)

0

Equations (7) and (8) show that tf>(z) in D 2 • This proves the theorem.

= / 1 (z)

in D 1 and tf>(z)

= / 2 (z)

We give the following particular case of Theorem 1.14.1.

Corollary 1.14.1. If a function f(z), analytic in the upper half-plane, has a boundary function on (a, b), which is almost everywhere zero; that is, lim f(x

+ iy) =

0

a.e.

!f-+o+

on (a, b), andf(z) is bounded in some rectangle a~ x then f(z) is identically zero in the upper half-plane.

~

b, 0
~

c,

The conclusion of the corollary continues to hold, with suitable modifications, if the upper half-plane is replaced by a circle and the interval by an arc. Actually a much better result than that in the corollary is known: If an analytic function in the upper half-plane has a boundary function which is zero on a set of positive measure, then the function is identically zero. A similar result is true for a more general domain than a half-plane or a circle (Lusin and Privalov [1 ]).

1.15. Maximum Modulus Theorem and Theorems of Phragmen-Lindelof A very important theorem in complex function theory is the following, the maximum modulus theorem.

Theorem 1.15.1. Let f(z) be analytic i.n a bounded domain D and continuous on D u fJD, fJD being the boundary of D. Then, writing M = maxseaD lf(S) I, (1) 1/(z) I< M for all z e D, unless f(z) is a constant of absolute value M. For the proof, see for instance Hille [1], p. 205.

I.

36

INTRODUCTION

Important extensions of the maximum modulus theorem were given by Phragmen and Lindelof (see Boas [1] or Titchmarsh [3]). Let f(z) be analytic in a domain D and on 8D except at one point P of 8D, 8D being a simple closed curve. Their theorems conclude, under mild conditions of f(z), the boundedness of f(z) at all points of D by M from the bound M of f(z) on 8D except at P. One of the typical theorems 1s:

Theorem 1.15.2. Let D be the domain between two straight lines which make an angle xfa at the origin. Let f(z) be analytic in D and on the boundary lines. If (2) 1/(z)l ~M on the lines and

f(z) = O[exp(rP)]

(3)

as I z I = r--+ oo, where {J < a uniformly in the angle, then (1) holds throughout the domain D. (See ,Titchmarsh [3], p. 177.) In order to investigate the matter systematically, or to study the behavior of a function as z goes to infinity in different directions, a function h(O) due to Phragmen-Lindelof is introduced. Suppose that f(z), z = rei0, is analytic in a ~ 0 ~ {J, I z I ~ r0 for some r 0 ~ 0, and thatf(z) is of order e. e > 0; that is lim sup[logl f(rei 0 ) lfre+•] = 0

(4)

r-+oo

uniformly in a

~

0

~

{J for any e

> 0, but not for any negative e. Define

h(O) = lim sup[logl f(rei 0 ) lfre].

(5)

r-+oo

The following theorem states a basic property of h(O).

Theorem 1.15.3. Let {J - a < xf e and let (6)

Let H(O) be a function of the form a cos eO+ b sin eO such that H(a) = h1 , H({J) = h2 • Then

h(O)

for

a~

0

~

{J.

~

H(O)

(7)

1.16.

37

INNER PRODUCT SPACE

The function H(O) in the theorem is obviously given by [h 1 sin e(P- 0)

+h

2

sin e(O- a)]/sin e(P- a).

One or both of h(a), h(e) may be -oo. In this case h(O) = -P for a < 0 < p. This is regarded as a particular case of the theorem. Finally we mention a theorem due to P6lya and Szego ([1], p. 147).

Theorem 1.15.4. Let f(z) be analytic in Re z ;;;; 0. Suppose that (i)

there are positive constants A and B such that 1/(z)

I ~ AeBizl,

in

Re z;;;; 0;

(ii) for r > 0, 1/(ir) I ~ 1,

1/(-ir) I~ 1;

(iii) lim sup[logl/(r) 1/r] ~ 0. Then in the entire half-plane Re z ;;;; 0, If(z)

I ~ 1.

1.16. Inner Product Space Consider a non empty space X of abstract elements (or objects). An operation called addition is defined for any pair of elements x, y, of X and is denoted by x + y. Another operation called scaler multiplication by a complex number is defined for any complex number a and any element x of X and is denoted by ax. Suppose that, for any elements x, y E X, x + y E X and that for any complex number a and an element x E X, ax E X. Those two operations are assumed to satisfy the following conditions for all elements x, y, z E X and all complex numbers a, p. (a) X+ y = y +X. (b) x + (y + z) = (x + y) + z. (c) a(x + y) =ax+ ay. (d) a(Px) = (ap)x. (e) (a+ P)x =(ax)+ (Px). (f) 1-x=x. (g) There is an element 0 such that x + 0 = x, O·x = 0. The element x + (y + z) is simply written as x + y + z and (ax) + (Py) = ax+ py. Such a space X is called a linear space. The element 0 is called the null element. If no confusion is expected, then 0

I.

38

INTRODUCTION

will be used for 0. We do not show here the immediate consequences of the above conditions. For example, 0 in definition (g) is unique. x - y stands for x + (-1 )y. Let X be a linear space. We suppose furthermore that, for each pair x, y of elements of X, a complex number denoted by (x, y) is defined with the following properties for x, y, z E X, and a complex number a. (i) (ii) (iii) (iv) (v)

(x,y)=(Y,x). (x + y, z) = (x, z) + (y, z). (ax,y) = a(x,y). (x,x);;;;O. (x, x) = 0 if and only if x = 0.

In this case, the space X is called an inner product space and (x, y) is called the inner product of elements x and y. Let X be again a linear space. Suppose that, for each element x E X, a nonnegative number denoted by II x II, satisfying the following conditions is defined: (i') (ii') (iii')

II x II = 0 if and only if x = 0 . II X+ y II ~II X II + II y 11II axil= Ia I llxll·

Such a space X is called a normed space and II x II is called the norm of x. Let X be an inner product space. We shall show that the following inequality relation holds for the inner product. For any x,y EX,

I (x, y) 12 ~ (x, x)(y, y).

(1)

This inequality is called the Schwarz inequality. If y = 0, then (1) is immediate. So we suppose y =1=- 0. For any complex number A, (2) (x + Ay, x + Ay) ;;;; 0

from (iv). From (i), (ii), and (iv), the left-hand side is

(x,

X

+ Ay) + (Ay, X+ Ay) = (x + Ay, x) + (x + Ay, .l.y) --- --- --= (x, x) + (.l.y, x) + (x, Ay) + (.l.y, .l.y) = (x, x) + .l(y, x) + (.l.y, x) + (.l.y, Ay) = (x, x) + .l(x,y) + A(x,y) +I A l (y,y). --:-----:-

2

(3)

1.16.

39

INNER PRODUCT SPACE

This must be nonnegative from (2). Since A. is arbitrary, we set A.= -(x,y)f(y,y). Then (3) takes the form

(x x) _ (y, x)(x, y) _ (x, y)(y, x) , (y,y) (y,y)

+

(x, y)(y, x) (y,y)

which must be equal to or greater than 0. This shows (1 ). The equality holds in (1) when and only when

X+

AJ = (}

for some complex number A.. Suppose that I (x, y) 12 = (x, x) A.= -(x,y)f(y,y). Then as before

(x

+

A.y, x

+

+ (y, y),

(4)

and

y =f=. 0.

Take

A.y) = (x, x) - [(x, y)(y, x)f(y, y)].

The right-hand side is zero by our assumption and hence (x + A.y, x + A.y) = 0 which yields x + A.y = 0. Conversely, x + A.y = () for some complex number A.. Then, writing a= -A., x = ay,

I (x, y) 12 = I (ay, y) 12 = I a I (y, y) 2 = (ay, ay)(y, y) = (x, x)(y, y). Hence, in (1 ), the equality sign holds. Now for any x of an inner product space, define Then this II x II satisfies (i') and also (ii') because

II x

+ y 11

2

II x II = (x, x)I 12 •

+ y, x + y) = (x, x) + (y, x) + (x,y) + (y,y) ~ II x 11 + I (y, x) I + I (x, y) I + II Y W, = (x

2

which is not greater than ~llxW+211xllllyll+llyW

[because of (1 )]

= (II X II

+ II y 11)

2•

This shows (ii'). Condition (iii') is easy to show. Thus we may say that any inner product space becomes a normed space with II x 11 2 = (x, x). An inner product space in which the norm of elements x are given by (x, x) 112 is called a normed inner product space.

I.

INTRODUCTION

Let X be a normed space. Then we may introduce the notion of convergence of a sequence of elements of X. A sequence {xn, n = 1, 2, ... } of elements of X is said to converge to an element x of X, as n --+ oo, if II Xn - x II --+ 0 as n --+ oo. If a sequence {xn, n = 1, 2, ... } of X is such that II Xm- Xn II-+ 0 as m,n --+ oo, then {xn} is called a Cauchy sequence. It is easy to show that any convergent sequence to some element is a Cauchy sequence. However, the converse is not always true. If any Cauchy sequence is a convergent sequence; that is, for any Cauchy sequence {xn }, there exists an x E X such that Xn converges to x, then the space X is called complete. A complete normed space is called a Banach space. A complete normed inner product space is called a Hilbert space. Let f(x) be a measurable function (generally complex valued) defined on (a, b), where a, or b, or both may be infinite. Let p be a positive number not less than 1. If

J: I

f(x)

IP dx <

(5)

oo,

thenf(x) is said to belong to LP(a, b). (This notation was used in Section 1.10.) In other words LP(a, b) is the class of measurable functions for which (5) holds, or LP(a, b) is the space of elements that are functions on (a, b) satisfying (5). It is easy to see that LP(a, b) is a hnear space with ordinary addition and multiplication by a complex constant. ) 11/11= (fbalf(x)IPdx. 1/p

(6)

Then from Section 1.10, Eq. (7) (Minkowski's inequality) we see that

II/ +g II~ 11/11

+ llg II

for J, g E LP(a, b) and, hence, actually (6) satisfies the norm condition (ii'). Condition (iii') is obvious. If the null element () stands for a function which is zero almost everywhere, then II f II really becomes the norm and LP(a, b) is a normed space. We agree that we do not distinguish between any two functions that are equal almost everywhere. Besides, we have LP(a, b) is a Banach space. That is, LP(a, b) is complete. (See, for example, Royden [1].)

1.16.

INNER PRODUCT SPACE

For £2(a, b), we define the inner product by (f,g)

=

J:J(x)g(x) dx,

J,g E £2(a, b). We easily see that this actually satisfies the conditions for the inner product. Hence £2(a, b) is a Hilbert space. Let (lP) be the space of all sequences ~ = {~n , n = 1, 2, ... } of real or complex numbers such that :L~ 1 I ~n IP < oo. Let p ~ 1. Define a~={a~n•

n=1,2, ... }

for any complex number a and ~

for

~. 'YJ E

+ 'Y} =

{~ n + 'YJn , n = 1, 2, ... }

(IP). Define the norm

We may then show that (IP) is a Banach space and (12 } is a Hilbert space, with inner product 00

(~, 'YJ} =

L ~nfin

n=I

for ~ = {~n}, 'Y} = {'YJn} E (IP). We give finally a brief review of operators. As ordinary functions, we consider functions defined on a set in an abstract space, which assume values (elements) in another abstract space. Letf(x) or fx be defined on a setS of a space X and letf(x) assume elements in a space Y. The word mapping is also used as a synonym for "function." It is said that f maps S in X into Y, or f(x) or f: S----+ Y is a map of S into Y. Also f(x) is called an operator or a transformation. An operator f defined on a linear space X which maps X into another space Y is called a linear operator if (7) for all x 1 , x 2 E X, where a 1 , a 2 are complex numbers. Suppose that X and Y are normed spaces. If there is a constant M such that (8) 11/(x) II ~ M II x II

I.

42

INTRODUCTION

for all x E X, then f(x) is called a bounded operator. Here II f(x) II is the norm in Y and II x II is the norm in X. If, for any E > 0, there is a 15 > 0 such that, for all x 1 , x 2 E X, with II X 1 - X 2 II < 15, II f(xl) - f(x 2 ) II < E, then the operator f is called uniformly continuous. A linear operator is bounded if and only zf it is uniformly continuous. The continuity of an operator at an element is also defined in an obvious way. We then find that a linear operator is uniformly continuous zf and only if it is continuous at an element. Letf(x) be a bounded linear operator. Let us write the infimum of M in (7) when x ranges over the whole space X by 11/11. (Note that this is different from 11/(x) II-) We can see that

II/II= supllf(x) II /II x II, :z;eX z#J

or

II/II= IIZII=I sup llf(x) II=

sup llf(x) IIZII;:i;l

II·

Now consider the space of all bounded linear operators. Then we have: The space of all bounded linear operators from a normed space X to a Banach space Y is a Banach space. (See Royden [1].)

II Fourier Series and Fourier Transforms

2.1. The Riemann-Lebesgue Lemma We shall begin with a very useful theorem in analysis called the Riemann-Lebesgue lemma, which is stated as follows:

Theorem 2.1.1. Ifj(x)

E

£1(-oo, oo), then

J~ooJ(x)eiAz dx--+- 0

(1)

as A-+- ±oo (l real). Since eiAz is bounded by 1 in absolute value, the existence of the integral in (1) is obvious. We shall prove this in a more general form:

Theorem 2.1.2. Suppose that 43

II.

K(x) is bounded over (-oo, oo),

(i)

J: K(x) dx

(ii) as A

--+

(iii)

FOURIER SERIES AND FOURIER TRANSFORMS

o(A)

(2)

V(-oo, oo).

(3)

=

±oo, and f(x)

E

Then lim Ioo J(x)K(J.x) dx = 0. A-+±oo

(4)

-oo

The existence of the integral in (4) is again obvious. Actually, this theorem generalizes the preceding theorem because eiz satisfies the conditions of K(x). Proof.

We shall prove first that

I~f(x)K(J.x) dx--+ 0

as

A.--+ ±oo.

(5)

If it happens that

f(x) = { ~ (constant)

over [a, p], elsewhere,

0

~

a

< p < oo

(6)

then (5) is easily seen to be true, for

I~ f(x)K(J.x) dx = =

c I: K(J.x) dx (cfA.) I:A K(u) du- (cfA.) I: K(u) du.

(7)

If a = 0, the last integral does not appear in (7) and the first integral on the right-hand side of (7), written by

(PcfPA.)

I

PA

0

K(u) du,

converges to 0 because of condition (ii). If a > 0, the same thing is true for the last integral in (7). Once this is determined, (5) holds for any step functionf(x) vanishing outside a finite interval, since a step function is a linear combination of functions of type (6). Now suppose that f(x) is any function in V(O, oo). Then for any e > 0, there is a step function g,(x) vanishing outside a finite interval

2.1.

45

THE RIEMANN-LEBESGUE LEMMA

such that

J~ I/(x)- g.(x) I dx < e.

(8)

We then write

I = J~ f(x)K(J.x) dx = J~ [J(x)- g.(x)]K(J.x) dx

+ J~ g.(x)K(J.x) dx

=11 +12 ,

(9)

say. Fixing e > 0 and letting .A.--+ ±oo, we see that / 2 --+ 0 by what was proved above. For / 1 , writing I K(x) I ~ C, C being a constant, we have

I I1 I ~ c f~ I f(x)- g.(x) I r1x < ec. So we have lim sup I I A-+oo

I ~ eC.

Since e is arbitrary this proves (5). In quite the same way we may show that for f(x) e V(-oo, 0),

(oof(x)K(J.x) dx--+ 0. This and (5), together, complete the proof of the theorem. The proof of this theorem is due to Bochner and Chandrasekharan [1 ]. Sometimes we will have occasion to deal with the improper integral (10)

Thus we shall give the following theorem which is comparable to Theorem 2.1.2. The special case of it with K(x) = eiz is due to Pringsheim [1].

Theorem 2.1.3. Suppose that (i) (ii)

K(x) is bounded over (-oo, oo) K(x) dx = 0(1), -oo
J:

(11)

and (iii) f(x) is locally integrable and of bounded variation over (- oo, -A] and [A, oo) for some A > 0, and

f(x)--+ 0,

as

x--+ ±oo.

II.

46

FOURIER SERIES AND FOURIER TRANSFORMS

Then the improper integral

J::f(x)K(.I.x) dx exists for every A. =F 0 and converges to zero as A.---+- ±oo. A function is called locally integrable if it is integrable over every finite interval.

Proof.

Suppose that

where M is a constant independent of A. Since f(x) is of bounded variation over [A, oo), f(x) may be written

f(x)

=

(12)

ft(x) - fz(x),

where fi(x), i = 1, 2, is nonincreasing and /i(x) ---+- 0, as x ---+- oo. Let s > 0 be arbitrary and let A. =F 0 be any number. Choose a positive number B so large that B ~ A and

0


x

for

~B.

(13)

Then, for X' > X ~ B, we have, from the second mean value theorem [Section 1.9, Eq. (12)], the fact that there is a C, X < C < X' such that

From this, we have

IJ:' fi(x)K(.I.x) dx I IJ =

1(X)A.- 1

J:

K(u) du

I

(I /i(X) 1/1 A. I) 2M < s/1 A. I. This shows that, for each A. =F 0, J-+"" f(x)K(.I.x) dx exists. The argument shows also that J-+-oof(x)K(.I.x) dx exists. ~

(14) same

Now take s = 1 and choose B for s = 1 in the above. We then find from (14) that

IJ:"" f(x)K(.I.x) dx I~ ,t IJ:"" / (x)K(.I.x) I~ 2/1 A. I. dx

1

Therefore, lim IAI-+oo

f-+oo f(x)K(.I.x) dx B

=

0.

(15)

2.1.

47

THE RIEMANN-LEBESGUE LEMMA

On the other hand, from Theorem 2.1.2, we have lim JB f(x)K(h) dx IAI->-oo

= 0,

(16)

0

because condition ( 11) implies condition (ii) in Theorem 2.1.2. Equations (15) and (16) give us the fact that lim IAI->-oo

J->-oo 0

f(x)K(h) dx

lim Jo IAI->-oo

=

f(x)K(h) dx

0,

=

0

+-oo

is obtained in a quite similar way. This completes the proof. We finally prove the following theorem which is sometimes called the generalized Riemann-Lebesgue lemma and is of use with K(x) = eiz.

Theorem 2.1.4. Suppose that conditions (i) and (ii) on K(x) in Theorem 2.1.3 are satisfied. Let f(x) E V(-oo, oo) and let h(x) be a function of bounded variation over [- oo, oo]. Then the integral

J~oof(x + a)h(x)K(h) dx

(17)

converges to zero as A.- ±oo uniformly with respect to -oo


There is no problem about the existence of the integral (17) because f ELI( -oo, oo) and h(x)K(h) is bounded from the conditions on h(x) and K(x).

Proof. The idea of the proof is the same as that of Theorem 2.1.2. We shall prove here that

J~ f(x

+ a)h(x)K(h) dx- 0

uniformly for - oo

< a < oo.

J:

00

J(x

as

JA.J- oo,

(18)

The integral

+ a)h(x)K(h) dx

is handled in the same way. We may suppose that h(x) is nonincreasing in [0, oo) since, for a general h(x) of bounded variation, we may represent h(x) as the difference of two nonincreasing functions and treat the problem separately.

II.

48

FOURIER SERIES AND FOURIER TRANSFORMS

Let e be any positive number and choose a step-function g.(x), vanishing in [B, oo) and (- oo, - B] in such a way that

J~oo I f(x)- g.(x) I dx < ef2MD,

(19)

where we suppose that

\J:

K(x)dxl

~ M,

-oo


and I h(x) I ~ D, M and D being constants. Let I K(x) constant. We then have

I~ f(x

+ a)h(x)K(.tx) dx =

I~ [f(x

I ;:;:;; C, C being a

+a)- g.(x + a)]h(x)K(A.x) dx

+ I:-« g.(x + a)h(x)K(A.x) dx (20) say.

I III~ MD I~oo lf(x + a)-g.(x +a) I dx = MD

I: I

f(x)- g.(x) I dx < ef2.

For / 2 , we apply the second mean value theorem and get

12 = h(O)

I:

g.(x

+ a)K(.tx) dx

for some 0 < C < B- a. Note that in (20) if B ;:;:;; a, then / 2 = 0 and so we supposed B > I a I· Let the step-function g.(x) be given by

g.(x) = ci, where -B

= t0 <

t1

m-1

/ 2

if

tm -

a

= h(O) i~ ci

< · · · < tn = B. Then we may write

It;+l-«

< C ~ tm+I -

_ 11 11

K(A.x) dx

+ h(O)cm ICA,._,. K(A.x) dx

a. Hence

(21)

2.2.

49

FOURIER SERIES

The last quantity is independent of a and hence we may choose Ao independently of a so that the quantity (21) is less than e/2 for I A I > Ao. Therefore, I I I ~ I 11 I + I 12 1< e for I A I > A0 • This proves (18).

2.2. Fourier Series A series of the form (1) with a constant a0 and An(x) = an cos nx + bn sin nx, with constants an , bn , n = 1, 2, ... , is called a trigonometric series. Its partial sum n

sn(x) = lao

+ k-1 L Ak(x)

(2)

is called the trigonometric polynomial of order n. Define

n > 0,

Cn = l (an- ibn),

(3)

Co= lao, Cn = l (a-n

+ ib-n),

n < 0.

If an and bn are real, then

n > 0, that is, they are the conjugate complexes of each other, and an and bn are given in terms of en by

an= Cn + C-n• bn = i(cn- e-n),

n ~o. n

> 0.

(4)

sn(x) takes the following form in terms of en: n

sn(x) = C0

+ l L [(ak- ibk)eil::t: + (ak + ibk)eikz] k-1

(5) The series

(6)

II.

50

FOURIER SERIES AND FOURIER TRANSFORMS

is called a trigonometric series in complex form. The sum (5) is called a trigonometric polynomial of order n in complex form. Now let us suppose that a function f(x) e Ll( -:n, :n) is given over ( -:n, :n). If in (1 ), an and bn are given by

n

an= :n- 1 J:,.f(x) cos nx dx, bn = :n-1

J:,. f(x) sin nx dx,

= 0, 1, 2, ... ,

n = 1, 2, ... ,

(7) (8)

.then the series (1) is called the Fourier series of f(x) and an and bn are called the Fourier cosine coefficients and Fourier sine coefficients, respectively. Together they are called the Fourier coefficients. In this case, the cn are given by

Cn = (Z:n)-1 J~ f(x)e-inz dx,

n = 0, ±1, ±2, ....

(9)

They are called Fourier coefficients in complex form. If f(x) is an even function, then bn = 0, n = 1, 2, ... , and an turns out to be

an = (2/n)

J:

f(x) cos nx dx,

n = 0, 1, 2, ... ,

(10)

and if f(x) happens to be an odd function, then

n = 0, 1, 2, ... ,

bn

= (2/:n)

J:f(x) sin nx dx,

n = 1,2, ....

In order to indicate that (1) is the Fourier series of f(x), we use the notation 00

f(x) """"' (a0 /2)

+ n-1 L (an cos nx + bn sin nx)

or, in complex form, 00

f(x} """"'

a., bn, and c" being given by

L Cneinz, n--co

(7), (8), and (9), respectively.

We note that

J"......

0, { cos mx cos nx dx = :n,

Z:n,

m =I= n, m = n =I= 0, m= n = 0;

(11)

2.2.

51

FOURIER SERIES

f_,. tc

•

s10

• mx s10 nx dx = { 0, :n:,

J~ sin mx cos nx dx =

0,

m 'i=n, m = n = 0, m=n'i=O for any pair

m, n.

(12) (13)

The following simple theorem follows from the orthogonality property (11 )-(13 ).

Theorem 2.2.1. If a trigonometric series (1) converges to a function f(x) uniformly for -:n: ~ x ~ :n:, then (1) must be the Fourier series of f(x). It is obvious that f(x) is continuous and periodic with period 2:n:. Proof.

Suppose that 00

(a0 /2)

+ n=l L (an cos nx + bn sin nx) =

f(x),

(14)

the series converging uniformly for [ -:n:, :n:]. Multiply both sides by cos mx and integrate them over ( -:n:, :n: ). We then see, using (11) and (13), that am should be the Fourier cosine coefficients. If we multiply both sides of (14) by sin mx and integrate them over ( -:n:, :n:), we find that bm must be the Fourier sine coefficients. This proves the theorem. This theorem suggests that if we consider the Fourier series when f(x) e V( -:n:, :n:) is given, then it might converge to f(x) at least under some conditions onf(x). Actually this is one of the major problems in the theory of Fourier series. Dealing with the Fourier series in complex form, we study the convergence of the symmetric sum (15)

when n - oo. The more general convergence of bilateral series is not considered, the relation (5) being taken into account. Ifj(x) is given over ( -:n:, :n:), we frequently make the periodic continuation, that is, we define a periodic function tfo(x) such that

t/J(x) = f(x),

tfo(x

+ 2:n:) =

tfo(x)

-:n:<x<:n:, for all

I x I ~ :n:.

lfj(x) is defined to be even at x = ±:n: andf(:n:) 'i= f(-:n:), then we have

II.

52

FOURIER SERIES AND FOURIER TRANSFORMS

to modify the values of f(x) at x = ±.n so that cp(.n) = cp( -.n). It is to be noted that the change of values of f(x) on a set of measure zero does not make any difference in the Fourier coefficients and hence the Fourier series. So, generally, we suppose, in the study of Fourier series, that a given function is periodic with period Z.n. If a function f(x) has period 2T, T > 0, and if f(x) E V(- T, T), then the Fourier series of f(x) is defined by f(x) "'(a0/2)

+L

(an[cos(n.nxfT)

+ bn sin(n.nxfT)])

(16)

or f(x) "'Len exp(in.nxfT),

where an= T- 1 I'l' f(x) cos(n.nxfT) dx,

n = 0, 1, 2, ... ,

(17)

bn = T- 1 I:'l' f(x) sin(n.nx/T) dx,

n

= 1, 2, ... ,

(18)

-'I'

and en= (2T)- 1 I:'l'f(x) exp(-in.nx/T) dx,

n = 0, ±1, ±2, .... (19)

2.3. The Fourier Transform of a Function in ! 1 (-oo,oo) Suppose that a given functionf(x) is defined over (-oo, oo) and belongs to L 1 (-oo, oo). We now introduce, as an analog of a Fourier coefficient in the case of a function defined on ( -.n, .n), the function

/(t) = (Z.n)-112

I:

e-i%tj(x) dx,

-oo < t < oo.

(1)

In the definition of Fourier coefficients [see Section 2.2, Eq. (9)], we had a constant factor (Z.n)-1, but here we take the constant factor (Z.n)- 112• This difference is actually not important at all. If a function f(x) is defined over the half-line (0, oo) and belongs to V(O, oo ), we define

/c(t) = (2/.n) 112 I: f(x) cos xt dx, / 8

(t)

=

(2/.n) 1' 2 I: f(x) sin xt dx,

0 < t < oo,

(2)

0 < t < oo.

(3)

2.4.

MAGNITUDE OF FOURIER COEFFICIENTS

53

They are analogs of Fourier cosine and sine series ;/(t),fc(t), and j 8 (t) are called, respectively, the Fourier transform, the Fourier cosine transform, and the Fourier sine transform of f(x). We formed the Fourier senes

based on Fourier coefficients en. The analog of the Fourier series will be the integral (4)

(except the constant factor), and we will naturally be interested m the convergence problem of this integral.

2.4. Magnitude of Fourier Coefficients; the Continuity Modulus Before investigating the convergence of Fourier series we shall study the rapidity with which Fourier coefficients converge to zero. Let an, bn be the Fourier coefficients of an integrable periodic function f(x) with period 2n and let en be the Fourier coefficients in complex form. First of all we mention the following theorem.

Theorem 2.4.1. If f(x) E LI( -n, n), then an, bn converge to zero as n -+ oo and en converges to zero as n -+ ±oo. This is just an immediate consequence of Theorem 2.1.1, because we may take, in place ofj(x) in Section 2.1, Eq. (1), the function lf>(x), which is identical with f(x) in ( -n, n) and is zero outside ( -n, n). Let (c 0 ) be the space of all sequences {an, n = 0, ±1, ... } of complex numbers such that an-+ 0 as n-+ ±oo. Then a sequence of Fourier coefficients of a periodic function of LI( -n, n) is an element of (c0 ). In other words, the Fourier coefficients define a mapping of LI( -n, n) to a proper subset of (c 0 ); that is, there is a sequence of (c 0 ) which is not a sequence of Fourier coefficients of any function belonging to LI( -n, n). Such an example will be given later. [See Section 4.1, Eq. (30).] Now we ask which conditions on f(x) will assure that the Fourier coefficients converge to zero more quickly. Suppose thatf(x) is periodic and absolutely continuous over (-n, n).

II.

54

FOURIER SERIES AND FOURIER TRANSFORMS

Then integration by parts gives us, for n =F 0, Cn = (2:n)-1

J:,. f(x)e-inx dx

= (2n)- 1 (e-inxf-in)f(x)

I:,.+

(2nin)- 1 J:,.f'(x)e-inx dx.

(1)

Note that f'(x) belongs to Ll( -n, n) since f is absolutely continuous. Then Theorem 2.1.1 shows that the last integral in (1) converges to zero as n ---. ±oo. The first term on the right-hand side of (1) vanishes because of periodicity. Hence Cn = o(1/n).

Repeating this argument we easily get the following theorem.

Theorem 2.4.2. If f(x) is periodic and the kth derivative of J<"l(x) is absolutely continuous, then when n ---. oo, k = 0, 1, 2, ... ,

(2)

and k

= 0, 1, 2, ....

(3)

This result suggests, roughly speaking, that the smoother a function becomes, the more quickly the Fourier coefficients go to zero. To achieve this idea, we introduce the continuity modulus of a function. In general, we take a function defined over a finite interval [a, b]. Let «5 > 0 be any number less than b - a. The quantity ro(d;f)=

sup

Jf(x2)-f(x1)J

(4)

lx 1 -x2 1::ad,a;:;axl,x 2;:;ab

is called the continuity modulus of f(x) in [a, b]. The ordinary continuity of f(x) on [a, b] is equivalent to the condition that ro( «5; f) ---. 0 as «5 ---. 0+. It is easy to see that if f 1(x) and f 2 (x) are two functions over [a, b], then (5) ro(d;f1 f 2) ;;;;; ro(d;f1) ro(d;f2),

+

+

ro().d;f) ;;;;; (). for any positive ). with ).«5

+ 1)ro(d;f)

(6)

< b - a. If ). is a positive integer we have

ro( ).«5; f)

~

Aro( «5; f).

(7)

2.4.

55

MAGNITUDE OF FOURIER COEFFICIENTS

(6) and (7) are seen in the following way. Let x 1 , x 2 be any two numbers such that I x 1 - x 2 I ~ A.15. Suppose x 1 < x 2 • We may find t 1 , t 2 , ••• , t; with the property that t 1 = x 1 + 15, tk+l = tk + 15 (k = 1, ... , j - 1 ), x 2 = ti ~ 15. The number of intervals (x 1 , ti), (t 1 , t 2 ), ••• , (ti, x 2 ) is at most A. if A. is a positive integer and [A.] + 1 if A. is not an integer, where [A.] denotes the greatest integer not larger than A.. Therefore,

Each term on the right-hand side is not greater than sup

I/(yl)-/(yz) I =

w(15;f),

I111-lla I :S"

and therefore, if if

A. is a positive integer, A. is not a positive integer.

Since [A.] + 1 ~ A.+ 1, we get (6) and (7) by taking the supremum of the left-hand side over I x 1 - x 2 I ~ A.l5. If f(x) satisfies the condition that, for any 15 < 0,

w(l5;f)

~

0
Kl5",

(8)

with a constant K, then f(x) is said to satisfy the Lipschitz condition of order a, and this is expressed by

f(x)

E

Lip a.

In fact Lip a is the class of functions satisfying the Lipschitz condition of order a. Now a is restricted in (0, 1] because if a > 1, then condition (8) implies that f'(x) = 0, and hence f(x), is identically a constant. If a function is defined over (a, b) and we talk about the modulus of continuity in a subinterval [a, {J] (a
w(l5;f; a, fJ) =

sup

If(x +h)- f(x) I,

(9)

O
where 15 is a positive number such that x ± 15 e (a, b), that is, 6 ~ min(a - a, b - fJ). Thus, if f(x) is periodic with period 2:n:, w(15;f) = w(15;f;-:n:, :n:) is

II.

56

FOURIER SERIES AND FOURIER TRANSFORMS

defined simply to be

w(tl;f) =

sup

I f(x +h)- f(x) I·

(10)

O
-oo
A more general notion of continuity is the integral modulus of continuity, which is defined in the following way. Letf(x) E LP(a, b), p > 0. The integral modulus of continuity ofj(x) in a subinterval (a, P) of (a, b) is defined by

w
(11)

IX

where tJ ~ min(a- a, b- p). For a periodic function f(x),

w
rI

f(x +h)- f(x)

IP dx.

(12)

-,.

It is obvious that if f(x) is continuous and periodic with period 2n,

2.5. More about the Magnitude of Fourier Coefficients Let f(x) be a periodic function with period 2n. We are going to find the bounds of Fourier coefficients.

Theorem 2.5.1. For Fourier coefficients of a periodic function f(x)

E

V(-n, n), we have

(1) and

for n = 1, 2, .... For Fourier coefficients of a continuous periodic function f(x), we have

I Cn I

~

!w((nfn);f),

I an I ~ w((nfn);f), for n = 1, 2, ....

(3)

Ibn

I ~ w((nfn);f)

(4)

2.5.

57

MORE ABOUT THE MAGNITUDE OF FOURIER COEFFICIENTS

Proof.

en= (Zn)- 1

J:,. j(x)e-inx dx

= -(2n)- 1

J:,. J[x + (nfn)]e-inx dx,

which is shown by the change of variable x in the first integral to x + (nfn). Adding these two forms of en and dividing by 2, we get

e11 = (4n)- 1

J:,. [J(x) -J(x +

: )] e-inx dx,

from which we have

I en I ~

(4n)- 1

J:,. lf(x) -f[x + (nfn)] I dx ~ (4n)- w< 1

0

((n/n),J). (5)

This is Eq. (1). Equations (2)-(4) can be obtained in a similar way. Now we shall show that, while the Fourier coefficients of any integrable function converge to zero, adding the further condition of continuity on the function does not give any information concerning the rate of decrease of Fourier coefficients. More precisely, we can show the following:

Theorem 2.5.2. For any given u11 > 0 with limn-+oo Un = 0, there exists a continuous periodic function f(x) such that its Fourier coefficients en satisfy

(6) n-+oo

Proof. We can determine n 1 < n 2 < · · · such that Choose, for instance, n. so that un. < v- 2 • Define f(x) by

L~ 1

un.

<

oo.

00

f(x)

=

L un. cos n.x, 1

which is absolutely and uniformly convergent. Hence f(x) is continuous and periodic. un. is the Fourier coefficient of f(x) by Theorem 2.2.1. Since f(x) is an even function,

2c-n =a" { Zen-

un.' 0,

(an: cosine coefficients, en: Fourier coefficients in complex form) n = n., n #n •.

This gives us lim sup e±nfun = 1. n-+oo

IJ.

58

FOURIER SERIES AND FOURIER TRANSFORMS

The theorem was shown by Lebesgue (see Zygmund [1 ], p. 38) and also by Randels [1] and Szasz [1]. However ifj(x) E Lip a, 0
Theorem 2.5.3. If a periodic function f(x) satisfies the Li1 "hitz condition of order a (0 < a < 1), then en= O(n-'"),

n--+

±=,

(7)

an= O(n-'"),

(8)

·This is an implication of Theorem 2.5.1, (3) and (4). The proof will be immediate. (7) and (8) remain true if a = 1. But the following theorem gives a slightly more general result since a function of Lip 1 is a function of bounded variation.

Theorem 2.5.4. Suppose that a periodic function f(x) is of bounded variation. Let the total variation of f(x) over [ -n, n] be T( -n, n). Then

I Cn I ~ (4n)-1 T( -n, n), I an I ~ (2n)- 1 T(-n, n), Proof.

(9) Ibn

I ~ (2n)- 1 T( -n, n).

(10)

From (5),

I en I ~ (4n)-1 J:,. If(x + (n/n))- f(x) I dx. For a periodic function, this integral is equal to (4n)- 1

J:

IJ[x

+ (knfn)] -f{x + [(k-1)n/n]} I dx

for any integer k. Adding these integrals fork = -n and dividing the sum by 2n, we obtain

I Cn I ~ (8nn)

r

t

-n k--n+l

~ (T(-n, n)f8nn)

+ 1, ... , 0, 1, ... , n

I f(x + (knfn))- J[x + ((k -1)nfn)] dx

J:,. dx =

T(-n, n)f4n.

A similar argument applies to obtain (10).

2.6.

59

SOME ELEMENTARY LEMMAS

2.6. Some Elementary Lemmas For later use, it will be convenient to give here some elementary lemmas.

Lemma 2.6.1. For any real a

leirz_11 More generally for any complex a

Proof.

~lal.

=

a

(1)

+ i{J, a, fJ real, for

fJ

~

0,

(2)

for

fJ

~

0.

(3)

It is sufficient to show (2) and (3) for a -=1= 0.

I eia -

1I=

Iia J:

eiau

du

I ~ I a I J: I

eiau

I du = I a I

J:

e-flu

du.

(2) and (3) are obvious from this.

Lemma 2.6.2. (i)

For any real a, b

IJ: (sin tft) dt I ~ C

(4)

for some absolute constant C. (ii)

f

-+00

0

(5)

(sin tft) dt = nf2.

Proof. It is sufficient to show (4) for 0 ~a ~b. Let 1 ~a
I:

(sin tft) dt

= a- 1

J:

sin t dt

Hence

= a- 1 (cos a - cos C)

II: (sin tft) dt I~ 2.

IfO~a~b~1

-

-

-' 0

~

J:

(sin tft) dt

~

J:

dt

~

1.

for some C.

(6)

~ a ~

For the case 0 1

s:

1

~

b,

(sin tft> dt 1 ~ 1

s: + s: 1

1

1

~ 1+ 2=

3.

Hence (4) has been shown with C = 3 for 0 ~a
I~ e-vt cos ut dt

=

vf(u 2 + v2),

v > 0.

Integrating both sides for u from 0 to 1, we find

I~ e-vt(sin tft) dt = arctan v- 1 •

(7)

Here the interchange of the order of integration is permitted since the repeated integral is absolutely convergent (integrable in the Lebesgue sense) and arctan means the principal value, -n/2 <arctan v < n/2. The left-hand side integral in (7) is uniformly convergent for v > 0. Because the application of the second mean value theorem gives us

I

I:

e-•t(sin tft) dt

I~ e-A•I (

(sin tft) dt

I

for some A < C < B and the last term is equal to or less than 2/ A by (6 ), therefore for any e > 0, there is an A 0 which is independent of v such that

II: e-• (sin tft) dt- arctan I< e v- 1

1

for A ;;;; A 0 • Letting v - O+ in this inequality, we get

II: (sin tft) dt- (n/2) I~ e

A ;;;; A 0 ,

for

which shows (5). Lemma 2.6.2 is now proved. We now give a more accurate value of C in (4) when 0 ~ a < b. Let x be any positive number. Let k be a nonnegative integer such that kn ~ x < (k + 1)n. Then

Ioz(sin tft) dt = ML Ili+Un . (sin tft) dt + Jz k-1

~

~

(sin tft) dt.

(8)

2.6.

61

SOME ELEMENTARY LEMMAS

Now

J,,..

(j+l)n

•

(s1n tft) dt

is positive or negative according as j is even or odd and

IJ::+lln

(sin tft) dt

I= J::+lln (J

sin t

Jft) dt

> I'~+Zln

(J+lln (J sin t J/t) dt (j+2)n I II(J+lln . (sin tft) dt .

If this relation is applied to (8), then we easily see that

I:

(sin tft) dt > 0

and that the maximum of words, we have shown 0<

J:

for

J: (sin tft) dt

(sin tft) dt ~

x > 0

(9)

is attained at x = :n. In other

I:

(10)

(sin tft) dt

for every x > 0. Using the left inequality,

I:

(sin tft) dt <

I:

(sin tft) dt

for any 0 < a < b. Thus we finally have

I:

(sin tft) dt

~

I:

(sin tft) dt

for any 0 < a < b.

~

I:

(sin tft) dt

for any 0 < a < b.

We also have

-I: Since

(sin tft) dt

I:

(sin tft) dt = 1.85194 ... ,

we have the result that

IJ: (sin tft) dt I~ 1.85195

for 0

~a<

b.

(11)

II.

62

FOURIER SERIES AND FOURIER TRANSFORMS

Lemma 2.6.3.

J:oo (sin xtft) dt =

X> 0,

n/2, { 0, -n/2,

X=

0,

(12)

X< 0.

The integral is boundedly convergent. This is immediate from Lemma 2.6.2.

Lemma 2.6.4. For A> 0, (2/n)

I xl >A, I xl =A, I xl
r~oo (sin Atjt) cos xt dt = { ~:1 0

•

(13)

where the improper integral on the left-hand side is boundedly convergent for A and x. This is easily seen from the preceding lemma because f ~oo (sin Atjt) cos xt dt = 0

l

f~oo

0

+l f

(sin(A

+ x)tft) dt

~oo

0

(sin( A- x)tft) dt.

2.7. Continuity and Magnitude of Fourier Transforms We are going to discuss, for Fourier transforms, mostly the things we have discussed for Fourier series. Write 11/11 1 = J:"oo 1/(x) I dx, when f(x) E Ll(-oo, oo).

Theorem 2.7.1. Letf(x) E Ll(-oo, oo) and let its Fourier transform be /(t), -oo < t < oo. Then (i) lime-+±oo/(t) = 0, (ii) 1/(t) I ~ (Zn)-112 11/111, and (iii) /(t) is uniformly continuous over (-oo, oo).

(1)

Proof. (i) is no more than the Riemann-Lebesgue lemma (Theorem 2.1.1). (ii) is obvious from the definition of /(t). For (iii),

/(t +h)- /(t) = (Zn)- 1' 2

J:

eilz(eihx-

1)/(x) dx.

2.7.

CONTINUITY AND MAGNITUDE OF FOURIER TRANSFORMS

63

Hence

lf
Theorem 2.7.2. For any given sequence u,1 > 0 such that lim,.-+oo u,. = 0, there is a functionf(x) eLI( -oo, oo), whose Fourier transform](t) satisfies the condition that lim sup(l](±n) 1/u,.)

> 0.

(2)

n-+oo

Proof.

Define, as in the proof of Theorem 2.5.2, f(x)

={

~ un, cos n,x

for

I xI ;;;;;;; n,

0,

(3)

where {n.} of the integers is determined in such a way that 0 < n 1 < n 2 < · · ·, :Liu,. < oo. Then

•

](t) = (Zn)- 1' 2

and ]( ±n) = (2n )- 1' 2

J:,. cos txf(x) dx ~1 u,.. f"_, cos nx cos n,x dx. 00

II.

FOURIER SERIES AND FOURIER TRANSFORMS

(The interchange of integration and summation is permitted since the series (3) is uniformly convergent.) Hence /( ±n) = {

~nf2)1 12un,

if n=n., if n=Fn.,

and lim sup[/(±n)fun] = (n/2)112, n-+oo

This shows the theorem.

Theorem 2.7.3. Ifj(x) e V(-oo, oo) is absolutely continuous and f(x)

J:oof'(u) du

=

with/'(x) e V( -oo, oo), then the Fourier transform/(t) ofj(x) satisfies

/(t) = o(l t l-1 )

as t--+ ±oo.

(4)

Moreover, if the 11th derivativej<•1(x) belongs to V( -oo, oo) for"= 0, 1, 2, ... , k, then

/(t)

=

o(1fl t II:)

as t--+ ±oo.

(5)

This is the analog of Theorem 2.4.2. The proof presented there is easily carried over if it is noted thatj<•1(x) e V(-oo, oo), "= 0, ... , k, impliesj<•1(±oo) = 0 for"= 0, 1, ... , k- 1. The analog of Theorem 2.5.1 is the following:

Theorem 2.7.4. If/(t) is the Fourier transform ofj(x) e V( -oo, oo), then (6) 1/(t) I ~ !(2n)-112 wUI((nfl t l);f), t =F 0, where w 111 (6;f)

=

sup O
Joo

lf(x +h)- f(x) I dx.

-co

2.8. Operations on Fourier Series Let/1(x) and/2(x) be periodic with period 2n and belong to V( -n, n) and let the Fourier coefficients of / 1(x) and / 2(x) be c~', c1:', n = 0, ±1, ... , respectively. We then easily see that:

2.8.

65

OPERATIONS ON FOURIER SERIES

(a) The Fourier coefficients of ad1 (x) + aJ2(x) with constants a 1 , a 2 are given by a 1c~, + a~~l, n = 0, ±1, .... Let en, n = 0, ± 1, ... , be the sequence of Fourier coefficients of a periodic function f(x) E £1( -n, n). Then we can easily show that: (b) The Fourier coefficients ofj(x- y) are given by cne-inv, n = 0,

±1, .... (c) The Fourier coefficients of f(x)eims are given by cn-m• n = 0,

±1, .... The space (c0 ) (Section 2.4) is a Banach space with norm

II c II

=

max

--oo
I Cn I,

where c

=

{en,

n

=

0, ±1, ... }.

If en, n = 0, ±1, ... , are Fourier coefficients ofje £1(-n, n), then for c = {en}, we have

II c II ~ (2n)-l f~ lf(x) I dx =

(2n)-1 11/llt·

(1)

From this fact and (a), we may say that the transformationf(x)--+ {en} from £1( -n, n) into (c 0 ) is linear and bounded. Next we discuss the integration of Fourier series. Let f(x) E £1( -n, n) be periodic, and let its Fourier series be 00

j(x) ,_

L Cneinz.

(2)

-oo

Let. (3)

F(x) = J:J(u) du. F(x) is not periodic in general. Now we consider

(])(x) = F(x) - CoX.

(4)

This becomes a periodic function (with period 2n). For, (])(x

+ 2n) =

J

z+2

__, f(u) du- CoX- 2nc0

= J:J(u) du

+ J~"" f(u) du- CoX- 2nc

Here the second term on the right-hand side is

J:

f(u) du = 2nc0 •

0•

II.

66

FOURIER SERIES AND FOURIER TRANSFORMS

Thus tP(x

+ 2n) =

F(x)-

C&X

= tP(x).

Now tP(x) is absolutely continuous and C/>'(x) = f(x) - c0 a.e. Let co

tP(x) """"

L

Cneinz.

n---oo

Then, for n =I= 0, en = (2n)-l

I:

tP(x)e-inz dx

= (2n)- 1 tP(x)(e-inzf-in)

~~ -(2:n)-1 I~

(f(x)- c0 )(e-inzf-in) dx.

The first member on the right-hand side vanishes because of the periodicity of tP(x) and therefore, en = (2:n)- 1

f: . J(x)e-inz dx (in)-

1-

(cof2:n)

I:. e-inz dx (in)-

1

or (5) For n = 0, C 0 = (2n)-1

I:

dx

tP(x)

~~ -(2:n)-1 I~ x[f(x)- c0 ] dx

=

(2n)-1xtP(x)

=

(2n)-1[ 2ntP(-:n)-

=

(2n)-1[ 2n2c0 -

J:,. xf(x) dx]

J:,. xf(x) dx].

Hence Co= (2n)-1

I:

(:n- x)f(x)

dx.

(6)

We write co

F(x)- C&X """"C0

+n---oo L' (cnfin) einz,

(7)

where ~' denotes the series in which the term with subscript 0 is missing.

2.9.

67

OPERATIONS ON FOURIER TRANSFORMS

In real form, (7) becomes 00

F(x)- !aoX ,....., C 0

+ n=l L (-bn cos nx + an sin nx)fn,

(8)

where C 0 is given again by (6). Later we shall show that the series (7) and (8) converge uniformly, without any further condition onf(x). (See Section 4.1, Theorem 4.1.5.) Next we form the termwise-differentiated series of the Fourier series. If we differentiate the Fourier series (2) termwise, we have the formal series L incneinz (9) 00

n=--oo

and in real form 00

L n(bn cos nx- an sin nx).

(10)

n=l

These are .not necessarily the Fourier series of some function of L 1 ( - oo, oo) because the coefficients en, nbn, -nan do not have to converge to zero as is seen from Theorem 2.5.2. If, however, f(x) is absolutely continuous, they are the Fourier series ofj'(x). The series (9) and (10) are sometimes called the derived Fourier series, but it must be kept in mind that they do not have to be Fourier series.

2.9. Operations on Fourier Transforms We shall discuss the Fourier transform analog of what we have done in the preceding section on Fourier series. Letf1 (x) and/2 (x) belong to V( -oo, oo) and their Fourier transforms be / 1 (t) and / 2(t), respectively. --;

The Fourier transform of ad1(x) + atf2 (x) with constants a1 and a1 1s given by ad1(t) + ad2(t). ~~

Let f(x) e V(- oo, oo) and its Fourier transform be /(t).

{bj

The Fourier transform of J(x- y) is given by j(t)e-i"1• ~J' The Fourier transform ofj(x)ei•z, s being real, is given by /(t- s). The space (C0 ) (see Section 2.7) is a Banach space with norm II/II= max IJ(x) -oo
1.

j(x)

E

(C0 ).

II.

68

FOURIER SERIES AND FOURIER TRANSFORMS

From Section 2.7, Eq. (1) and (a) above, we conclude that the Fourier transformation from £1(-oo, oo) to (C0 ) is linear and bounded. Now suppose thatf(x) E £1(-oo, oo), xf(x) E £1(-oo, oo), and

I~oof(x) dx = 0.

(1)

Then

F(x) = J: 00 J(u) duE £1(-oo, oo) because, using (1 ),

I

I~ I F(x) I dx = I~ dx I~ f(u) du

I

~I~ dx I~ lf(u) I du =I~ lf(u) I du I: dx -I~ u I f(u) I du < oo. Similarly, we may show that

(oo I F(x) I dx < oo. Then the Fourier transform of F(x) is given by (2n )-112[/(t) fit]

P(t)

~ { -(Zn)-'" J: •/(•) du

for

t ::;t: 0

for

t = 0.

This is seen by using integration by parts. Actually, for t ::;t: 0,

I~oo e-ilz dx J:oo /( U) du = (e-ilzJ-it) J: 00 J(u) du

~~oo -(-it)- 1 I:J(x) e-ilz dx

= /(t)fit

and, for t = 0,

I:

dx (oof(u) du

=-I~ dx I~ f(u) du + (oo dx J:oof(u) du = - I : uf(u) du-

=-I: uf(u) du.

(oo uf(u) du

(2)

2.10.

69

COMPLETENESS OF TRIGONOMETRIC FUNCTIONS

If condition (1) and the condition that xf(x) e £1(-oo, oo) are not supposed, then F(x) = J:ooJ(u) du is not necessarily integrable and we cannot define the Fourier transform of F(x). However, sometimes we consider

(2n)- 1' 2 J

[J(t)fit]e-itz dt Ill>"

for

<5

> 0.

Here to avoid the singularity at t = 0, which may occur, we exclude a neighborhood of the origin. But usually we consider the more convenient formal integral (3) (Zn}- 112 J:J(t)[(eitz -1)/it] dt. Now, if f(x) e £1(-oo, oo) is absolutely continuous and f'(x) e £1(-oo, oo), then the Fourier transform of /'(x) is given by it /(t). This is easily seen and as a matter of fact it was used in the proof of Theorem 2.7.3 although it was not explicitly given. Finally, we mention that if (1 +I x i)f(x) E £1(-oo, oo), then the Fourier transform of xf(x) is given by i]'(t). This is immediate because

can be differentiated under the integral sign since xf(x)e V(-oo, oo).

2.1 0. Completeness of Trigonometric Functions We now consider a unicity problem. Do Fourier coefficients determine the function ? This question is very basic in nature. If it happened that two different functions had the same Fourier series, then we should have a very unpleasant situation, in so far as we are interested in the problem of the expansion of a function into Fourier series. This question relates several important problems in analysis. Our problem is: Do the

Cn = (2n)-112 f:J(x)e-inz dx,

n = 0, ±1, ... ,

(1)

uniquely determine/( x) ? More extensively we may think of the following two problems:

II.

70

FOURIER SERIES AND FOURIER TRANSFORMS

(i) Let {c11 } be a given sequence. Under what conditions does there exist a function f(x) which satisfies (1)? (ii) If there is a function f(x) satisfying (1 ), is f(x) determined uniquely? Such a problem is called a trigonometric moment problem. More generally, suppose that we are given a sequence {cpn(x), n = 1, 2, ... } of functions of a certain class A of functions. Also, a sequence {c,, n = 1, 2, ... } of real or complex numbers is supposed to be given in advance. We assume another class B of functions. (i)

Is there a function f(x)

Cn = (ii)

E

B such that

Jf(x)cpn(x) dx,

n

=

1, 2, ... ?

If there were, would such a f(x) be uniquely determined in B?

Of course, the classes A and B must be such that the integral Jfcp exists for fEB, cp EA. The integrals may sometimes be replaced by the Stieltjes integral. Here we are not concerned with such a general moment problem. Later we shall study some of the special cases. For the Stieltjes integral case, see Section 11.1. For the most general case see, Banach [1 ], Davis [1 ], and Kaczmarz and Steinhaus [1]; and, for special moment problems, see Akhiezer [1] and Shohat and Tamarkin [1 ]. We introduce the following terminology. Let cp 11 (x) E Lq(a, b), n = 1, 2, ... , a, b being finite or infinite, and (1/p) (1/q) = 1, p ~ 1, q ~ 1. If

+

J:J(x)cp 11 (x) dx

=

0,

n

=

1, 2, ... ,

(2)

for a f(x) E LP(a, b) implies that f(x) = 0 almost everywhere, then {cp 11 (x)} is called complete in LP(a, b). For such a completeness problem, the reader may refer to Kaczmarz and Steinhaus [1] and Davis [1]. Here it is interpreted that L""(a, b) means the class B(a, b) of bounded functions over (a, b). Here we just consider the simplest case, stated at the beginning of this section, and prove the following theorem.

Theorem 2.10.1. A sequence {e-inz, n V(-:n:,:n:).

=

0, ±1, ... } 1s complete in

2.10.

COMPLETENESS OF TRIGONOMETRIC FUNCTIONS

In other words, if, for f(x)

E

(,. f(x)einz dx

71

£1( -n, n),

=

0,

n

= 0, ±1, ... ,

(3)

then f(x) = 0 a.e. Proof. Since cos nx and sin nx are linear combinations of einz and from (3) we have (,. f(x) cos nx dx = 0, (,.f(x) sin nx dx

=

n = 0, 1, 2, ... , n

0,

=

1, 2, ... .

e-inz,

(4a) (4b)

Now we may suppose without loss of generality thatf(x) is real-valued. From (4a) and (4b) we should have (,.f(x)T,(x) dx = 0

(5)

for any trigonometric polynomial (of real form) T,(x). Suppose first that (5) holds for a continuous functionf(x) over ( -n, n) apd suppose that f(x) is not identically zero. Then there is a point C, -n < C < n, such that/( C) #- 0. We may suppose that/( C)> 0, unless we may take -f(x) instead ofj(x). Since f(x) is continuous, there is an interval I = ( C- b, C+ b), -n < C- b < C+ b < n, in which f(x) > e > 0 for some e > 0. Construct the function p(x)

= 1 + cos(x + C) - cos b

which satisfies the conditions that p(x)

~

1

I p(x) I < 1

m /,

(6)

outside /.

(7)

Define (8)

which is a trigonometric polynomial of order n. (5) holds for this particular T,(x). We now break up the integral

f,.

-n

f(x)T,.(x) dx =

J+J I

I

c

II.

72

FOURIER SERIES AND FOURIER TRANSFORMS

and observe that

frJ(x)Tn(x) dx ~ e fr T,.(x) dx ~ e fr dx = 2e«5, and that lim J c f(x)T,.(x) dx = 0

n-+oo

I

since lim T,.(x) = limp"(x) = 0, in view of (7), and f(x)T,.(x) is uniformly bounded over Jc. Therefore we may choose n so large that

Hence

I

(,.J(x)T,.(x) dx

I> -I Ire I~ fr

2e«5- e«5 = e«5

>

0.

This contradicts (5). Thus we have shown that ifj(x) is continuous, then f(x) = 0 identically. Now suppose (5), with f(x) e Ll( -n, n). Define

F(x) =

J:

J(u) du.

Note that F( -n) = F(n) = 0. The latter equality follows from (4a) with n = 0. Integration by parts shows that

J:

F(x)einz dx

=

F(x)(einzfin) /:,. -(in)- 1 J:J(x)einz dx

=0, Set tP(x) us

and also

=

F(x) -a, where a

J:

n -F 0.

for

=

(Zn)- 1 J~,. F(x) dx. Then (9) gives

IP(x)einz dx = 0,

J:

(9)

n-F-0,

tP(x) dx = 0,

by the definition of tP(x). Since tP(x) is continuous, from what we have shown above, tP(x) = 0 identically, namely, F(x) =a identically. But since F( -n) = 0, a should be zero, which gives us F(x) = 0 identically, which in turn implies that f(x) = 0 a.e. This concludes the proof.

2.11.

UNICITY THEOREM FOR FOURIER TRANSFORMS

73

So far we have shown that {e-inz, n = 0, ±1, ... } is complete in £1(-:n, :n). Note that {e-inz, n = 0, ±1, ... } is not complete in LI( -:n-!5, :n + !5), !5 > 0. This is shown by a counterexample:

-1 f(x)

o:

={

1,

< < <

-:n - f5 -:n f5 :n - f5

+

X

X X

< -:n + !5, < :n - !5, < :n + !5.

(10)

(See Levinson [4], p. 2.) It is easy to verify. Actually we may say that {e-inz, n = 0, ± 1, ... } is not complete in £1 over an interval of length greater than 2:n. The completeness properties were studied by Levinson [4] in great detail.

2.11. Unicity Theorem for Fourier Transforms The analog of Theorem 2.10.1 for the Fourier transform is: ~

Theorem 2.11.1. If f(x) e LI(- oo, oo) and its Fourier transform is identically zero, then f(x) .Proof.

= 0 almost everywhere .

Suppose

J~oo e-ilz f(x) dx

= 0.

(1)

Then (2)

where c/J(x) is defined to be f(x- a), where a is any real constant. Now multiply both sides of (2) by sin tuft and integrate over (- T, T) with respect to t, where u is any positive constant and T is any positive number. We then obtain

Inverting the order of integration, which is obviously permitted,

r:

c/J(x) dx

rT

(sin tuft)e-itz dt = 0.

We may let T--+ oo inside the outer integral since

IT (sin tuft)ritz. dx -T

=

IT -T

.

(sm tuft) cos tx d t,

(3)

74

II.

FOURIER SERIES AND FOURIER TRANSFORMS

which is uniformly bounded, as is seen from Lemma 2.6.4. Furthermore, because of the same lemma, T (sin tuft) cos tx dt = 2 IT (sin tuft) cos tx dx I -T 0

converges to D(x, u) as T---+ oo, where D(x, u) = {

~

I xI > u, I xI < u.

for for

(4)

Therefore we have

I:

cp(x)D(x, u) dx

=0

for

u > 0,

which is no more than

I~ cp(x) dx = 0; that is,

I

u+a -u+a

J(x) dx

= 0.

Since a is arbitrary, this means that J~f(x) dx = 0 for every pair a, fJ such that fJ >a. This implies thatf(x) = 0 a.e. This completes the proof.

2.12. Fourier Series and Fourier Transform of Convolutions Let f(x) and g(x) belong to V( -oo, oo). Consider I:J(x- y)g(y) dy.

(1)

This integral does not have to exist for every x, since the product of two functions belonging to V(-oo, oo) is not necessarily integrable. However, we can show that (1) is well defined if we exclude some set of x of measure zero. Consider the function of two variables x, y lf(x- y)g(y)

I,

(2)

which is of course measurable as a function of x or of y or of x, y. The

2.12.

75

FOURIER SERIES AND TRANSFORM OF CONVOLUTIONS

repeated integral of (2) is

J~oo

dy

J~oo I f(x- y)g(y) I dx = J~oo Ig(y) I dy J~oo I f(x- y) I dx =

J: I

g(y) I dy

J: I

f(u) I du,

(3)

which is the product of two finite integrals and so

J~oo dy

J: If(x- y)g(y) I dx < oo.

Then the Tonelli theorem gives us

J~oo dx

J: I f(x- y)g(y) I dy < oo,

which is equal to (3). Also, in particular,

J~oo I f(x- y)g(y) I dy is finite for almost every x. Thi;; shows what we wanted to show, namely:

Theorem 2.12.1. Ifj(x), g(x) h(x)

E

£1(-oo, oo), then

= J~00 j(x- y)g(y) dy

(4)

exists as the Lebesgue integral for almost every x and h(x) Furthermore

J: h(x) dx

=

E

J~00 j(x) dx J~oo g(x) dx.

£1(- oo, oo ). (5)

The function h(x) is called the convolution ofj(x) andg(x), and is denoted by (/ • g)(x) or simply f • g if no confusion can result. If f(x) and g(x) are defined over (0, oo), then h(x)

= J:J(x-y)g(y)dy =f•g,

X> 0,

(6)

is called the convolution ofj(x) andg(x). This is precisely (4) ifj(x) and g(x) are extended to (-oo, oo) with f(x) =g(x) = 0 for x < 0, and hence h(x) E £1(0, oo). If /(x), g(x) are periodic functions (with for instance, period ~) be-

II.

76

FOURIER SERIES AND FOURIER TRANSFORMS

longing to £1( -:n, n), then (7)

h(x) = J:J(x-y)g(y) dy =f•g

is the convolution, which is again periodic with period 2n and belongs to £1. We now consider the case J, g E £1(- oo, oo ). We see that

(8) for

h(x) = J:J(x- y)g(y) dy = J~oof(z)g(x- z) dz. If/1,/2./a, all belong to £1( -oo, oo), then, since/1 • fa we may form (/1 • fa) • fa and we easily see that

E

£1( -oo, oo),

(9) Hence the convolution operation is commutative and associative.

Theorem 2.12.2. If J, g e £1(- oo, oo) and their Fourier transforms are/, g, then the Fourier transform off • g is (2:n)112](t)g(t). Proof.

(2n)-1ta

J:

e-itz dx J:J(x- y)g(y) dy

= (Z:n)-1/a J:g(y) dy J:J(x _ y)e-itz dx

(the interchange of the integrations is permissible)

= (Z:n)-112 J : g(y)e-itv dy J~00 j(x _ y)e-it
J:

g(y)e-itll dy J:J(z)e-itz dz,

which proves the theorem. The relation (9) is also true for the case in which J, g E £1(0, oo) or J, g E £1( -:n, n), periodic. The result in Theorem 2.12.2 is also true for Fourier series.

Theorem 2.12.3. If 00

j(x) ,....,_ L --<>0

00

Cneinz,

g(x) ,....,_

L dneinz, --<>0

2.12.

77

FOURIER SERIES AND TRANSFORM OF CONVOLUTIONS

then 00

(f • g)(x) _,

L 2ncndneinz.

(10)

We shall offer a few remarks. Let B(a, b) denote the class of functions bounded over (a, b). If f(x) E B(-oo, oo), g(x) E £1(-oo, oo) or f(x) E £1(-oo, oo), g(x) E B(-oo, oo), then

= J:J(x- y)g(y) dy

h(x)

is defined for every y and h(x) e B( -oo, oo). Furthermore h(x) is uniformly continuous, since, for the case in which f e £1(- oo, oo) g(x) E B(-oo, oo),

I

I h(x + h)- h(x) I = J~oo

[f(x

+ h - y)- f(x- y)]g(y) dy I

~ C J~oo lf(x + h-y) -J(x- y) I dy, where I g(x)

I ~ C. The last integral is C

J: I

f(z

+ h)- f(z) I dz,

which does not involve x at all and converges to zero as h -+ 0 (Theorem 1.9.3). We also have

~C J~oolf(z)ldz=CIIflll·

lh(x)l

(11)

However, h(x) is not necessarily in £1(-oo, oo), as is easily seen by a simple example: f(x)

E

£1( -oo, oo),

g(x)

= 1.

If f(x), g(x) both belong to L 2 ( -oo, oo), then h(x)

= J~00 j(x- y)g(y) dy

is defined for all x and h(x) is bounded and uniformly continuous and

I h(x) I ~ II/ ll2ll g ll2· If f(x)

E

LP(-oo, oo),

g(x) e L"(-oo, oo),

P > 1, q > 1, we have a similar result.

(12) (1/p)

+ (1/q) =

1,

78 EXAMPLE.

II. Let f(x)

E

FOURIER SERIES AND FOURIER TRANSFORMS

£1(0, oo) and construct L(x)

= J~ e-zvj(y) dy.

(13

If x > 0, (13) exists. The function L(x) is called the Laplace transform of f(y). (See Chapter VII). Write y = e-f}, x = e", -oo < u, v < oo. Then (13) turns out to be

If we set g 1 (u)

=

exp( -e"), g 2(u) L(e")

= f(e-")e-",

then

= J~oo g 1(u- v)g 2(v) dv = g 1 • g 2 •

In this case, g 1 E B( -oo, oo), g 2 (u) E £1( -oo, oo). If f(x) E B(-oo, oo) in (13), then we may take g 1 g 2 E B(- oo, oo ). In this case, we may also take

E

£1(-oo, oo),

where

h1 (x)

=

h1

E

£1( -oo, oo),

h2 (x)

E

B(-oo, oo).

exp( -eZ) eZ,

h2(x) = f(e-z),

NOTES 1. In connection with Theorem 2.1.3, the following theorem is known (Prasad [1]): Suppose that the conditions (i) and (ii) on K(x) in Theorem 2.1.3 are satisfied. If f(x) is locally integrable and for some A > 0, c/> 1 (x) = x- 1 f~ uf(u) du, x > A and c/> 1 (x) = x- 1J;;A uf(u), x < -A are of bounded variation over [A, oo) and (-oo, -A], respectively, then the integral

f

-+oo f(x)K(Ax) dx -+-co

exists for A of:- 0 and converges to 0 as A = ± oo. The condition thatf(x) e V(-oo, oo) implies the condition onf(x) in this theorem. 2. In the above theorem (Note 1), the condition ofj(x) can be replaced by,the condition that e-z f~ e"f(u) du and e" f;;A r"f(u) du are of bounded variation over [A, oo) and (-oo, -A], respectively. This result generalizes both the theorem in Note 1 and Theorem 2.1.3 (Cassar [1]).

79

NOTES

3. For conclusions similar to the one in Theorem 2.1.3, some other conditions onf(x) are known. See Cassar's paper quoted above and also Izumi [1].

4. If f(x) e V( -n, n) and periodic with period of period

m,

m and

K(x) is a bounded function

then lim

J"

f(x)K(J..x) dx

=

(m)- 1

A-+±oo -n:

J"

f(x) dx

-n:

J"

K(x) dx

-n:

This was given by Fejer [1]. (See also Bary [1], p. 67.) = g(x) sin(px + q) where p, q are constants and g(x) converges to zero monotonically as x -+ ±co, then the integral

5. If f(x)

f

-+00

f(x)ethe dx

-+-oo

exists for J. # p and converges to zero. (See Bochner [5].)

6. The Fourier coefficients of a function of bounded variation are 0(1/n) as n-+ co as was shown in Theorem 2.5.4. This additional condition that the function (Keogh [1 ]). In other words, there is which n(!,. # o(1 ), (!,. = (a,. • + b,. 2 } 1' 1 ,

result cannot be improved to o(1/n) even if the be continuous is imposed (Riesz [2]). Also see a continuous function of bounded variation for a,., b,. being Fourier coefficients.

7.

In connection with NotJ 5, Wiener's theorem states that a necessary and sufficient condition for a function of bounded variation to be continuous is that

.

~ k(!~c = o(n) 1

(Wiener [1]). Another necessary and sufficient condition is that

.

~ l!t

=

o(log n)

1

(Lozinskii [1]). For a generalization of these results see Matveev [1].

8. Regarding the results in Note 6, it is to be noted that if the Fourier coefficients of a function of bounded variation are of o(1/n), then it is continuous. (See Zygmund [1], p. 29.) 9.

Another very simple proof of Theorem 2.5.4 with the bound T( -n, n)/1 n given by Taibleson [1].

I for

I c.. I was

10. Regarding the completeness of {e-tnz, n = 0, ± 1, ... }, it is easy to see that if we remove from this sequence any single term, then the sequence ceases to be complete even in C (the class of continuous functions).

11. Each of {cos nx, n

= 0, 1, ... } and {sin nx, n = 1, 2, ... } is complete in V(O, n).

12. {etnz, n > 0} is complete in L 1 over any interval of length less than 2. This result and its generalization to the sequence {eiA,.z, J.,. > 0} have been made by Levinson, where J.,. is not necessarily an integer. One of his results is: {etA,..,, J.,. > 0} is complete

II.

80

FOURIER SERIES AND FOURIER TRANSFORMS

in V over an interval of length L if lim inf(n/J.,.)

> L/2n

ft-+00

(see Levinson [4], p. 3, Theorem II. He also discussed LP-completeness in detail).

13. One of the most general forms of unicity theorems of the type of Theorem 2.11.1 will be the following theorem due to Offord [1]. Also see Offord [5, 6]. If lim ..4.-+oo

holds for every x, then t/>(x)

=

J.A-..4. tf>(x)ei"'' dx =

0 a.e.

0

Ill Fourier-Stieltjes Coefficients, Fourier-5tie/tjes Transforms and Characteristic Functions

3.1. Monotone Functions and Distribution Functions Let us denote by M = M(- oo, oo) the class of all bounded, nondecreasing functions over (-oo, oo) and by M 0 = M 0 (-oo, oo), the subclass of M, the functions F(x) of which satisfy F(- oo) = limz-+-oo F(x) = 0. Also define the class D = D( -oo, oo) offunctions of M such that F(- oo) = 0, F( oo) = 1. The class offunctions of bounded variation over [a, b] is denoted by V(a, b), where V = V(-oo, oo) is the class of functions having bounded variations over the whole line (- oo, oo ). Obviously D c M 0 c M c V. The class M(a, b) means the class of non decreasing functions defined over [a, b]. Let (.Q, B, P) be a given probability space. A random variable X(w), which is synonymous with a measurable function with respect to the a-field B, has the distribution function defined by

+

P[X(w)

< x] 81

= F(x).

(1)

82

III.

FOURIER-STIELTJES TRANSFORMS

F(x) then belongs to D. Besides, it has the property that it is left-continuous. If a monotonic function F(x) satisfies the relation F(x) = l[F(x

+ 0) + F(x- 0)]

(2)

at its discontinuities, then the function is said to be standardized. In an analytical treatment, particularly in the use of Fourier-Stieltjes coefficients or transforms, it is quite convenient to think of standardized monotone functions. As a matter of fact, for any Borel measurable function g(x), the integral

J~"' g(x) dF(x), fn so far as it exists, remains the same if we modify the nondecreasing iunction F(x) eM to be standardized at its discontinuities. This is obvious if we go back to the definition of the integral (which is the Lebesque-Stieltjes integral). Actually, let p be the completed measure of Borel measurable sets on the real axis generated by F(x). Then by definition,

But the measure generated by F(x) is invariant under the modification of F(x) at its discontinuities. Hence, we suppose, unless explicitly stated otherwise, that every function of M is standardized. Even when we talk about distribution functions, they are supposed to be standardized. So the ordinary conditions of left-continuity of a distribution function are ignored here except when we should emphasize some probabilistic meaning, where the specification will be made. Thus any function of D (which is now standardized) is called a distribution function. However, when we deal with a function of M(a, b) we do not change the 'lJalues of a function at the finite end points a and b. In other words, a nondecreasing function is defined only on [a, b], and not outside of it; the standardization is made only for the discontinuities interior to (a, b). If F(x) e D is absolutely continuous and F(x) =

j:_ p(u) du,

p(x) is the probability density function or more simply the probability

3.1.

MONOTONE FUNCTIONS AND DISTRIBUTION FUNCTIONS

83

density. p(x) is defined except on a set of measure zero and p(x) ;;;:; 0 there. Let F(x) belong toM, F(co) = B, F(-co) =A. If F(x)

+ F( -x) =

constant

for any

x,

(3)

then F(x) is called symmetric. The constant on the right-hand side must be A + B. In other words, a function F(x) of M is symmetric if and only ifthe graphy = F(x) is symmetric with respect to the point (0, t(B-A)) on the vertical axis. A point x is called a point of increase or a point of the spectrum of F(x) E M if, for any 15 > 0, F(x + 15) > F(x- 15). The set of such points is called the spectrum of F(x). The set of continuity points of F(x) E M, which are points of increase, is called the continuous spectrum. The set of discontinuities of F(x) E M is called the point spectrum. We denote by S(F) and P(F) the spectrum of F(x) and the point spectrum of F(x), respectively. It is obvious that a function belonging to M that is not identically constant has a spectrum. It is easy to see that the spectrum of a nonconstant function belonging to M is a closed set. A distribution function that has only a point spectrum contained in a sequence of equidistant points is called a lattice distribution function In other words, a lattice distribution function is a step-function extended from -co to co such that the distances between two discontinuities are integral multiples of some positive constant. A distributien function defined by U(x) = {

~:

X< 0, X> 0,

(4)

[U(O) = !] is called the unit distribution function. U(x- a) is called the unit distribution function with point spectrum {a}. A distribution function F(x) such that F(x)

={

~

for for

x < -:n:, x ;;;:; :n:

(5)

is called a circular distribution function. This terminology will be recognized if we take into account the following matter. Let X( w) be a random variable representing the angle between a random radius of a unit circle

III.

84

FOURIER-STIELTJES TRANSFORMS

and the negative real axis. Points are distributed on the circumference by random radii. In other words, the ranges of X(w) are (-n, n]. In this case, the mass placed on the point e+in ( = e-in) is placed on n, the distribution function F(x) of X(w) satisfies F(n) = 1. F(-n) = 0 and is continuous at x = -n. However, we here agree to call any distribution function F(x) with F(n) = 1, F(x) = 0 for x < -n, a circular distribution function. Let Y( w) be any random variable. Define X( w) by X(w)

= Y(w)

(mod 2n),

-n

< X(w)

~

n.

(6)

If G(x) is the distribution function (not standardized), then the distribution function F(x) of X(w) is given by 00

F(x) ·=

L

[G(2kn

k~-oo

+ x)- G((2k -1 )n)],

(7)

which is a circular distribution function. The standardization is made for the points in ( -n, n).

3.2. Fourier-Stieltjes Series Let F(x) Cn

E

M( -n, n). Form

= (Zn)-1 J~

e-inx

dF(x),

n

= 0, ±1, ±2, ....

(1)

They are called the Fourier-Stieltjes coefficients of F(x) and we sometimes use the representation

(2) The series in (2) is called the Fourier-Stieltjes series. It is obvious that

I Cn I ~

(Zn)-1[F(n)- F( -n)].

(3)

Hence c = {cn , n = 0, ± 1, ... } is a bounded sequence. If the norm of an element (sequence) of the space (m) of bounded sequences is introduced as (4) II c II = sup I Cn I -oo
and the norm of a function F(x)

II F II

=

E

J~ dF(x)

M( -n, n) is defined by = F(n)- F(-n),

(5)

3.2.

85

FOURIER-STIELTJES SERIES

then (3) is expressed by

II c II

~

(Zn)-1 II F II.

(6)

The mapping (1) from M(-n, n) to (m) is then linear and bounded. Different from the case of ordinary Fourier coefficients, en, defined by (1 ), does not have to converge to zero as n -+ ±oo as a simple example shews: If we take the unit distribution function U(x- a), -n
(J}(x) = F(x) -

err,

(8)

where

c0 = (2n)- 1 [F(n) - F( -n)]. Here, (J}(x) is a function of bounded variation. Since (J}(n) = (I}( -n), we can continue (J}(x) over the whole line. If the resulting periodic function is also denoted by (J}(x), the ordinary Fourier coefficients of (J}(x) are given by

en= (1/2)

J:,. (J>(x)e-inz dx

=

J:,. (e-inzJ-in) d(J>(x) (1/Znin) J:,. e-inz dF(x)- (c f2nin) J:,. e-inz dx

=

cnfin

=

(1f2n)(J>(x)(e-inzJ-in) ~~ -(1/2n) 0

for n =I= 0,

(9)

where we used integration by parts for the Stieltjes integral. For n = 0, we easily see that

C0 = F(-n)

+ (2n)-

1

J:,. (n- x) dF(x).

(10)

Therefore, we may write 00

(J}(x) ,..... Co

+ 2:' n---oo

(cnfin) einz.

(11)

86

III.

FOURIER-STIELTJES TRANSFORMS

Hence the Fourier-Stieltjes series of F(x), without a constant term, 00

L' Cn einz ....00

is just the derived Fourier series of
3.3. Average of Fourier-Stieltjes Coefficients Let F(x) belong to M( -n, n) and its Fourier-Stieltjes coefficients be en, n = 0, ±1, .... We saw that en does not have to converge. However, we may show that the average (N)- 1 L~N en does converge. More precisely, we have:

Theorem 3.3.1. If en, n = 0, ±1, ... , are the Fourier-Stieltjes coefficients of a function F(x) eM( -n, n), then N

lim (2N)- 1 N-+oo

·': ; )

L

cneinz

n--N

(2n)- 1 [F(x + 0)- F(x- 0)] . . { (2n)- 1 [F(n)- F(n- 0)] <' = (2n)- 1 [F(-n + 0)- F(-n)]

-

0

for for for for

I xI < n, I xI > n.

Because it will be used frequently, we state first the following formula.

Lemma 3.3.1.

f

einu

n--N

= { sin(N

+ l)u/sin(u/2),

2N + 1,

uo:f:.O, U=

0.

(2)

In real form, ~ _ { sin(N + l)u/sin(u/2), 1 + 2 .t.., cosnu- 2N+ 1 1 n-1

u#-0, 0.

U=

(3)

This is easy to show. Since the series in (2) is a geometric series with common ratio eiu, it is equal to

3.3.

87

AVERAGE OF FOURIER-STIELTJES COEFFICIENTS

which is seen to be the right-hand side of (2) for u =j::. 0. The case u = 0 is obvious. From formula (2), we obtain

LN

Cneinx

=

n=-N

LN

n=-N =

f"

(2n)-1 - ein<x-ul dF(u) "

J:,.

(2n)-1

= n- 1

L~N ein<x-ul] dF(u)

J:,. [sin(N + i)(x-u)f2sin(u-~f2)] dF(u).

(4)

The integrand of this integral is called the Dirichlet kernel. In this formula, the kernel is interpreted to be N i if x - u = 0, ±2n.

+

Proof of Theorem 3.3.1.

Because of the lemma,

f"

N cneinx = (2Nn)- 1 (2N)- 1 L DN(x- u) dF(u), -,. n--N

where D). (u) = sin(N

(5)

+ i)u/2 sin(u/2). We point out that

. (2N: )- 1D ( ) _ { 0, l1m n N u (2n )- 1,

N-+oo

u =j::. 0, ±Zn _ 0 ±2 U , n.

(6)

Since (2Nn)- 1DN(u) is uniformly bounded, the bounded convergence theorem applies to the right-hand side of (5), which gives us N

lim (2N)- 1 N-+oo

L

clleinx = n--N

r

D(x- u) dF(u),

(7)

~

where D(x- u) = 0, x-u =j::. 0, ±2n, and D(x- u) = (Zn)- 1 for x-u= 0, x-u= 2n. The integral in (7) is the integral of D(x- u) with respect to the measure p. generated by F(x). Therefore, the last integral is equal to (2n)- 1p.{x }, where p.{x} means the measure of a set {x} of a single point x. If I xI < n, then p.{x} = F(x 0)- F(x- 0) and, at the end points, x =nand -n, p.{x} is F(n)- F(n- 0) and F( -n 0) - F( -n), respectively. This gives us the first three relations in ( 1). The fourth relation in ( 1) is obvious since D( x - u) = 0 for I x I > n. This completes the proof.

+

+

We shall remark that the same conclusion of Theorem 3.2.1 is valid if F(x) is replaced by any function of V( -n, n) since a function of bounded variation is represented by the difference of two nondecreasing functions.

88

Ill.

FOURIER-STIELTJES TRANSFORMS

3.4. Unicity Theorem for Fourier-Stieltjes Coefficients We shall show that the Fourier-Stieltjes coefficients determine the function uniquely up to additive constants. More precisely, we have:

Theorem 3.4.1. Let F(x) and G(x) be two functions belonging to M( -:n, :n). If

J:,. e-inz dF(x) = J:,. e-inz dG(x) then for every -:n

<

x

for

n = 0, ±1, ... ,

(1)

< :n, F(x) = G(x)

for a constant a. For the end points x F(:n) - F( -:n)

+a

(2)

= ±n,

= G(:n) - G( -:n).

(3)

Remember that F(x) and G(x) are standardized for x interior to ( -:n, :n). If the standardization is not made, the conclusion reads: F(x) = G(x)

+a

at interior continuity points of F(x) and G(x) must be continuous at the continuities of F(x). Proof. Then

From (1) with n

=

0, we have (3). Write H(x)

J:,. e-inz dH(x) = 0

for

=

F(x)- G(x).

n = 0, ±1, ....

Integration by parts shows that

H(x)e-inzl:,.

+ in J:,. H(x)e-inz dx =

0.

The integrated term is zero because of (3). Therefore,

J:

H(x)e-inz dx

=0

Let a = (2:n)- 1

for

n = ±1, ±2, ....

J:,. H(x) dx;

then, from this,

J:,. [H(x)- a]e-inz dx = 0,

n=0,±1, ....

(4)

3.5.

89

FOURIER-STIELTJES TRANSFORM AND CHARACTERISTIC FUNCTION

Note that this holds, including the case where n = 0. Hence by the completeness of {e-inz, n = 0, ±1, ... } (Theorem 2.10.1), H(x) =a a.e. Since H(x) is a function of bounded variation, H(x + 0) and H(x- 0) exist at every point x, and by letting Xn ---+ x + 0 or Xn ---+ x - 0, through the set where H(x) =a, we find that H(x) = a must be true for every -n < x < n (even at discontinuities, because of standardization). This proves the theorem. We note that if F(x) = G(x) +a for some constant a and (3) is satisfied, then ( 1) holds for every n = 0, ± 1, ....

3.5. Fourier-Stieltjes Transform and Characteristic Function Let F(x) EM. The Fourier-Stieltjes transform of F(x) is defined by

f(t) =

J:

eizt dF(x),

-oo


oo.

(1)

This is an analog for a continuous parameter of a Fourier-Stieltjes coefficient. Also, we may define the Fourier-Stieltjes transform (1) with F(x) E V. If F(x) is an indefinite integral of p(x), then this reduces to the ordinary Fourier transform up to the constant factor (2n)- 112 and the sign of the parameter t. For convenience, we shall not discuss this and shall confine ourselves to considering the Fourier-Stieltjes transform of F(x) EM. We take the form (1) for the transform, omitting the constant factor in accordance with a rather traditional use of (1 ), particularly in probability theory. If F(x) ED, namely, F(x) is a distribution function, then (1) is called the characteristic function of F(x) or if X( w) is a random variable having the distribution function F( x ), then ( 1) is called the characteristic function of the random variable X( w ). To begin, we make the following obvious statements. (a) If a > 0, then with F(x) EM, F(ax +b) EM for every real b. The Fourier-Stieltjes transform of F(ax +b) is given by

e-itbtaj(tfa) if F(x) has the Fourier-Stieltjes transform f(t). (b) If/1 (t) and/2 (t) are, respectively, the Fourier-Stieltjes transforms of F 1(x) and F 2 (x), both belonging to M, then ad1(t) + atf2 (t) is the

III.

90

Fourier-Stieltjes transform of a 1F 1 (x) positive constants.

FOURIER-STIELTJES TRANSFORMS

+ a F (x), 2

2

where a 1 and a 2 are

For a characteristic function, we have (c) If / 1 (t) and / 2 (t) are the characteristic functions of F 1 (x), F 2 (x) e D, respectively, then atf1 (t) + aJ2(t) is the characteristic function of a 1F 1 (x) + a 2F 2 (x), where a 1 , a 2 are positive numbers such that al + a2 = 1.

Theorem 3.5.1. then

If/(t) is the Fourier-Stieltjes transform of F(x) eM,

(i) /(0) = F(oo)- F(-oo), (ii) 1/(t) I ~/(0), (iii) /( -t) = f(t),

(2) (3)

(4)

and (iv) f(t) is uniformly continuous over ( -oo, oo). Proof. (i)-(iii) will be very easy to show. Condition (iv) is proved in just the same way as in the proof of Theorem 2.7.1 (iii). Actually, lf(t +h)- f(t)

I ~I: I eihz -11 dF(x)

and the right-hand side does not involve t at all and converges to zero as h -+ 0 by the bounded convergence theorem.

Theorem 3.5.2. The Fourier-Stieltjes transform of a function belonging to M is real-valued if and only if the function is symmetric. We recall that a function F(x) is symmetric if and only if F(x)

+ F( -x) =constant.

(5)

Proof. Suppose that F(x) is a symmetric distribution function. Then (5) is true. Letf(t) be its Fourier-Stieltjes transform. f(t)

= I~oo eitz dF(x) =

I~"" cos tx dF(x) + i I~"" sin tx dF(x)

=I~ cos tx d[F(x)- F(-x)]

+ i I~ sin tx d[F(x) + F(-x)]

(6)

3.5.

FOURIER-STIELTJES TRANSFORM AND CHARACTERISTIC FUNCTION

91

The last integral vanishes because of (5), which proves that f(t) is realvalued. Next we suppose that f(t) is real-valued. Then from (6),

I~ sin tx dH(x) = 0

for all

(7)

t,

+

where H(x) = F(x) F( -x). Multiply both sides of (7) by cos tuft, u being any positive number, and integrate it with respect to t over (0, A), A being any positive number. Then we have 0=

J:

(cos tuft) dt

=I~ dH(x)

J:

I~ sin tx dH(x)

(cos tu sin txft) dt.

Let A go to infinity. Then from Lemma 2.6.4, the inner integral converges boundedly to a function D,.(x) = U(x- u) [U(x) is the unit distribution function], where

D,.(x) =

l

l,

X> U,

l,

X=

0,

0

u,

~X< U.

Hence we have

I~ D,.(x) dH(x)

= 0,

which means

HH(u

+ 0)- H(u)] + H(oo)- H(u + 0) =

0.

If we choose u to be a continuity point of H(x), then H(u) = H(oo). Therefore, F(u) F( -u) =constant at continuities of F(u) and F( -u). Since F(u) is standardized, F(u) F( -u) will be constant for all u > 0. This proves the theorem.

+

+

We now show the unicity theorem for the Fourier-Stieltjes transform.

Theorem 3.5.3. If F(x) and G(x) belong toM and

I:

eitz

dF(x) =

I:

eitz

dG(x),

then F(x)- G(x) is a constant for every x.

-00

< t < oo,

(8)

92

III.

FOURIER-STIEL TJES TRANSFORMS

The proof is carried out just the same way as in the proof of Theorem 2.1.1. Recall that F and G are standardized.

+ a),

Proof. Write H(x) = F(x) - G(x), H 1(x) = H(x constant. We then have -00

a being any

< t < oo.

Multiplying both sides by sin tuft, u being any positive constant, integrate over (- T, T) with respect to t and let T---+ oo. We then get

I~oo D(x, u) dH1(x)

= 0,

where, as in Section 2.11, D(x, u)

~ {:

for for for

I xI > u, I xI= u, I xI < u.

Therefore, we have

u--O I --tHO

dH 1(x)

+ t I {u}

+ t I{--u}

dH 1(x) = 0,

+ t I {--u+al

dH(x) = 0,

dH 1(x)

or

u+a--O I --u+a+O

dH(x)

+ t I {u+a}

dH(x)

(9)

where {z} is a set of a single point z. If we note that

I I

{z}

b--0

a+O

dH(x) = H(z

+ 0) -

H(z- 0),

dH(x) = H(b- 0)- H(a

+ 0),

we easily see from (9), noting that H is standardized, that H(u

+ a) -

H( -u

+ a) =

0.

Since a is arbitrary and u is an arbitrary positive number, H({3) = H(a) for every pair ({3, a) such that {3 > a. Letting {3---+ oo, we get H(a) = = H(oo) =constant. This shows the theorem. Actually Theorem 3.5.3 holds without any change of proof, if F(x) and G(x) are supposed to belong to V[-oo, oo]. In particular, if F(x)

3.6.

93

PERIODIC CHARACTERISTIC FUNCTIONS

and G(x) are absolutely continuous,

F(x) =

J:

00

J(u) du,

G(x) =

J:oo g(u) du,

withf(x),g(x) E £1(-=, =),then the theorem becomes essentially the same as Theorem 2.11.1. In other words, the above statement implies Theorem 2.11.1.

3.6. Periodic Characteristic Functions Let F(x) E D, that is, let F(x) be a distribution function. Let f(t) be its characteristic function. The following results, with suitable modifications, are easily generalized to a function belonging M 0 and its FourierStieltjes transform. But for simplicity and convenience in probability theory, we discuss the characteristic function only. Suppose that F(x) is a lattice distribution function, whose point spectrum is contained in a set {2knft0 , k = 0, ± 1, ±2, ... }, t 0 > 0. Denote the jump of F(x) at 2knft0 by PA-( ~0). Then the characteristic function f(t) is given by 00

f(t) =

L

PA- exp(2knit/t0 ).

(1)

k--oo

Obviously

I:~oo h

= 1. Therefore, 00

f(nt 0 ) =

Lh =

-oo

1,

n being any integer. In other words,J(t) assumes its maximum value of 1 at t = nt0 , n = 0, ± 1, .... The function f( t) is a periodic function with period t 0 • We are now interested in the converse problem and shall prove the following theorem.

Theorem 3.6.1. Suppose that F(x) ED. Let f(t) be its characteristic function. If

1/{to) I = 1

(2)

for some t 0 =I= 0, then F(x) should be a lattice distribution function with the point spectrum contained in the set {2kn wft0 , k = 0, ± 1, ±2, ... }, where w is a certain real number.

+

Ill.

94

Proof.

FOURIER-STIELTJES TRANSFORMS

From (2), there is a real w such that f(t 0 )eiw = 1, namely,

Taking the real part of the left-hand side, we have

J~oo cos(toX + w) dF(x) = 1. Since

J:'oo dF(x) =

1, this can be written

J~oo [1- cos(toX + w)] dF(x)

=

0.

Since 1 - cos(toX + w) ~ 0, the spectrum of F(x) must be contained in the set of x's such that loX+ w = 2kn, k = 0, ±1, .... This shows the theorem. If we have the assumption that Ref(t0 ) = 1, in place of (2), then w is taken to be zero, or:

Corollary 3.6.1. If the characteristic functionf(t) of F(x) ED satisfies Ref(t 0 ) = 1

(3)

for some t 0 =I= 0, then F(x) is a lattice distribution function with the point spectrum contained in the set {2knjt 0 , k = 0, ± 1, ... }. Another simple corollary of the theorem is the following:

Corollary 3.6.2. satisfies

(i)

If the characteristic function f(t) of F(x)

E

D

(4) for some 11 =I= t 2 , t 1 =I= 0, t 2 =I= 0 such that t 1/t 2 is irrational, then F(x) should be the unit distribution function U(x- a) for some a. (ii) If Ref(t) = 1 in some interval a < t < b, then F(x) = U(x).

Proof. (i) If (4) is satisfied and there are at least two points in the point spectrum, then, from Theorem 3.5.1, the distance between two points of a point spectrum is an integral multiple of 2njt0 and at the same time of 2njt 1 • Since t 0 jt 1 is irrational, this is impossible. Hence the point spectrum of F(x) consists of a single point.

3.7.

95

SOME INEQUALITY RELATIONS

(ii)

is easily shown from

J~oo (1

-COS

toX) dF(x) = 0

for a< t 0 < b

as in the proof of Theorem 3.6.1, or from the statement just before Corollary 3.6.1. Corollary 3.6.1 can be stated in the following way.

Corollary 3.6.3. The characteristic functionf(t) of F(x) E Dis periodic with period t 0 if and only if F(x) is a lattice distribution function whose point spectrum is contained in the set {2knft0 , k = 0, ± l, ... }. The "if" part was given at the beginning of this section and the "only if" part is no more than Corollary 3.6.1 since periodicity gives /(0) =/(to) = 1. Finally we remark that if f(t 0 ) = -1, then f(t) is periodic and the origin is not contained in the point spectrum. The last statement is seen from the fact that in the proof of Theorem 3.6.1, w is never zero. Actually it can be (2m+ 1)n for any integer m and nothing else.

3.7. Some Inequality Relations for Characteristic Functions We shall give here some inequality relations of elementary character. All the theorems that will be stated in this section hold for the FourierStieltjes transforms of at least M 0 , with obvious modifications. For simplicity we state them in terms of characteristic functions. If we want to get the results for the function of M 0 , we have only to replace F(t) by F(t)/F( oo) and f(t) by f(t)/F( oo ).

Theorem 3.7.1. Let f(t) be the characteristic function of F(x) e D. Then (i)

1 f(t

+ h) -

f(t)

1

~

211 2 (1 - Re f(h)]lta

(1)

for every t and h; {ii)

1 - Ref(2t) ~ 4[1- Ref(t)]

(2)

for every t; (iii)

I Imf(t) I ~ {l[1 - Ref(2t)]}1'2

(3)

II I.

96

FOURIER-STIEL TJES TRANSFORMS

for every t; (iv)

I(2h)-l r::f(u)dul ~ {[1 + Ref(h)]/2}1

(4)

12

for every t and h. Items (1) and (4) were given by Raikov [1], and (1) also by Wiener [2], (2) by Khinchin [3].

Proof of (i). lf(t +h)- f(t) 12

IJ:

=

~

eilz(eihz

[J: I

eihz

-1) dF(x)

-11 dF(x)

r

r

~ J~oo I eihz -112 dF(x) (by the Schwarz inequality)

= J~oo 2(1- cos hx) dF(x) = 2[1- Ref(h)]. Proof of (ii).

Note that

1 - cos 2tx = 2(1 - cos2 tx) Then 1 - Ref(2t)

=

J:

~

4(1 - cos tx).

(1- cos 2tx) dF(x)

~ 4 J~oo (1- cos tx) dF(x) Proof of (iii).

I Imf(t) I =

=

4[1- Ref(t)].

IJ: sin tx dF(x) I

~ [ J~oo sin2 tx dF(x)

r 2

(by the Schwarz inequality)

= [t

J:

(1 -cos 2tx) dF(x)

r

2

= [!(1- Ref(2t)]li2.

Proof of (iv). Note that sin 2yfy 2 ~ (2 sin !y cos !y)l/4 sin I !y

=

cos2 !y

=

!(1

+ cosy).

3.7.

97

SOME INEQUALITY RELATIONS

We then see that

I

(2h)- 1

J::

J(u) du

I=I =

(2h)- 1

s:::

du

IJ~oo eitz(sin

~ [ J~oo [sin

2

J:

eiux

dF(x)

hxfhx) dF(x)

I

I

r r

hxf(hx) 2 ] dF(x)

2

(by the Schwarz inequality)

~ [ l J~oo (1 + cos hx) dF(x) =

[t[1

+ Ref(h)]]l

2

12 •

(1) shows that the uniform continuity of f(t) is obtained from the continuity of f(t) at the origin. Using (2), we can show that each of the following functions is not the characteristic function of any distribution function.

(a) (b) (c) (d) (e)

exp( -t4 ), exp(-1 t 1"), a> 2, exp( - t4 - t 5 ), 2[exp tk + exp(-tk)]-1, k = 2, 3, ... , (1 + t 4 )-I, etc.

For instance, we take (b): 1- Ref(2t)

= 1- exp(-l2t I") 2t I" I 2t-12+ " =I ... 1! 2!

(5)

and

(6) Comparing (5) and (6) for small t, the inequality (2) does not hold for

a> 2. Another application of the theorem is the following corollary.

Corollary 3.7.1. The distribution function whose characteristic functionf(t) satisfies (7) as t-o, J(t) = 1 + o(t2 ) is the unit distribution function.

98 Proof.

Ill.

FOURIER-STIELTJES TRANSFORMS

Taking the real parts of both sides of (7), we have

1 - Ref(t) = o(t2 ). Hence we may write

1 - Ref(t) = t 2tf>(t), where t/J(t)-+ 0 as t-+ 0. Since Ref(t) is even, t/J(t) must be even and nonnegative. From (2),

that is,

t/>(2t)

~

tf>(t)

for every

t.

Using this inequality (n - 1) times,

Since t is arbitrary, (8)

Fix t and let n -+ oo. Then we have

tf>(t) = 0

identically.

In other words, Ref(t) = 1 This implies that f(t)

=

t.

= 1 identically, because from (1 ),

f(t +h) -f(t) = 0 Hence f(h)

for every

1, taking t

=

for every t

and

h.

0.

The corollary can be used to prove more simply than in the above, that (a), (b), (c), and (e) are not characteristic functions. Suppose now that f(t) is a periodic characteristic function 00

f(t) = where Pn

~

L Pn exp(2nnitft0 )

n--oo

0, L Pn = 1, and t 0 is the period. We then see that lim sup f(t) = 1.

(9)

ltl-+oo

This is obvious since 1/(t)

I ~ 1, and f(kt 0 ) = 1, for any integer k.

3.7.

99

SOME INEQUALITY RELATIONS

If (9) does not hold and if f(t) is strictly less than 1 in a vicinity of infinity, then the behavior of f(t) near t = 0 is restricted in a certain way. Precisely we have:

Theorem 3.7.2. If f(t) is the characteristic function of F(x) e D and if, for some b > 0, ~a

lf(t)l

for

It I ~ b,

(10)

where 0
for It I
j(-t)

= J~oo e-izt dF(x) = J~oo eizt d[1- F(-x)], I

f(t)

12

=

f(t)f( -t)

is the characteristic function of

J~oo F(x- u) d[1- F(-u)], which is a function of D. This is a particular case of Theorem 3.10.1. (which will be given later). It is sufficient to show (11) for 0 ~ t < b since lf(t) I is an even function. Apply Theorem 3.7.1, Eq. (2) to lf(t) 12 • We then have 1-

I

J(2t)

12

~

4(1 -

I

f(t)

12 ).

(12)

The left-hand side is not less than 1 - a2 for b/2 ~ t < b [from (10) with b/2 for b and with I f(2t) 12 for f(t)]. (It is easy to see that f(2t) is again a characteristic function and so is 1/(2t) 12.) Therefore, we have 1 - a2

~

4[1 - lf(t)

12 ]

or

lf(t)

12 ~

1 - 1(1 - a 8 ).

100

Ill.

In the same way we have that, for b/2 2 1- a 2

~ ~

FOURIER-STIELTJF.S TRANSFORMS

~

t < b/2,

4[1 - 1/(2t) jl1] 42 [1 - 1/(t) 12 ]

[using (12) again], which yields I f(t) 12 ~

1 - 4-2 (1 - a 2 )

for

b/2 2 ~ t

< b/2.

Continuing this procedure, we have

Since 1/2"-1 > tfb, the last inequality shows that

for b/2" ~ t < b/2"-1 • Since n is any positive integer, the last inequality gives us our conclusion. Theorem 3.7.2 implies the following corollary. Corollary 3.7.2. Letf(t) be a characteristic function. Suppose lim sup I f(t) I < 1. rtr-+oo

(13)

Then for any e > 0, there is a constant c = c( e) such that 1/(t) I ~ c < 1

(14)

for It I ~e. Proof. From (13) there are a and b such that (10) holds. Now suppose I t I ~ e. Then, for e ~ I t I < b,

1/(t) I

~

1 - (1 - a 2 )(t2/8b2 )

~

1 - (1 - a 2 )(e2/8b 2 ).

Define For this c, (14) holds. In particular, we have:

Corollary 3.7.3. If f(t) is the characteristic function of a function

3.7.

101

SOME INEQUALITY RELATIONS

belonging to D, which has a probability density, then for any e > 0, there is a constant c such that (14) holds for It I ~ e. We conclude this section by showing the following results (Loeve [1], p. 196).

Theorem 3.7.3. If f(t) is the characteristic function of F(x) e D, then for t > 0

I

x 2 dF(x) ~ (3/t 2 )[1- Ref(t)],

IZI
{"' 1 ~ 111 Proof.

dF(x)

~

(7ft)

I:

(15)

[1- Ref(u)] du.

(16)

Equation (15) follows from 1-

Re f(t) = I~oo (1 -cos tx) dF(x)

~I

l:z:l
(t 2x2 f2)[1- (t 2x2 f12)] dF(x)

~ (llt2 /24)

I

l:z:l
(since 1 - cos u

~ lt 2

I

l:z:l
x 2 dF(x) ~

!u2 [1 - (u 2f12)])

x 2 dF(x).

Equation (16) follows from t- 1

I' [1- Ref(u)] du = I' du Ioo t- 1

0

0

-oo

(1- cos ux) dF(x)

= I~oo [1 -(sin txftx)] dF(x) ~I

l:z:l~1/t

[1 -(sin txftx)] dF(x).

Since 1 - (sin ufu) ~ 1- sin 1 > 1/7 for 1 ~ u < n/2, and ~1 - (2/n) for u ~ n/2, we have

1 - (sin ufu) Hence t- 1

~

1 - (2/n)

~

I' [1- Ref(u)] du ~ (1/7) I 0

1/3.

lxl~l/t

dF(x).

Ill.

102

FOURIER-STIELTJES TRANSFORMS

3.8. Average of a Characteristic Function It is the purpose of this section to show a theorem on characteristic functions analogous to Theorem 3.3.1 and the related results.

Theorem 3.8.1. Let F(x)

E

D and its characteristic functi<m be f(t).

Then for any x, lim(2T)- 1 T-+oo

(2T)- 1

f(t) ritz dt

(1)

-T

+ 0) -

exists and is equal to F(x Proof.

IT

F(x - 0).

I:T f(t) ritz dt = (2T)- rT rizt dt I: eitv dF(y) = (2T)- I~oo dF(y) rT eilv-zlt dt 1

1

I:

=

[sin T(y- x)fT(y- x)] dF(y). (2)

Since lim[sin T(y-x)fT(y- x)]

for y =F x, for y = x,

~

= {

T-+oo

we have the conclusion of the theorem, taking the limit under the integral sign in (2).

Corollary 3.8.1. Let F(x)

D and letf(t) be its characteristic function. If x is a continuity point of F(x), then E

lim (2T)- 1

IT

T-+oo

J(t)

ritz

dt

= 0.

(3)

-T

Now we write

.L(J) = lim (2T)- 1 T-+oo

IT

f(t) dt

-T

(4)

for any function f(t) if the right-hand side exists. Suppose that the spectrum of F(x) E D 1s the point spectrum P(F) = {xn, n = 0, ±1, ... }. Then 00

J(t) =

L

Pneitzn,

n--oo

where Pn = F(xn

+ 0) -

> 0,

(5)

F(xn - 0).

(6)

Pn

3.8.

103

AVERAGE OF A CHARACTERISTIC FUNCTION

Here note that {xn} is not necessarily arranged in the order of magnitude. P(F) may have finite limiting points. Now we show the following theorem.

Theorem 3.8.2. For a characteristic function f(t) of a function F(x) e D with the point spectrum {xn, n = 0, ± 1, ... } (7) n=-=-oo

where Pn is given by (6). Equation (7) means that 1(1 f 12 ) always exists and is always equal to the right-hand side of (7). Proof.

(2T)-l

f~T

I

f(t)

12

dt

= (2T)-l f~T f(t)f(t) = (2T)- 1 J~T =

J""

dF(x)

J~oo eizt dF(x) J~oo e-ivt dF(y)

dt

J""

-oo

dt

-oo

[sin T(x- y)/T(x-y)] dF(y). ( 8)

As in the proof of Theorem 3.8.1, the inner integral converges boundedly to F(x + 0) - F(x - 0) as T - oo. Hence the bounded convergence theorem assures that (8) converges to

J~oo [F(x

+ 0)- F(x- 0)] dF(x),

which is no more than

L [F(xn + 0) -

F(xn - 0))2.

This shows the theorem. In particular, if f(t) is periodic and Xn is given by 2nnft0 , t 0 > 0, n

= 0, ±1, ±2, ... , then, since it is easily seen that

we have (9)

II I.

104

FOURIER-STIELTJES TRANSFORMS

This is called the Parseval relation for a periodic characteristic function. The same equality holds for any Fourier series of a periodic function of L2, which will be discussed later in Section 6.1. From Theorem 3.8.2, the following corollary is immediate.

Corollary 3.8.2. Let F(x) For

E

D and f(t) be its characteristic function. (10)

it is necessary and sufficient that F(x) be everywhere continuous. Finally we remark that for the average of a characteristic function over a finite interval, Raikov's inequality [Theorem 3.7.1, Eq. (4)] gives some estimate.

3.9. Convolution of Nondecreasing Functions First we introduce the notion of the vector sum of two point sets. The vector sum of two point sets A and B is the set of all points that are represented as a b, a E A, b E B. If one of A and B is empty, the vector sum is interpreted to be empty. The vector sum of two sets A and B is written A (+)B. It is easy to see that if A and B are bounded closed sets, then A ( +) B is also bounded and closed. The following facts are easily seen

+

A(+)B=B(+)A

A ( +) (B ( +) C)

= (A (+)B) ( +) C.

(1) (2)

If B = {x} is a set of single point x, then A ( +) {x} is a translation of A by x. We show the following fact. If A and B are closed sets and one of A and B is a bounded set, then A ( +) B is closed. Suppose that A is bounded. Let a finite number z be a limiting point of A ( +) B and {xn + Yn} (xn E A, Yn E B) be a sequence converging to z. Since A is bounded and closed, {xn} has a subsequence {xnk} that converges to x, which is a finite number of A. Then z - Ynk converges to x E A and therefore Ynk must converge to a certain finite number y which belongs to B since B is closed. Thus z = x + y E A (+)B.

3.9.

105

CONVOLUTION OF NONDECREASING FUNCTIONS

The above result is not always true if both A and B are not bounded. Let A = {-1, -2, -3, ... } be the set consisting of all negative integers. Let {Pn, n = 1, 2, ... } be a sequence of positive numbers such that

limp,.= a,

n

0 < Pn < Pn+I•

n->-oo

= 1, 2, ... , 0 < a <

1.

Define B by the set of points {n + p,., n = 1, 2, ... }. Then for any positive integer m and any integer j < m, j + Pm is a point of A ( +) B, since j + Pm = ( -m + j) + (m + p,.). Conversely, any point of A (+) B is easily seen to be of the form j + Pm (j < m, m > 0). Now let m --+ oo. j + a is a limiting point of A ( +) B, but it is not a point of A ( +) B since j + a is not of the form j + Pm. Now we are going to study the convolution of two bounded nondecreasing functions. We recall here that every function M is standardized. Let F 1(x), F 2 (x) belong toM. Since a function of M is a Borel function, (3)

is well-defined. G(x) is called the convolution of two functions of M. G(x) is also described by

It is obvious that F 1

(F1 (F1

•

•

F 2 is again in M, and that

F 2 )(oo)

=

F 2)( -oo)

=

•

F 1 (oo)[F2(oo)- F 2(-oo)], F 1 ( -oo)[F2(oo)- F 2(-oo)].

(4) (5)

Integrating by parts,

J~oo F 1 (x- u) dF2 (u)

=

F 1(-oo)F2(oo)- F 1 (oo)F2(-oo)

+ J~oo F (x- u) dF (u). 1

2

(6)

Hence F 1 • F 2 is not necessarily equal to F 2 • F 1 • So from now on in this chapter we assume throughout that F 1 , F 2 E M 0, that is, F 1 (-oo) = F 2(-oo) = 0, in order to have a simpler formula for (6). In this case (6) simply becomes (7) of the following theorem.

Theorem 3.9.1. If F 1 (x), F 2 (x), and F 3 (x) belong to M 0 , then

(F1

•

Fl • F2

=

F2 • Fl'

(7)

F 2) • Fa

=

F1

(8)

•

(F2 • Fa)·

Ill.

106

Proof.

FOURIER-STIELTJES TRANSFORMS

The proof of (8) is also simple.

(FI • F 2 )

•

F 3 = J~oo (F1

J:

(F2

•

F 2)(x- u) dF3 (u)

•

F 1 )(x- u) dF3 (u)

J~oo dF3 (u)

J:

[by (7)]

F 2(x- u- v) dF1 (v)

J~oo dFI(v) J~oo F 2 (x- u- v) dF3 (u) (by the Tonelli theorem) [by (7) again]. Because of this theorem, the convolution operation on functions belonging to M 0 is commutative and associative. We next talk about the spectrum of the convolution. S(F) means the spectrum of F as defined in Section 3.1.

Theorem 3.9.2. Let F 1 (x) and F 2 (x) belong to M 0 • Then (9)

E means the closure of a set E. Proof.

First we show that (10)

Write F = F 1

•

F 2 • Since S(F) is closed, it is sufficient to show that

S(F1 ) ( +) S(F2 ) c S(F). Let w = x + y, x E S(F1 ), y E S(F2 ). We are going to show that w is a point in S(F). Let (a, b) be any interval containing w. Then x E (a - y, b-y) and hence (11)

3.9.

CONVOLUTION OF NONDECREASING FUNCTIONS

107

For any !5 > 0,

~

J

Y+!d/2)

y-(d/2)

[F 1(b + !5-z)-F1 (a-!5-z)] dF2(z).

Since

F 1(b+!5-z)- F 1(a-!5-z) > F 1 [b-y+(!5/2)]- F 1 [a-y-(!5f2)] for y- (!5/2)

< z < y + (!5/2),

F(b + !5)- F(a- !5)

~

{F 1 [b- y + (!5/2)]- F 1 [a- y- (!5/2)]} X {F2 [y + (!5/2)]- F 2 [y- (!5/2)]}.

From (11) andy E S(F2 ), both factors on the right-hand side are positive and hence F(b + !5) - F(a - !5) > 0. Since (a - !5, b + !5) is an arbitrary interval containing w, this shows that w E S(F). We next show that S(F) c S(F1 ) ( +) S(F2 ).

(12)

If the set K

= S(F1 ) ( +) S(F2 ) = (-oo, oo ),

there is nothing to show. So we may suppose that there is a point w which does not belong to K. We are going to show that wE S(F), which proves (12). Suppose that w were a point in S(F). Since K is a closed set, the complement Kc is open and wE Kc. Therefore, there is a neighborhood (a, b) of w such that (a, b) c Kc. Since we are supposing wE S(F), F(b)- F(a) > 0, namely,

Hence there is a point x belonging to S(F2 ) such that

Then (a - x, b - x) contains a point y belonging to S(F1 ). In other words x + y E (a, b), and since x E S(F2 ), y E S(F1 ), x + y E S(F1 ) ( +) S(F2 ) ( c K). This contradicts (a, b) E Kc.

Ill.

108

FOURIER-STIELTJES TRANSFORMS

REMARK. If one of S(F1 ) and S(F2 ) is a bounded set, then the righthand side of (9) is replaced by S(F1 ) ( +) S(F2 ).

This is true because, in this case, S(F1 ) ( +) S(F2 ) is closed as we have seen in the first italicized statement of this section. We refer the reader to Wintner [3] concerning the material presented in this section.

3.1 0. The Fourier-Stieltjes Transform of a Convolution and the Bernoulli Convolution We shall give a fundamental theorem on the Fourier-Stieltjes transform of a convolution.

Theorem 3.10.1. Let F(x) be the convolution F 1 • F 2 of functions F 1 and F 2 belonging to M 0 • Letf(t),f1 (t), andf2 (t) be the Fourier-Stieltjes transforms ofF, F 1 , and F 2 , respectively. We then have -00


oo.

(1)

This is easily seen from the following lemma.

Lemma 3.10.1. Suppose that G(x) eM, H(x) is any bounded Borel function, and F(x, y) is a bounded Borel function of y for each x and belongs toM for each y. Then

J~oo H(x) d[f: F(x, y) dG(y)] =

J:

dG(y)

J:

H(x) dzF(x, y),

(2)

where dz indicates that the inner integral is taken with respect to the variable x.

Proof.

Note that

J:

F(x, y) dG(y)

is in M. Now let us choose a sequence of simple functions hn(x) that converges boundedly to H(x) everywhere. Then by the bounded convergence theorem,

J:

H(x) d[

J:

F(x, y) dG(y)] =

}~~

J:

hn(x) d[ J~oo F(x, y) dG(y)].

3.10.

109

TRANSFORM OF A CONVOLUTION

Suppose hn(x) = ck on Ek> k = 1, 2, ... , m(n), Ek n Ej = ¢, k =/=j, ck and Ek depending on n. Then the integral on the right-hand side is

If we can show that

I d[ I: F(x, y) dG(y)] Ek

then the sum of both sides [k

=

I~oo dG(y)

I

Ek

dJi'(x, y),

(3)

= 1, 2, ... , m(n)] becomes

which converges to the right-hand side of (2) since hn(x) converges boundedly to H(x) and the bounded convergence theorem applies. So it is sufficient to show (3). If Ek is an interval (a, b), both sides reduce to

I: [F(b- 0, y)- F(a + 0, y)] dG(y)

and hence (3) is true for intervals. Now consider the class B' of sets E for which (3) holds with E in place of Ek. From what we have shown above, B' contains all open intervals. Let E 1 , E 2 , ••• E B' (at most a countable number of Ek's ). Then (3) holds for these E/s and it still holds for U:,1 Ek> which is easily seen by an application of the bounded convergence theorem. Also, it is easily shown that Ec E B' with E E B'. Therefore B' is a a-field. But the Borel field B is the minimal a-field over the class of all open intervals. Hence B' :::> B. In other words (3) holds for any Borel set Ek. This completes the proof. For later use, we remark the following: Let X be a measurable space with finite measure!-'· F( x, w) is a bounded measurable function of w E X for each x and is a function belonging to M for each w. Then

I~oo H(x) d[

I F(x, w) dft] =I X

X

dft

I~oo H(x) dxF(x, w)

for any bounded Borel function H(x). The proof of Lemma 3.10.1 is carried over without any change.

(4)

III.

110

FOURIER-STIELTJES TRANSFORMS

Proof of Theorem 3.10.1.

J~oo eixt dF(x) =

J:

eixt d[

= J~oo dF2(y)

J~oo F 1(x- y) dF2(y)]

J:

eixt dxFl(x- y)

(by the lemma) =

I~oo dF2(y) I~oo ei
= J~oo eivt dF2(t) J~oo eiut dFl(u) =

f2(t)fl(t).

If F 1 , F 2 , ••• , Fn all belong to M 0 and F is the n-fold convolution F1 • F 2 • · · · • Fn, then the Fourier-Stieltjes transform ofF is the product of the Fourier-Stieltjes transforms of Fk, k = 1, 2, ... , n.

Consider a very special distribution function

0,

t. {J(x)

1,

=

! i

X< -1, -1 <X< 1, 1 < x, at X= -1, at X= 1.

(5)

{J(x) is sometimes called the Bernoulli distribution function (see Fig. 3.1). The characteristic function of {J(x) is

J~oo eitx d{J(x)

=

cost.

• _,

l'

FIGURE

3.1.

Then the characteristic function of {J(xfa) is cos at for a > 0 [see Section 3.5 (a)]. Let a 1 , a 2 , ••• , an> 0. From Theorem 3.10.1, the characteristic function f(t) of the n-fold convolution (6)

[which means that if we write {J(xfak)

= {Jk(x), an(x) is ({J1 • {J2 •

• • •

111

NOTES

• fln)(x)] is given by

f(t) =

ncos akt. n

(7)

k=l

The spectrum of an(x) is the point spectrum P(an) cons1stmg of all points of the form :L~- 1 ±ak. an(x) is called the Bernoulli convolution. Suppose ak = 3-k. Then the point spectrum P( an) consists of L~=l ±(1J3k). For every n ~ 1, it is seen that an(x) = !, for -t < x < j-, where

1

1

1

1

(6 = 3-3"2- 33 - .. ·, Also an(x) is found to be constant in each interval contiguous to a Cantor set obtained by trisection of [-!, H Hence, if we consider

a(x) = lim an(x), n_,.oo

which can be shown to exist and to be a distribution function, then it is a singular function whose spectrum constitutes a Cantor ternary set made over [- !, !]. Many interesting examples are obtained by appropriate choices of {ak}. This situation was extensively studied by Wintner [3]. We shall come back to this topic later in Section 13.4. NOTES 1. Corollary 3.6.3 is true if F(x) e D is replaced by F(x) e V. In other words, the Fourier-Stieltjes transform f(t) of a function F(x) e V is periodic if and only if F(x) is a step function (extending from -oo to oo) with discontinuities contained in a sequence of equidistant points. This was shown by Ramachandran [1].

3.

Corollary 3.7.1 can be shown in a simpler way, if we use the idea of moments, because (7) implies the vanishing of the first and the second moments, which concludes that f(t) 1. For the moment of a distribution function, see the later Section 11.1.

=

4.

In connection with Corollary 3.7.1, we mention the following result. If

Pr

=

J:

I X 1• dF(x) <

F(x) e D,

oo,

then

It - J I ~ c.p. I t 1·.

1

~

r

~

2,

Ill.

112

FOURIER-STIELTJES TRANSFORMS

where f(t) is the characteristic function of F and c. is a constant depending only on r [actually it is given by 3.38/(2.6)']. This was given by van Bahr and Esseen [1].

5.

Let F(x)

E M

and be absolutely continuous with F(x) =

roo

p(u) E V(-oo, oo),

p(u) du,

Boas and Kac [1] proved the following inequality: If f(t) is the Fourier-Stieltjes transform, and if f(t) = 0

for

It I~ A,

A being any positive constant, then we have

If(t) I ~ f(O) cos[n/( {A/t} + 1)], where {a} means the least integer not less than a. The inequality is the best possible. In particular, if A = 1, t = l, the inequality becomes

If(!) I~ lf(O). In this case, there is a simple example which shows that it is the best possible: Let p(u) = 27f-lu-• sin1 u; then f(t) = 1 - I t I for I t I < 1 ; = 0 for I t I > 1, and hence f(O) = 1 and j(l) = !. See Boas and Kac [1] and Boas [2]. Boas has shown an inequality of similar kind for the ordinary Fourier transform: If f(t) =

J:

e1" 1t/J(x) dx,

where t/J e V(- oo, oo) and is real, and if f(t)

lf(t) I ~ (n/4) for

I t I ~ A/3.

=

J:

0 for

I t I ~ A,

I t/J(x) I dx

then

IV Convergence and Summability Theorems

4.1. Convergence of Fourier Series We are going to give elementary convergence criteria for Fourier series. Much is kno~n about the convergence of Fourier series and tremendous progress has been made recently, but these are beyond the scope of this book. Readers are referred to Bary [1 ], Hardy and Rogosinski [1 ], Zygmund [1, 2], or Sunouchi [1]. / --: Letf(x) be periodic with period 2n and belong to Ll(-oo, oo). Let its Fourier series be 00

f(x) ,...., i-a 0 or

+ n-1 L (an cos nx + bn sin nx),

(1) (2)

-oo

The partial sum sn(x) of (1) or (2) is n

sn(x)

=

i-a0

+ k-1 L (ak cos kx + bk sin kx) 113

(3)

IV.

114

CONVERGENCE AND SUMMABILITY THEOREMS

or (4) -n

So when we talk about the convergence of the series (2), we talk about the limit of L~n c1.eikz when n -+ oo. Now we calculate s11 (x), keeping in mind that a11 , b11 , or C11 are Fourier coefficients of f(x).

s11 (x) =

Eeikz(2n)- J: f(t)e-ikt dt (2n)-1 f j(t) L eik dt. 1

k--n

=

n

n

n

--,.

k--n

From Lemma 3.3.1, we see that

s11 (x)

= n-1

=

n-1

J:

f(t){sin(n

+ l)(t- x)/2 sin[(t- x)/2]} dt

J:J(x + u) D (u) du

(5)

11

since the integrand is periodic, where

D 11 (u) = sin(n

+ l)u/2 sin !u.

(6)

This function in (6), introduced in Section 3.3, is called the Dirichlet kernel. The formula (5) is substantially no more than (4) in Section 3.3, where we dealt with the Stieltjes integral. The integral on the right-hand side of (5) is called the Dirichlet integral. If it happens thatf(x) = 1 identically, then (5) reduces to

1 = n-1

J:

(7)

D11(u) du.

Note that D 11 (u) is an even function of u. We may then write S11

(x) = n-1

J: J:

= (2/n)

[J(x

+ u) + f(x- u)]D (u) du 11

tf>(u)D11 (u) du,

where

tf>(u) = tf>(u, x) = l[f(x

+ u) + f(x- u)].

(8) (9)

From (5) and (7), we have, for any c, that

s11 (x)- c = (2/n)

J:

VJ(u)D 11 (u) du,

(10)

4.1.

115

CONVERGENCE OF FOURIER SERIES

where 1p(u) = 1p(u, x, c) = if>(u) - c.

(11)

Suppose a point x is given and we are going to discuss the convergence of Fourier series at this point x. Since if>(u) and 1p(u) are integrable, if>(u)/(sin !u) and 'P(u)/(sin !u) are integrable over (<5, n) where <5 is any positive but fixed constant. Hence the Riemann-Lebesgue lemma (Theorem 2.1.1 ) shows that sn(x)- c = (2/n)

J:

'P(u)Dn(u) du

+ o(1 ).

(12)

In other words:

Theorem 4.1.1. The convergence property of a Fourier series at a point x depends only on the function in a neighborhood of x. This is the so-called "local property of convergence." Actually, the Fourier coefficients are determined by the values of the function over the whole interval ( -:n;, n), but the convergence at x depends only on the values of the function around a point x. Since Dn(u) = sin(n _ sin(n

-

+ !)u + [ 2 sm_1!u

u

_

_!_] sin(n + !)u u

+ ! )u + rn (U')

(14)

u

and

rn(u) = g(u) sin(n

+ !)u,

(13)

g(u)

= O(u),

where g( u) is independent of n. Hence as n---+- oo, again by the Riemann-Lebesgue lemma. Hence we get the following theorem.

Theorem 4.1.2. The necessary and sufficient condition for the Fourier series of f(x) to converge at x to c is given by

J:

'P(u)[sin(n

+ !)ufu] du---+- 0

where 15 is any positive fixed constant.

as n---+- oo,

(15)

IV.

116

CONVERGENCE AND SUMMABILITY THEOREMS

From this it is not difficult to obtain the following theorem. We of course suppose that f(u) E £1( -n, n) and is periodic.

Theorem 4.1.3. (i) If, for some c and for some !5 > 0,

J: [I

f(x

+ u) + f(x- u)- 2c lfu] du <

(16)

oo,

then the Fourier series converges at x to c. (ii) If f(u) is of bounded variation in a neighborhood of x, [x- !5, x + !5], !5 > 0, then the Fourier series converges at x to t[f(x + 0) + f(x- 0)]. Criterion (i) for the convergence is called Dini's test, and criterion (ii) is called Jordan's test. It is to be noted that in (ii), the existence of f(x + 0), f(x- 0) is implied by the condition that f(u) is of bounded variation in a neighborhood of x.

Proof. (i) Since u-t,p(u) E £1(0, !5) because of (16), the RiemannLebesgue lemma gives us the conclusion. (ii) For convenience we write n + ! = A.. Take c = i[f(x + 0) + f(x- 0)]. Then 1p(O+) = 0 since 1p(u) is of bounded variation in a neighborhood. 1p(u) is written (17) where h 1(u) and h 2 (u) are nondecreasing, nonnegative functions with h 1 (0+) = h 2 (0+) = 0. Let e > 0 be an arbitrary number. Choose !5 so small that 0 ~ hk(u) < e for 0 < u ~ !5; 1p(u) is of bounded variation and has the representation (17) over [0, !5) [define 1J1(0) = 1p(O+) and hk(O) = hk(O+), k = 1, 2 if necessary]. Now fix !5 > 0 and choose N so that the term o(1) on the right-hand side of (12) is less than e in absolute value for n ~ N. We then get

I sn(x)- c I ~ ktii (2/n)

J:

hk(u)(sin ufu) du

I+ e.

Because of the second mean value theorem

I J: (2/n)

hk(u)(sin A.ufu) du

I= (2/n)hk(d) IJ:k ~

(2/n)e·C

(sin A.ufu) du

I• = k

1, 2

(18)

and from Lemma 2.6.2, C is an absolute constant and 0 ~ Ck ~ !5. Hence

4.1.

117

CONVERGENCE OF FOURIER SERIES

we have shown that for n

~

N,

I sn(x)- c I < (4/n)e·C +e.

(19)

The proof is now complete. In Dini's test (i), if x is a continuity point or is a discontinuity point of the first kind of f(u), then c must be either f(x) or l[f(x 0) f(x - 0)], respectively. The following corollary is obvious from (i).

+

+

+

Corollary 4.1.1. If f(x u)- J(x) = 0(1 u I") for some a > 0, then the Fourier series converges at x tof(x). In particular, if/'(x) exists at x, then the Fourier series converges to f(x). As to the generalizations of criteria given in Theorem 4.1.3 and other criteria, the reader is referred to Zygmund [2], Vol. 1, or Bary [1], Vol. 1. In connection with Jordan's test, (ii) of Theorem 4.1.3, we mention the following theorem of Dirichlet.

Theorem 4.1.4. If f(x) is continuous everywhere, periodic of period Zn, and of bounded variation over [-n, n], then the Fourier series converges uniformly to f(x). Take c =f(x) in the proof of Theorem 4.1.3. Sincef(x) is uniformly continuous, we may choose t5 to hold independently of x in (18) (the decomposition (17) is to be made throughout [-n, n]) and the proof of

(2/n)

J:

tp(h) sin l.u du,

going to 0 in Theorem 4.1.3, is carried over to get its uniform convergence to zero. Another point is to verify that the term o(l) in (12) and

J:

tp(u)rn(u) du = o(l),

r,(u) being defined by (14), hold uniformly. These facts were shown by the use of the Riemann-Lebesgue lemma. If, instead of the RiemannLebesgue lemma, the generalized Riemann-Lebesgue lemma (Theorem 2.1.4), with finite interval as integration range and with K(x) =sin x

or

K(x) = cosx,

IV.

118

CONVERGENCE AND SUMMABILITY THEOREMS

is used, we ·may without difficulty verify that the above limits hold uniformly; this gives us the theorem. Now let

F(x) =

I:

f(u) du.

Then as is easily seen (actually we observed this in Section 2.8),

F(x) - (aoX/2) is periodic and 00

F(x)- (aoX/2) '""'""'C0

+ L (-bn cos nx + an sin nx)fn,

(20)

n-1

where C0

= (2n)-1

I:

(n- x)J(x) dx.

Obviously F(x) - (aoX/2) is absolutely continuous and hence it is of bounded variation over [-n, n]. Therefore by Theorem 4.1.4, the series in (20) converges uniformly to

F(x)- (aoX/2). Hence we have the following theorem.

Theorem 4.1.5. If an, bn are Fourier coefficients of a function f(x), then the series 00

L (-b n-1

11

cos nx

+ an sin nx)fn

converges uniformly. The sum of (21) is

F(x) - (aoX/2) - C0 • An obvious particular case with x = 0 is the following:

Corollary 4.1.2. If bn are Fourier sine coefficients, then

always converges.

(21)

4.1.

119

CONVERGENCE OF FOURIER SERIES

In complex form, Theorem 4.1.5 can also be described in the following form.

Theorem 4.1.6. Let F(x)

=

J:,.J(u) du,

where f(x) is periodic and belongs to £1( -:n, :n). If c,. are the Fourier coefficients of f(x) in complex form, then N

F(x) = lim

L

(einz- 1 )c,.fin,

(22)

N-+oo n--N

where (einz - 1 )fin is interpreted to be x when n = 0. The series on the right converges uniformly over [-:n, :n]. This is immediate, because, as in the proof of the foregoing theorem, N

F(x)- CrrX = C0 +lim

L'

c,.einzJin,

(23)

N-+oo n=-N

where I:' means that the term corresponding to n = 0 is omitted. C0 is the same number as in (20). The above formula (23) is obtained from Section 2.8 (7). Let x = 0 in (23). We then see that N

C0 =lim

L'

cnfin.

N-+oo n--N

Putting this into (23), we have (22). Fin&lly we shall give some simple examples of Fourier series. Letf(x) be a periodic function with period 2:n (see Fig. 4.1), defined by 0 <X< m, X= 0, m.

f(x) = { (x - x)/2, 0,

FIGURE

4.1.

(24)

IV.

120

CONVERGENCE AND SUMMABILITY THEOREMS

Computing the Fourier coefficients and noting that f(x) is of bounded variation, we find 00

f(x) = :n:- 1

L sin n:'(fn,

-oo

<X<

(25)

oo.

n-1

Adding g(x)

= xf2:n: to both sides, we get (see Fig. 4.2)

(:n:/2) + m:n:, m:n: { lx + L sin nxfn = ' n=1 (:n:J 2) - m:n:, 00

-m:n:, m

2m:n: < x < 2(m

+ 1):n:,

X = 2m:n:, -(2tr, - 1):n; < X= -2m:n:,

X

(26) <: -2m:n:,

= 0, 1, 2, .... Compare (26) with Lemma 2.6.3. r---

+ + :: r----j r-----i

I I

-4rr:

-2rr~

I

1

'

:

:

I I

I I

2rr

4rr

FIGURE

10 ~

4.2.

r--

L...-....l

The partial sum of (25) N

L sin nxfn n-1

is uniformly bounded. This is seen from the following: Since it is periodic, we may suppose I x I ~ :n:. Thus

L.v (sin nxfn) = L IX cos nx dx = IX [sin(N + l)ufsin lu] du- !x. N

n-1

n-1

0

0

(27)

The left-hand side of (26): 00

!x

+ L (sin nxfn) 1

itself is not a Fourier series because of the term ~x. If we expand the periodic function h(x) of period 2:n: which is identical with the above

4.2.

121

CONVERGENCE OF FOURIER-STIELTJES SERIES

function in ( -:n:, :n:), then we get (see Fig. 4.3) oo

2 L [sin(2n

+ 1):n:xf(2n + 1)] =

n=o

{

0 < x
!n,

0,

-in,

(28)

We give another example, which is obtained by a direct computation. 00

L (cos nxfn) =

-log(2 sin

n=l

ll x J),

This series does not converge at x

-:n; ~X~

:n:,

X

=j::. 0.

(29)

= 0. ------; I

FIGURE

4.3.

I

I 7TI

.....___ I

Finally we mention that 00

L (sin nxflog n)

(30)

2

is not a Fourier series, because L;:' 1f(n log n) does not converge, which contradicts Corollary 4.1.2.

4.2. Convergence of Fourier-Stieltjes Series A formula similar to (22) in Section 4.1 holds for Fourier-Stieltjes series. To show this is the purpose of this section. Let F(x) E M( -:n:, :n:). Form the function (x) = F(x) - CoX,

(1)

where c0 is given by (2:n:)- 1 [F(:n:)- F( -:n:)]. As we saw in Section 3.2, (x) is periodic with period 2:n: and of bounded variation over [-:n:, :n: ], and hence by Theorem 4.1.3, the Fourier series of (x) converges everywhere. On the other hand, the Fourier coefficients of (x) are, by Sec-

IV.

122

CONVERGENCE AND SUMMABILITY THEOREMS

tion 3.2, Eqs. (9) and (10), given by

en = cnfin, C0

n oft. 0,

= F(-n)

+ (2n)-

1

J:

(2)

(n- x) dF(x)

(3)

if en are the Fourier-Stieltjes coefficients of F(x). Hence N

:L' (cneinz/in).

F(x)- CoX= C0 +lim

(4)

N-+oo n--N

Let x = 0. Then N

:L' (en/in). N-+oo n--N

F(O) = C0 +lim

Eliminating C0 with this and (4), we have N

:L' [cn(einz -1)/in].

F(x)- F(O) =CoX+ lim

N-+oo n--N

Hence we get the following theorem.

Theorem 4.2.1. If F(x)

E

M( -n, n), then for every -n ;:;:;; x ;:;:;; n, N

F(x)- F(O) =lim

L

[(einz -1)cnfin],

(5)

N-+oo n--N

where en are the Fourier-Stieltjes coefficients of F(x) and (einz - 1 )fin is interpreted to be x when n = 0. · If F(x) is continuous as well as in M( -n, n), then (5) converges uniformly over [-n, n ]. We again mention that F(x) is standardized. The second statement follows from Theorem 4.1.4. The theorem for a characteristic function, which is analogous to this theorem, is quite important in the theory of probability. It is known as the inversion formula for a characteristic function (see Theorem 4.4.1 ). If the standardization is not adopted, the conclusion should read that (5) holds if x and 0 are continuity points of F(x), and that N

F(x)- F(x') =lim

L

[(einz- einz )cnfin]

(6)

N-+oo n--N

holds for any continuity points x, x' of F(x). The term in (6) with n = 0 is interpreted as x - x'.

4.3.

123

FOURIER'S INTEGRAL THEOREMS

4.3. Fourier's Integral Theorems; Inversion Formulas for Fourier Transforms Let f(x)

E

£1( -oo, oo) and let its Fourier transform be

f(t)

(2n)-11 2

=

J:

e-iztj(x) dx.

(1)

This is, of course, thought of as an analog of Fourier co~fficients in the periodic case. Then the analog of the Fourier series in Section 4.1, Eq. (4) will be

s(x, A.)

= (2n)-11 2

J~.t ](t)eitz dt.

(2)

So we may expect that s(x, A) converges to f(x) as A.- oo under some conditions of f(x). If this should be the case, we may write

f(x)

= (2n)- 112

~~m J~A ](t)eitz dt.

(3)

A~oo

This formula gives us the representation of f(x) in terms of its Fourier transform. This kind of formula is called an inversion formula. Inserting (1) into (2), we see that

s(x, A.) = (2n)- 1 = n- 1

r.t

J:

eizt dt

J:

e-iluj(u) du

[sin A.(u- x)f(u- x)]J(u) du.

(4)

This is similar to Eq. (5) of Section 4.1 and is also called the Dirichlet integral. The kernel of the integral (4) is called the Dirichlet kernel as well. Equation (4) is also written

s(x, A.)= n- 1 J~oof(x

+ u)(sin A.ufu) du.

(5)

Theorem 4.3.1. Suppose that f(u)/(1 (i)

+ I u I) E £1( -oo, oo).

If for some c and for some

J:

[Jf(x

c5

(6)

> 0,

+ u) + f(x- u)- 2c Jfu] du <

oo,

(7)

IV.

124

CONVERGENCE AND SUMMABILITY THEOREMS

then lim n- 1 A-+oo

foo

-oo

f(x

+ u)(sin A.ufu) du =

c.

(8)

If f(u) is of bounded variation in a neighborhood of a point x,

(ii) then

lim n- 1 A-+oo

foo

f(x

+ u)(sin A.ufu) du =

l[f(x

+ 0) + f(x- 0)].

(9)

-oo

This theorem is called Fourier's single integral theorem. Since (6) is supposed to hold, there is an A > 0 so that

Proof.

If

lui>A.

f(u

+ x)(sin A.ufu) du I~ f

lui>A

(I f(u + x) 1/1 u I) du

can be made as small as desired. The convergence of

A f(u + x)(sin A.ufu) du- c = fA tp(u)(sin A.ufu) du, f -.t 0

where tp(u) = l[f(x + u) + f(x- u)- 2c], is equivalent to the convergence of Eq. (10), Section 4.1. [It is immaterial that A is different from n in Eq. (10) of Section 4.1.] So Theorem 4.1.2 gives us the required conclusion. Condition (6) of Theorem 4.3.1 can be replaced by condition (10) in the following theorem.

Theorem 4.3.2. Suppose that f(u) is locally integrable and that f(u)fu-+ 0

monotonically as

u-+ oo and u-+ -oo.

(10)

Then the statements (i) and (ii) of Theorem 4.3.1 hold with the improper integral f:_':, in (9). Proof. For the proof, it is sufficient to show that J;oo and J~oo can be made as small as desired. This was substantially shown in the proof of Theorem 2.1.3. f(u) in Theorem 2.1.3 may stand for f(x + u)fu in our theorem. More precisely, supposing that f(u)fu is nonincreasing for A < u, the second mean value theorem applies to obtain, for B >A'

4.3.

125

FOURIER'S INTEGRAL THEOREMS

> max(A, A - x), f(x + u) sin AU du

J

B .A'

u B

= J .A'

-_

f(x X

f(x

+ u) x + u

X+

A') ++ A'

U

U

Jc .A'

. A d U U

SID

x +u . -U- Sln

1

IIU

d

(for some

u

A' < C < B)

f

= f(x +A') xJc sin AU du + f(x +A') c sin AU du X+

A'

_ f(x + A') + A' X

.A' X

u

X

+ A'

.A'

JCA sin v d + f(x + A') v A' .A.'A V X+

_!_( 1 11

1 A') 1,. _ cos ll'o cos ILL1 ,

where the integral in this expression is bounded (Lemma 2.6.2). This converges to zero as A', B --+- oo. For the integral J-+-oo, the same thing is seen to be true. This shows the theorem. From the proofs of Theorems 4.3.1 and 4.3.2, it must be mentioned that the essential point that distinguishes the convergence in the i<~ourier single integral theorem from the convergence of Fourier series lies in the study of the conditions which make

I

lui>.A.

f(x + u)(sin Auju) du

or

J-.A. -+-00

+ J-+oo f(x + u)(sin Aufu) du .A

small when A is large. Fourier's single integral theorem may be described in the form of the following theorem if f(x) is supposed to belong to V(-oo, oo). It is called Fourier's double integral theorem.

Theorem 4.3.3. Suppose thatf(u) E V(-oo, oo). (i) If for some c and for some ~ > 0,

J:

[if(x

+ u) + f(x- u)- 2c ifu] du <

oo,

(11)

then

-+oo foo .:n:- 1 f 0 dt _00 f(u) cos t(u- x) du =c.

(12)

(ii) If f(u) is of bounded variation in a neighborhood of a point x, then .:n;- 1

J:oo dt J:J(u) cos t(u- x) du = i[J(x + 0) + f(x- 0)].

(13)

IV.

126

CONVERGENCE AND SUMMABILITY THEOREMS

Proof The condition that f( u) e Ll(- oo, oo) is imposed for the existence of the inner integrals on the left-hand sides of (12) and (13). We have

n-1 J.t dt 0

Joo

-oo

f(u) cos t(u- x) du = n- 1 Joo f(u) du -oo

= n-1

J.t cos t(u- x) du 0

J:J(u)[sin A.(u-x)f(u-x)]du.

Hence our theorem reduces to the Fourier single integral theorem. Let us define, for f(u) e Ll(-oo, oo), A(t) = n-1 J:J(u) cos tu du,

0 < t < oo,

(14)

B(t) = n-1 J : f(u) sin tu du,

0 < t < oo.

(15)

These are a little different from cosine and sine transforms (see Section 2.3) in that u extends over (- oo, oo) and the constant factors are not (2n) 1' 2 , but are the analogs of Fourier coefficients. Then we may write n-1

J.t dt Joo 0

f(u) cos t(x- u) du =

J.t [A(t) cos tx + B(t) sin tx] dt,

(16)

0

-oo

which is the analog of Fourier series of real form. So we have from Theorem 4.3.3:

Corollary 4.3.1. Under the conditions of Theorem f.3.3, (i) or (ii),

J:oo [A(t) cos tx + B(t) sin tx] dt = i[f(x + 0) + f(x- 0)] under condition

where c = B(t) are given by (14) and (15), respectively.

c,

(ii), and A(t) and

Another important version of Theorem 4.3.1 (ii) is that it gives us the inversion formula for the Fourier transform.

Theorem 4.3.4. Suppose that f(x) e Ll( -oo, oo) and is of bounded variation over every finite interval. Then lim (Zn)-111 '1'-+oo

J'l'

/(t)ei:zt dt = i[f(x

-'1'

where /(t) is the Fourier transform of f(x).

+ 0) + f(x- 0)],

(17)

4.3.

127

FOURIER'S INTEGRAL THEOREMS

This is immediate from Theorem 4.3.1 if (4) is kept in mind. The integral (Zn)-111

J:

g(t)ei:zt dt,

when g(t) e Ll(-oo, oo), is sometimes called the inverse Fourier transform. It is, of course, a Fourier transform with -x in place of x. Now we notice that if the Fourier transform/(t) ofj(x) E Ll(-oo, oo) happens to belong to Ll(- oo, oo ), then

J

oo

o1

(Zn)-112 -oof(t)eizt dt = f(x)

a.e.

(18)

In other words, the inverse Fourier transform of the Fourier transform is the given function almost everywhere. This statement is an immediate consequence of a theorem given later (Corollary 4.8.1). Finally, we mention the following inversion formulas for the cosine and sine transforms.

Theorem 4.3.5. Let f(x) e Ll(O, oo). Let the cosine and sine transforms ofj(x) be/c(t) and/s(t), respectively [defined by Eqs. (2) and (3) of Section 2.3]. If f(x) is of bounded variation over every finite interval [0, a], 0
J-+oo Jc (t) cos xt dt = o

(2/n)-112

0

J-+oo /s(t) sin xt dt = o

{ l[/(x

+ 0) + f(x- 0)],

i/(0+), { Hf(x

0,

+ 0) + f(x- 0}],

0

< x < oo,

X=

0

0, (19)

< x < oo,

X=

0. (20)

Equation (19) is a corollary of Theorem 4.3.4 with an evenf1 (x) being defined to be the given function f(x) for x > 0 and f( -x) for x < 0. For (20), it is the same. But a direct proof is also simple, because the left-hand side of (19) is (2/n}

J:

cos xt dt

= (1/n}

J~ f(u) cos tu du

J~ f(u)[sin l(u- x)f(u- x)] du

+ (1/n) J~ f(u)[sin l(u + x)f(u + x)] du. Define g(u) = f(u) for u

~ 0,

g(u) = 0 for u < 0. Then the above two

IV.

128

CONVERGENCE AND SUMMABILITY THEOREMS

integrals are the Dirichlet integrals of g( u) over (- oo, oo ), and so from Theorem 4.3.1 it is easy to see that (19) follows. For (20), similar reasoning is used.

4.4. Inversion Formula for Fourier-Stieltjes Transforms We are going to give a result for Fourier-Stieltjes transforms which is an analog of Theorem 4.2.1.

Theorem 4.4.1. Letf(t) be the Fourier-Stieltjes transform of a function F(x) eM. Then F(x)- F(O)

f'-T [(ri"'e -1)/-it]f(t) dt, -oo < x < oo.(1)

= lim(2n)- 1

T-+oo

We recall that F(x) is standardized. Equation (1) means that the limit on the right-hand side always exists and the equality holds. This theorem is known as the Levy inversion formula for a characteristic function when F(x) ED. Proof.

(T [(e-izt -1)/-it]f(t) dt = (2n)- (T [(e-izt -1)/-it] dt I: eitu dF(u) = (Zn)- I: dF(u) I:T [(e-it
(Zn)- 1

1

1

=I: dF(u){-n- I: [sin t(u- x)ft] dt + n- I: (sin tuft) dt}. 1

1

From Lemma 2.6.3, the expression in the braces converges boundedly as T--+ oo to - U(u- x) + U(u), where U(x) is the unit distribution function (standardized). In other words, lim(Zn)- 1 T-+oo

IT

[(rilt -l)f-it]f(t) dt

-7'

= -I~oo U(u- x) dF(u)

+ I~oo U(u) dF(u).

The first Stieltjes integral is

-[F(oo)- F(x

+ 0)] + i[F(x- 0)- F(x + 0)]

and the second is this with x = 0, which gives the required result.

4.4.

INVERSION FORMULA FOR FOURIER-STIELTJES TRANSFORMS

129

An implication of the inversion formula is that a function of M is uniquely determined up to additive constants by its Fourier-Stieltjes transform, and in the case of a distribution function, it is determined uniquely by its characteristic function. This was already given by Theorem 3.5.3.

If F(x) say that

E

M is not necessarily standardized at discontinuities, we should

F(x 2 ) - F(x 1 ) = lim(2:n)- 1 IT [(e-izat- e-iz,t)f-it]J(t) dt T-+oo -T

(2)

for continuity points x 1 and x 2 of F(x). Taking x 2 = x, x 1 = -x in (2) (with standardization), we have F(x)- F(-x) =lim :n;-1 IT (sin xtft)f(t) dt. T-+oo -T

(3)

The right-hand side is the symmetric improper Dirichlet integral. Another consequence of the inversion formula is the following:

Theorem 4.4.2. If F(x) EM and if its Fourier-Stieltjes transformf(t) belongs to £1( -oo, oo), then F(x) is absolutely continuous and F(x)- F(-oo) =I~ p(u) du,

(4)

where p( u) ~ 0, p( u) E £1(- oo, oo) and also p( u) belongs to ( C 0) and hence p( u) E L•(- oo, oo) for every r ~ 1. [See Section 2. 7 for ( C 0 ).] Proof. Suppose that f(t) transform of f(t) by

E

£1(-oo, oo). Write the ordinary Fourier

(Z:n:)-112 I~ooj(t)e-itz dt = (2:n)ll2p(x). Then p(x) belongs to (C0). We form f(t) dt Iz e-itu du I 0z p(u) du = (Z:n:)-1 I"" -oo 0

= (Z:n:)-1 I~oo [(ritz- 1)/-it]j(t) dt. Hence from (1), F(x)- F(O)

= I: p(u) du.

(5)

IV.

130

CONVERGENCE AND SUMMABILITY THEOREMS

Differentiation of both sides of this equation gives us p(u) ~ 0 a.e. Since ~ 0 everywhere. From (5), we easily see that

p(u) is continuous, p(u) p(u) E £1(-oo, oo).

Finally, we remark that there is another form of the inversion formula for a characteristic function (Zolotarev [1 ]).

Theorem 4.4.3. Letf(t) be the characteristic function of a distribution function F(x). Then for every x,

! - :n;-1

F(x) =

I-+oo [Im(f(t)e-it.z)/t] dt. 0

(6)

(6) means that the right-hand improper integral exists and (6) holds. F(x) is of course assumed to be standardized.

Proof. From Theorem 4.4.1, F(x)- F(O)

= lim(2:n)-1{ IT [(e-i.zt- l)f-it]f(t) dt o

T-+oo

+I: [(ei.zt- l)fit]f(t) dt} [by the property f(t) = f( -t)], F(x)- F(O) = lim(2:n)-1 IT {[--e-i.z~(t) o

T-+oo

+ e+i.z~(t)]fit} dt

+ lim(Z:ni)-1 ITo {[f(t) -f(t)]ft} dt

(7)

T-+oo

(where the second limit will be shown below to exist and the separation of limits is permissible), and F(x)- F(O) = lim(Z:n)-1

IT {Im[f(t)e-i.zl]jt} dt 0

T-+oo

+ lim(2:ni)-1 IT t-1 dt Ioo T-+oo

0

[eit.z- e-it.z] dF(x).

-co

The last member of (8) is lim :n;-1 Ioo dF(x)

T-+oo

-co

IT0 (sin txft) dt =

:n;-1

[from Lemma 2.6.3,

:nU(x)

l :n;, -!:n = { -! :n;, 0,

Ioo-co [:nU(x)- !:n] dF(x)

0,]

X> X< 0, x=O

(8)

4.5.

131

SUMMABILITY

and lim :n:-1

T-+oo

I"" dF(x) IT (sin txft) dt 0

00

= i Iooo+ dF(x)- i IodF(x) -oo = t [1-F(O+)- F(O-)] = i - F(O). Inserting this into (8), we have (6), from (7).

4.5. Summability Consider an infinite matrix A

~ (aa) ~ ( =~ ~:

..)

(1)

Consider a series (2) Let sn be the partial sum sn

=

L~=o ck.

Form

00

a .. =

L anJ.:Sk' k-o

n = 0, 1, 2, ... ,

(3)

where the series is supposed to converge for every n = 0, 1, 2, .... If a11 converges to a limit s, then the series (2) is said to be summable to s. If the matrix satisfies the conditions: 0

for each k = 0, 1, 2, ... ,

(4)

L ! ank I ~ C

for each n = 0, 1, 2, ... ,

(5)

lim

ani.·=

n-+oo 00

k=O

where Cis a constant independent of n, 00

lim

Lank= n-•oo k-o

1,

(6)

then it is called the Toeplitz matrix. If (2) is summable to s by the Toeplitz

IV.

132

CONVERGENCE AND SUMMABILITY THEOREMS

matrix, (2) is said to be T -summable to s, or summable by the Toeplitz method to s. In this book we shall not go into the details of summability methods. We shall just apply a few methods of summability to Fourier series and analogs of them to Fourier transforms. Readers who have more interest in the theory of summability are referred to Hardy [2]. Here we will be content only with showing the following theorem of Toeplitz [1 ].

Theorem 4.5.1. Let {sn, n = 0, 1, 2, ... } be a given sequence. If sn converges to s, then it is T -summable to the same sum s. Proof.

The series (7)

is convergent because of (5) and the convergence of sn. Writes"' = s + E~~:, where E~~: --+ 0 as k --+ oo. Then 00

Un

=

s

00

L ~k + k-0 L anli:Ek • k-0

From (6) it is sufficient to show that 00

Un'

=

L anli:Ek --+ 0 11:-o

as

n- oo.

(8)

Given any E > 0, choose k 0 so that I E11: I < E/ (2C) for k ~ k0 , where C is a constant in (5). Take N, depending on k0 , so large that I ank I < E/ {2 :E~o I E~~: I) for n ~ N, k = 0, 1, ... , k 0• This is possible since an11: --+ 0 as n --+ oo for each k. We then have

I u,/ I

This completes the proof.

4.5.

133

SUMMABILITY

Because of Theorem 4.5.1, we may say that the T-summability method is a generalization of the concept of convergence. This property of summability is called regular. In other words, T -summability is a regular summability. The integral analog is possible. Let f(x), x > 0 be locally integrable. Write s(x) =

J:

f(u) du. t

< oo, 0 < x < oo. Form

J~ a(t, x)s(x) dx,

(9)

Let a(t, x) be a given function over 0

~

where this integral is supposed to exist for every t as a Lebesgue integral or an improper integral. If (9) converges to s as t--+ oo, the formal integral f(u) du is said to be summable.

J:

If a(t, x) satisfies that lim a(t, x) = 0,

for almost all

X> 0,

(10)

for every

t >0,

(11)

t-+oo

J~ I a(t, x) I dx ~ C, and lim 1-+oo

J"" a(t, x) dx =

(12)

1,

0

then (9) converges as t--+ oo to s, provided that s(x) converges to s as X-+oo.

The proof is carried out in a way similar to that of the preceding theorem. The summability is also applicable for f~zf(u) du = s1 (x) or f~zf(u) du = s2 (x). Now consider a very special matrix A, in which ank

={

(n

0,

+ 1)-I,

k = 0, 1, .. . , n, k >n.

(13)

In this case, (14)

sn being the partial sum of (2). The summability defined by this is called (C, 1 )-summability (Cesaro-summability of order 1). The Toeplitz theorem gives us the following:

IV.

134

CONVERGENCE AND SUMMABILITY THEOREMS

Theorem 4.5.2. Any convergent series is (C, 1)-summable to s. (C, 1) refers to the first average of sit. The converse is not true as is easily seen from a simple example: L,oo ( -1 )n. It is remarked that an in (14) is also written n

an

L [1- k(n + 1)-1]ct.

=

(15)

It-o

We shall consider another special case of the Toeplitz matrix. Let {rn} be a sequence of positive numbers 0 < rn < 1 such that r11 increasingly converges to 1, as n--+ oo. Take

Then it is easy to see that ank satisfies (4)-(6). Hence (ad) is a Toeplitz matrix. We have 00

an= (1- rn)}: s~.:rnk•

(16)

k=O

which is assumed to exist. Note that for 1 > r > 0, 00

L spit=

00

(1- r)- 1

1.:-o

L s~.:(r"- r"+I) 1.:-o 00

= (1 -

r)- 1

L s~.:r"- (1 1.:-o

00

r)- 1

L s~c_ 1 r" 1.:-1

00

= (1 -

r)- 1

L c"rk; k-0

that is, 00

L spit = 1.:-o

in so far as

00

(1 - r)-1

L cp",

(17)

k-o

L skrk is convergent. Hence (16) can be written 00

an =

L c~.:r/'. k-0

(18)

We then define the summability, which requires the convergence of an in (18), for any choice ofthe sequence r11 (0 < r11 < 1) such that rn--+ 1 - 0 to a fixed numbers. This method of summability is called Abel-summability, which is the same thing as saying:

4.6. (C,

1)-SUMMABILITY FOR FOURIER SERIES

135

Suppose I:k..o cf
lim

L ckrk =

s

,__.1-o k-o

exists, the series (2) is called A-summable (Abel-summable) to s. From the Toeplitz theorem we have the following:

Theorem 4.5.3. If I;:,0 en converges to s, then it is A-summable to s. This theorem is known as Abel's continuity theorem and is useful in complex function theory as well as in analysis generally. A repetition of the proof of the Toeplitz theorem for 00

(1 - r)

L skrlc,

0 < r < 1,

k-0

gives us the proof of the theorem, where the convergence of I: sf
where s~11 = theorem.

L~=o

s•. Using this relation it is easy to show the following

Theorem 4.5.4. A series which is (C, 1)-summable to s is A-summabie to s. The converse statements of Theorems 4.5.3 and 4.5.4 are not true. Results as to the converse with suitable sufficient conditions are called Tauberian theorems and will be given in Section 8.6.

4.6. (C, 1 )-Summability for Fourier Series We are going to apply the summability methods to Fourier series. Letf(x) E £1( -n, n), periodic with period 2n. Let sn(x) be the partial sum of the Fourier series of f(x) and write

an(x) =

1

n

+ 1 k-o L sk(x). n

(1)

IV.

136

CONVERGENCE AND SUMMABILITY THEOREMS

From Section 4.1, Eq. (5), 1J"'-n f(u) 2(n + 1) sin 1 Hx- u) [n·( =n k~O SlO k + 21) (x-u) ] du

O'n(x)

=1

n

J"'-n f(u)Kn(x- u) du,

(2)

where

K (u)" - 2(n

1 ~ sin(k + + 1) sin tu t'::o 1

n

1 2)u

cos ku- cos(k

+ 1) sin tu k~o 2 sin tu 1 4(n + 1) sin2 tu [1- cos(n + 1)u],

+ 1)u

2(n

that is, K ( )

" u = 2(n

1

+ 1)

[ sin(n_ + 1) sm lu

Taking a special function f(u)

tu ]2·

(3)

= 1 (identically), we see that (4)

Kn(u) is called Fejer's kernel and the integral on the right-hand side of (2) is called the Fejer integral. The Fejer kernel has the following properties.

(i)

Kn(u) ~ 0, Kn(u) is even,

(5)

(ii)

Kn(u) ~ Cfnu 2 for all n ~ u > 0,

(6)

(iii)

Kn(u)

~

Cn for all u

> 0,

(7)

where C is an absolute constant. These are easy to show. From these properties we can show the following theorem, which is due to Fejer.

Theorem 4.6.1. (i) If a periodic function f(x)

E £1( -n, n) is continuous at a point x, then the Fourier series is (C, 1)-summable to f(x) at x. (ii) Ifj(x) is continuous over [-n, n], the Fourier series is uniformly ( C, 1)-summable to f(x ).

4.6. (C,

1)-SUMMABILITY FOR FOURIER SERIES

Proof.

(i)

137

Because of (4) and (2),

a11 (x)- f(x)

J:,. [J(u)- f(x)]K (u- x) du

n- 1

=

11

= (2/n)

I:

tp(u)K11 (u) du,

(8)

(9)

where tp(u) = l[f(x Let

8

+ u) + f(x-

(10)

u) - 2/(x)].

> 0 be arbitrary and let " be chosen so that

I w(u) I <

8

(11)

I u I ;:;;; "·

for

This is possible since f(x) is continuous at x. Then

I a,.(x)- f(x) I ;:;;; I(2/n) ;:;;; 8(2/n)

I: I:

tp(u)K11 (u) du K,.(u) du

I+ I(2/n) I: tp(u)K,.(u) du I

+ (2Cfnn)

[by (11) and (6 )]. Hence

I an(x)- f(x) I ;=;;;

8

+ (2Cfnn)

I:

u- 2 1 tp(u) I du

I: [I tp(u) l/u

2]

du.

(12)

Taking N so large that the last member is less than 8 for n ;;;; N (this is possible since J~ [I tp(u) 1/u2 ] du < oo for a fixed "), we have

I O'n(x)- f(x) I < 28

for

n;;;; N.

(ii) is seen from the proof of (i), noting that the choice of " and N is made independently of x since a continuous function over [ -n, n] is uniformly continuous there. REMARK.

The (C, 1)-summability of f(x), at a point x, to sis equiva-

lent to n-1

I

lul
(sin 2 nufnu 2 )f(u

as n ~ oo for any fixed " > 0. For, we may easily see that the kernel

+ x) du ~ s

(13)

IV.

138

CONVERGENCE AND SUMMABILITY THEOREMS

satisfies conditions (5)-(7) for K(u), and that the difference

n-1 (sin 2 nufsin 2 u) - n-1 (sin 2 unfu 2 ) uniformly for I u

I~

=

O(n- 1 )

d.

An important special case of Theorem 4.6.1 is the following corollary.

Corollary 4.6.1. Any continuous periodic function is uniformly approximated by a trigonometric polynomial. More precisely, if e is an arbitrary positive number, and f(x) is any continuous periodic function, there is a trigonometric polynomial Tn(x) (which depends on e) such that

1/(x)- Tn(x) I <

e,

-ex><

X<

(14)

ex>.

We may take an(x) for Tn(x) with a suitably large n. A generalization of Theorem 4.6.1 is given by:

Theorem 4.6.2. Letf(x) be a periodic function belonging to LI( -.n, .n). (i)

If at a point x,

I: I f(x

+ u)- f(x) I du =

as

o(h)

h--+ 0,

(15)

then the Fourier series of f(x) is (C, 1)-summable to f(x). (ii) Every Fourier series of an integrable function is (C, 1)-summable to f(x) almost everywhere. Proof. (ii) is a consequence of (i) since, from Theorem 1.9.1, (15) holds almost everywhere for any J(x) ELI. So it is sufficient to show (i). Write

CTn(x)- f(x)

= (2/.n)

Ittn tp(u)Kn(u) du 0

+ (2/.n) Id

Vn

+ (2/.n) I; tp(u)Kn(u) du = /1

+ lz +Is,

say. Without any condition on f(x) except integrability, I 18 1

< e/2

for a sufficiently large n. This was shown in (12).

tp(u)Kn(u) du

4.5. (C,

139

1)-SUMMABILITY FOR FOURIER SERIES

Now write

I: I

X(u) =

VJ(t) I dt.

Then by (15) X(u)

=

for u -o.

o(u),

(16)

We then see, using (7) that

I /1 I ~ (2Cnfn) I0 I VJ(u) I dt = lin

(2Cnfn)X(n- 1 ).

Hence 11 = o(1)

n- oo.

as

If d has been chosen so small that X(u) ~ e I u

I

for

Iu I <

(17)

d

[which is possible by (16)], then by the use of (7) and integration by parts,

I /2l ~ (2Cfnn) I~

lin

I V'(ull du 2 u

~

(2Cfnn)[X(d)fd 2 ]-(2Cfnn)n 2X(1fn)+(4Cfnn)

~

(2Cfnnd 2 )X(d)

~ [2CX(d)fnnd 2 ]

+ (4Cefnn)

I~

1/n

u- 2 du

I~lin

[X(u)fu 3 ] du

[by (17)]

+ (4Cfn)e.

Taking n large enough, this could be made as small as desired. This shows the theorem. An implication of Theorem 4.6.2 (ii) is this: The sequence of trigonometric functions is complete in £1( -n, n). This statement was made and proved in Theorem 2.10.1, but if Theorem 4.6.2 is used, it is almost obvious because if n = 0, ±1, ... ,

then the Fourier coefficients are all zero and then un(x) = 0 for every n. Since un(x) converges almost everywhere to f(x) by Theorem 4.6.2, f(x) = 0 almost everywhere.

IV.

140

CONVERGENCE AND SUMMABILITY THEOREMS

4.7. Abei-Summability for Fourier Series Let f(x)

E

£1( -n, n) and be periodic. Let its Fourier series be 00

f(x) '"""(a0 /2)

+ L (an cos nx + hn sin nx)

(1)

n-1

or 00

j(x) '"""" L Cneinx.

(2)

-oo

We now study the Abel-summability of this series; in other words, we are concerned with the problem of the convergence of 00

+ L (an cos nx + bn sin nx)rn

f(r, x) = (a 0 /2)

(3)

n-1 00

L

=

Cnein:I:rlnl,

(4)

n--oo

according to Section 4.5, Eq. (18), which leads to the definition of Abelsummability. (3) and (4) converge for 0 < r < 1 because the coefficients are o(1) as n---+ oo or In I---+ oo. We deal mainly with (3). It will be convenient for later use to state the following lemma.

Lemma 4.7.1. For 0 1

P(r, 0) =

2

~

r < 1,

oo

+ J:/n cos nO=

1-r2 2(1- 2r cos 0

+ r2)

,

oo r sin 0 Q(r, 0) = r~ rn sin nO= 1- 2r cos 0 + r2.

Proof.

(5) (6)

Write z = rei6 • We have

1

2

1+z + L rn(cos nO+ i sin nO)= -21 + n=l L zn = 2 (1 _ z) 1 - r + i2r sin 0 = 2(1- 2r cos 0 + r oo

oo

n-1

2

2) •

From this the lemma follows.

(7)

4.7.

141

ABEL-SUMMABILITY FOR FOURIER SERIES

P(r, 0) and Q(r, 0) are periodic functions of 0. (5) and (6) give us P(r, 0) dO = 1,

;rr;-1 [ , .

J:,. Q(r, 0) dO =

:rr:- 1

(8) (9)

0.

We now evaluate f(r, x).

f(r, x) = (2n)- 1 = ;rr;-1

f(t) dt

[,.

J:,.J(t)[ ~

+ 11~1 rn;rr;-

1

J:,. f(t) cos n(t- x) dt

+ ,~/n cos n(t- x)] dt.

Hence from (5), we have

f(r, x)

=

:rr:- 1

J:,. P(r, t - x)f(t) dt.

(10)

This integral is called the Poisson integral, and P(r, 0) is called the Poisson kernel. The function Q(r, 0) is called the conjugate Poisson kernel. This name comes from the following consideration. Consider the Fourier series of complex form and write

z = reiz,

0

~

r < 1,

(11)

en being the Fourier coefficients. This is thought of as a Laurent series. z = 0 is a singular point. However, if we take

0

~

r < 1,

then it is a power series which defines an analytic function in 0 Suppose that f(x) is real-valued. Then (12) becomes

(12) ~

r < 1.

00

L cnrneinz

n-o

and its real part is given by (3), while the imaginary part is 00

/(r, x) =

L (-bn cos nx + an sin nx)rn, n=1

(13)

IV.

142

CONVERGENCE AND SUMMABILITY THEOREMS

which is equal to

~1

J:,. f(t) sin nt dt + sin nx J:,. f(t) cos nt dt] r f(t) f sin n(t- x)rn dt

[-cos nx n-1 = n-1

-n

= n-1

I:

n-1

n-1

Q(r, t - x)f(t) dt

(14)

as is seen from (6). In other words, the integral (14) is the conjugate part of the Poisson integral in (10). In general, a real-valued function, which is the real or imaginary part of a function that is analytic in I z I < 1, is called a harmonic function inlzl
f(r, x) and /(r, x) are harmonic functions in real-valued function f(x).

I z I = I r I < 1, for a

We do not intend to discuss harmonic functions. Our concern is to show the Abel-summability theorems.

Theorem 4.7.1. (i) If a periodic function f(x)

E V( -n,n) is continuous at a point x, then the Fourier series is A-summable to f(x) at x. (ii) If f(x) is continuous over [-n, n ], then the Fourier series is uniformly A-summable to f(x).

Theorem 4.7.2. Letf(x) be a periodic function belonging to V( -n, n). (i)

If at x,

I:

lf(x

+ u) -f(x) I du =

o(h)

as h -o,

then the Fourier series of f(x) is A-summable to f(x). (ii) Every Fourier series of an integrable function is A-summable to f(x) almost everywhere.

As a matter of fact we do not have to prove these theorems. Because of Theorem 4.5.4, they are consequences of Theorems 4.6.1 and 4.6.2. A direct proof will be carried out in a way quite similar to those of Theorems 4.6.1 and 4.6.2, if the following properties of the Poisson

4.7.

143

ABEL-SUMMABILITY FOR FOURIER SERIES

kernel are noticed : P(r, 6)

~

0,

P(r, 6) is an even function of 6,

(15)

P(r, 6)

~

C(1 - r)/r6 8 ,

I 6 I ~ n, C = constant,

(16)

P(r, 6)

~

1/(1 - r).

(17)

The proofs of these properties are easily shown from 1-~

1-~

2(1-2rcos6 + r 8 ) - 2[(1-r) 8 + 4rsin 8 l6]

~

1-~ 1 2(1-r) 8 ~ 1-r'

1- r8 > 1 - r. 8 ~ (1 - r 8 )n8 2[(1-r) + 4r sin 2 l6] = Sr sin 2 l6 8r62

l6j

~n.

Theorem 4.7.2 (ii) may be stated in the following form in terms of the Poisson integral: The Poisson integral (10) of f(x) e Ll( -n, n) converges to f(x) almost everywhere as r --+- 1 - 0. More can be said as to the convergence of the Poisson integral. Suppose in (10) that r, x depend on a parameter r = r(a), x = x(a), 0 ~a < 1 and as a --+-1, r(a) --+-1- 0, x(a) --+-x0 • If x(a)/[1 - r(a)]

(18)

is bounded for a < 1, then it is said that (r, x)--+- (1, x0 ) nontangentially. Write (r, x) as reiz. Then point reiz in the unit circle in the complex

plane is considered to approach a point eizo on the circle in such a way that the path along which reiz moves is kept in an angular domain with

FIGURE

4.4.

vertex at eizo, where the angular domain is symmetric to the radius through eizo and each side of the domain makes an angle of less than n/2 with the radius (see Fig. 4.4). We also say more properly in the above case that reiz approaches e'zo non tangentially.

IV.

144

CONVERGENCE AND SUMMABILITY THEOREMS

We may show the following:

Theorem 4.7.3. The Poisson integral (10) of f(x) E V( -n, n) converges to f(x 0 ) as reiz tends non tangentially to eizo for almost all x 0 , -n < x 0 < n. Proof. f(r, x)- f(x 0 )

say; note that r

I 11 I ~

=

r(a), x

[1 - r(a)]- 1

~ [1- r(a)]- 1

=

Write y

y(a)

I I

= 1-

y-1

=

J

=

I~ P(r, u)[f(u

+ x)- f(x

0)

]du

x(a). From (17)

IUI<1-r(IZ)

lUI <1-J'(IZ)

If(u + x)- f(x 0 ) I du

I f(u + x - X0 + x 0 ) - f(x 0 ) I du.

(19)

r(a). The last member is

I/+(Z-Zo)

z-zo

n- 1

If(v + Xo)- f(xo) I dv If(v + X0 ) - f(x 0 ) I dv.

(20)

The first integral is, if x =f=. x 0 , _!__ IZ+(Z-Z0 )

y

-

...!._

=

Iz-zo

y

0

y

+ (x-x

0 0)

_

y

x0

X -

y

1 X-X 0

Iz-z

0 •

o

(21) [y + (x - x 0 )]/y and (x - x 0 )fy lire bounded by a constant, since reiz approaches eizo non tangentially. The remaining factors in (21) converge to zero for almost all x 0 • Hence the first integral of (20) converges to zero for almost all x 0 • The same thing is true for the second integral.

I.

/2 = n-1 "

+ :n-1 s-" -..

'

4.8.

145

SUMMABILITY THEOREMS FOR FOURIER TRANSFORMS

where, because of (16),

I

:n:-1 I: P(r, u)[f(u

+ x)- f(x

du

0 )]

~ [C(1- r)fr] I: u-2 1 f(u

I

+ x)- f(x

0) I

du,

C being a constant. Writing

X(u)

= X(u, x) = I: If(v

+ x)- f(x I dv, 0)

the right-hand side of the above inequality is [C(1- r)fr] { [X(u)/u 2

]1:-

I: [X(u)fu 3 ] du}.

(22)

Now as before, X(u)

IH(:~HJ:0 ) = I0 lf(w

+x

0 )-

f(x 0 ) I dw-

Iz.-z I f(w + x 0

0

0 )-

f(x 0 ) I dw

and I u + (x- x 0 ) I ~ u + C 1u, I x - x 0 I < C 1u since I x - X 0 I ~ Cl)' ~ C 1u. Therefore X(u) = o(u), from which it is easy to show that (22) converges to zero for almost all x 0 • The same thing is true for the second integral of / 2 • This completes the proof.

4.8. Summability Theorems for Fourier Transforms We are now interested in (C, 1)-summability theorems for Fourier transforms. Letf(x) belong to £1( -oo, oo) and/(t) be its Fourier transform. We have shown before (Theorem 4.3.4) that (1)

converges to l[f(x + 0) + f(x- 0)] if f(x) is of bounded variation on every finite interval. We now deal with aT(x) = (2:n:)- 1' 2 T- 1

I du I"-uf(t)e'."dt. T

0

1

(2)

If O'p(x) converges to s, we say that (1) is (C, 1)-summable to s. Generally, if, for a locally integrable function g(t),

lim T- 1 IT du I" g(t) dt = s T-+oo

0

-u

IV.

146

CONVERGENCE AND SUMMABILITY THEOREMS

exists, then the integral

is said to be (C, 1 )-summable to s as T--+ oo. We do not have to use the following lemma now, but for completeness, we give it.

Lemma 4.8.1. The integral J:Tg(t) dt is (C, 1)-summable to s as T

--+

oo if and only if (3)

I:T [1 - (It l/T)]g(t) dt

converges to s as T--+ oo, where g(t) is supposed to be locally integrable. This is the analog of what Section 4.5, Eq. (15) tells us.

Proof.

Interchanging the order of integration, we get

T-1 IT du I" g(t) dt = '1'-1 IT g(t) dt IT du 0

-u

t

0

= T-1 I: [1 -

+ T-

1

+ T-1 Io

-T

g(t) dt IT du -t

(t/T)]g(t) dt

J:T (1

+ (t/T)]g(t) dt,

which is the same thing as (3) and the lemma is proved. Now we compute O'T(x). From (2), 0'7'(x) = (2nT}-1 IT du I" eizt dt Ioo f(v)e-it'" dv 0

-u

-oo

= (2nT)-1 I:J(v) dv I: du I:

eiCz-v>t

dt

= (nT)- 1 I : [f(v)f(x- v)] dv I: sin(x- v)u du = (nT)- 1

I~oof(v){[l- cos T(x- v)]f(x- v) 1 } dv.

Hence we get

O'T(x) = n- 1 I : [sin 1 .A(x- u)f.A(x- u} 2]f(u) du,

(4)

where .A = Tf2. This integral is again called the Fejb integral. The kernel

K(u, .A)= sin 1 .Auf.Au1 is called the Fejb kernel. This, of course, is the analog of K 11 (x) in Sec-

4.8.

SUMMABILITY THEOREMS FOR FOURIER TRANSFORMS

147

tion 4.6, Eq. (3) and has properties similar to it. n;-1

I: K(u, J.) du

= 1,

(5)

K(u, J.)

~

0,

K(u, J.) is even for u,

K(u, J.)

~

CfJ.u2 ,

for all

K(u, J.)

~

Cu,

for all u > 0.

). > 0, u > 0,

(6) (7) (8)

The proofs are quite easy. The analogs of Theorems 4.6.1 and 4.6.2 are the following:

;, Theorem 4.8.1. Let f(x) e £1(-oo, oo) and let /(t) be its Fourier transform. Then: (i)

If f(x) is continuous at a point x, then fT(x) = (Zn)-112 I:T /(t)eizt dt

is (C, 1)-summable to f(x) at x. (ii) Ifj(x) is continuous and bounded over ( -oo, oo),fT(x) is (C, 1)- )' 2 summable tof(x) uniformly over (-oo, oo). \ '

Theorem 4.8.2. Let/(~) e £1(-oo, oo) and let /(t) be its Fourier transform. (i) If

I: lf(x

+ u)- f(x) I du =

(9)

o(h)

ash-+ 0, then fT(x) in Theorem 4.8.1 is (C, 1)-summable to f(x) at x. (ii) For any f(x) E £1(-oo, oo), j,l'(x) is (C, 1)-summable to f(x) almost everywhere. Actually these theorems reduce to Theorems 4.6.1 and 4.6.2, respectively, if the following is noticed:

aT(x)- f(x)

=

(1/n) I~oo (sin 2 J.ufJ.u 11 )[f(u

=

(2/n) I~ VJ(u)K(u, J.) du,

where J. = Tf2, VJ(u) = l[f(x + u)

+ f(x -

+ x)-f(x)] du

u) - 2f(x)], and

II: VJ(u)K(u, J.) du I~ J.- I: [I VJ(u) l/u 1

11]

du.

IV.

148

CONVERGENCE AND SUMMABILITY THEOREMS

Hence the (C, 1)-summability problem now is equivalent to the consideration of (2/n) tp(u)K(u, .A.) du. The apparent difference of this integral and that of Section 4.6, Eq. (9) is of no importance. A consequence of this theorem is:

J:

Corollary 4.8.1. If the Fourier transform/(t) belongs toLl( -oo, oo), then (Z:n)-112 I:/(t) eizl dt = f(x) a.e. Because of the integrability of /(t), the left-hand side integral in the corollary exists and on the other hand, from the foregoing theorem,

is (C, 1)-summable to f(x) almost everywhere. Hence the conclusion must hold, since convergence implies (C, 1 )-summability. For the A-summability of the Fourier transform, we will have a similar situation. Consider g(z) = (2n)-112 I~ ](t)eizl dt,

y > 0,

(10)

where z = x + iy, x, y being real, which is the analog of Section 4.7, Eq. (12). It is easy to see that g(z) is really defined for y > 0, -oo < x < oo, and is an analytic function in the upper half-plane y > 0. In fact, for any h = e + iTJ, such that y + 'YJ > 0, [g(z

+ h)- g(z)]fh =

(Z:n)-1' 2h- 1 I~ /(t)(ei 111 - 1)eizl dt.

(11)

Hence [g(z

+ h)- g(z)]/h ~ 2n-1' 2 I~ I/(t) I te'"~' 1e-111dt

for -y <

'YJ

< 0

and

~ (Z:n)-112 I~ I/(t) I te-111 dt

for

'YJ

>0

by (2) and (3) of Lemma 2.6.1. For any fixed y > 0 and for 'YJ < y/2, the above integrals are bounded by Cte<-111 2 11 (C being some constant), which belong to Ll( -oo, oo).

4.8.

149

SUMMABILITY THEOREMS FOR FOURIER TRANSFORMS

Hence the dominated convergence theorem allows us to let h ---+ 0 inside the integral in (11 ). In other words g'(z) exists and is given by i(2n)- 1 ' 2 J~ /(t)teizt dt. This implies the analyticity of g(z) in y > 0. Now for y > 0, g(z)

= (2n)-1

Joo eizt dt Joo 0

= (2ni)- 1 J~oo

j(u)e-itu du

-oo

[J(u)f(u- z)] du.

We have thus shown the following theorem.

Theorem 4.8.3. If g(z) is given by (10), then g(z) Js analytic in Im z > 0 and is represented by g(z)

= (2ni)- 1 J~oo

[J(u)f(u- z)] du,

for

Im z > 0.

(12)

The integral in (12) is called the Cauchy integral. The real and imaginary parts of 1fi(u - z) are, respectively, yf[(x- u) 2 + y 2 ] and (x- u)f[(x - u) 2 + y 2 ]. Consider f(x, y)

= n- 1 J~oo {y/[y2 + (x- u) 2 ]} j(u) du,

j(x, y) = n- 1 J~oo {(x- u)/[y 2

+ (x- u)

2 ]}

f(u) du.

(13) (14)

They are called, respectively, the Poisson integral and the conjugate Poisson integral of f(x). The multipliers of f(u) in the integrals in (13) and (14) are called, respectively, the Poisson kernel and the conjugate Poisson kernel. More substantial discussions concerning (12)-(14) will be made later. Note that (15)

This is seen directly. Since g(z) is analytic in Im z > 0, assuming that f(t) is real-valued, f(x, y) and j(x, y) are harmonic functions in y > 0.

IV.

150

CONVERGENCE AND SUMMABILITY THEOREMS

The A-summability for the inversion of the Fourier transform takes the following form analogous to that of Section 4.7, Eq. (4) J(x, y) = (2n)-1!2

J:

e-ltlveiztj(t) dt,

y > 0,

(16)

and we ask when this integral converges to f(x) as y--+ 0+. Actually, this f(x, y) is seen to be coincident with f(x, y) in (13) because (16) is

= x- 1

J:J(u) du J~ e-tv cos t(x- u) dt

= x-1

J:

{y/[(u- x) 2

+ y 2]} f(u) du.

The analogs of Theorems 4.7.1 and 4.7.2 are true. We state them in a simpler form in Theorem 4.8.4. The proof is seen if the following relations for the Poisson kernel P(:te, y) = yf(x2

+ y2)

(17)

hold: P(x,y) ~ 0 P(x, y)

~

yfx2

P(x, y) ~ 1/y

Theorem 4.8.4. Let /(x)

E

for y >0,

(18)

for y >0,

(19)

for y > 0.

(20)

£1(-oo, oo).

(i) Ifj(x) is continuous at x, then (16) converges tof(x) asy--+ 0+. (ii) (16) converges to f(x) as y--+ 0+ almost everywhere.

As in Theorem 4.7.3, part (ii) of this theorem can be stated as: (ii') (16) converges to f(x 0 ) for almost all x 0 as x non tangentially.

+ iy tends to x

0

That is, (x, y) tend to (x0 , 0) in such a way that x--+ x 0 , y--+ 0+, and I x - x 0 lfy is kept bounded.

4.9.

DETERMINATION OF ABSOLUTELY CONTINUOUS COMPONENT

151

4.9. Determination of the Absolutely Continuous Component of a Nondecreasing Function We discuss the (C, 1)-summability problem for Fourier-Stieltjes transforms. The analogs of the corresponding problems for Fourier series and Fourier transforms lead to a different type of problem. Let F(x) eM and its Fourier-Stieltjes transform be

J(t) = J~oo

eizl

dF(x).

(1)

Differing from what we studied in Section 4.4, we consider the integral (2)

and discuss its (C, 1)-summability. Before doing this, we give some remarks on a function of M or V. Let F(x) e M. Then the decomposition

F(x) = F.(x)

+ Fd(x) + F.(x)

(3)

is possible, where Fd(x) is the purely discontinuous component, Fa is the absolutely continuous component, and F 1 (x) is the continuous singular component. F., Fd, and F. belong to M. Fd(x) and F 8 (x) have derivatives equal to zero almost everywhere. If F(x) ED, it would be convenient to write (4)

where a 1 , a 2 , and a 3 are nonnegative constants such that a 1 + a 1 + a 3 = 1. Then F.(x), Fd(x), and F 8 (x) are distribution functions. Here we deal with (3). Write Fa(x) = Fd(x) + F 8 (x). Then Fa'(x) = 0 a.e. Now we mention a few properties of functions of V that will be used. Let F(x) e V. F(x) is almost everywhere differentiable (Theorem 1.7.2). The total variation of F is written as

T(F) =

J: I

dF(x)

I·

Then, for a Borel function f(x),

IJf(x) dF(x) I~ J: If(x) I d[ J: I dF(u) 1].

(5)

IV.

152

CONVERGENCE AND SUMMABILITY THEOREMS

The right-hand side is also written as

J:

lf(x)

II dF(x) I·

The total variation has the elementary property T(F +G)~ T(F) + T(G),

(6)

where F, G e V.

Lemma 4.9.1. For a function F(x) e M,

fzz+h I d[F(u)- Fa'(x)u] I -0

h-1

as h -o

for almost all x, where F'(x) exists and Fa'(x) = 0.

Proof.

fz

z+h

Write F(u)

= Fa(u) + Fa(u). Then, from relation (6),

I d[F(u)- F'(x)u] I ~ Jz+h I d[Fa(u)- F'(x)u] I + Jz+hl dFa(u) I· z z (7)

Since F (u) is nondecreasing,

f

z+h

z

I dFa(u) I =

Fa(x

+ h)- Fa(x) =

o(h)

if xis a point where Fa'(x) = 0. Also, we have inequality (7) with oppositive inequality sign and with a minus in the last term. Hence

J:+h I d[F(u)- F'(x)u] I = J:+h Id[ J: Fa'(v) dv- Fa'(x)u] I+ o(h) because F'(x) = Fa'(x) at the point x and Fa(u) is absolutely continuous and is written as

Therefore, we have

fzz+h I d[F(u)- Fa'(x)u] I = fx+h z I Fa'(v)- Fa'(x) I dv + o(h). Since Fa'(v) is integrable, the first integral on the right is o(h) for almost all x, which shows the lemma.

4.9.

DETERMINATION OF ABSOLUTELY CONTINUOUS COMPONENT

153

Now we shall prove the following theorem, which was given by van der Vaart [1].

Theorem 4.9.1. Let F(x) EM and its Fourier-Stieltjes transform be f(t). Let Fa(x) be its absolutely continuous component. Then

(2n)- 1 (T [1-

(I t 1/T)]e-ixnt) dt- Fa'(x)

(8)

as T - oo for almost all x. In other words,

is (C, 1)-summable to Fa'(x) almost everywhere. Also we could say that the derivative of the absolutely continuous component, and hence the component, is determined by the Fourier-Stieltjes transform. Proof.

Let x be a point at which Fa'(x) exists and Fa'(x) = 0. aT(x) = (2n)- 1 (T [1-

(I t 1/T)]f(t)e-ixt dt

. du = (2nT)- 1 I T dt It f(u)e-'xu 0

-t

= (2nT)- 1 IT dt It e-ixu du Ioo eiuv dF(v) 0

=

-t

-oo

(Zn)- 1 Ioo dF(v) T- 1 IT dt It e-i<x-v)u du -oo

= n- 1

I:

0

-t

[sin 2 l.(x- v)fl.(x- v) 2 ] dF(v),

where I. = Tf2. Now aT(x)- Fa'(x)

=

n- 1 I~oo [sin 2 l.(x- u)fl.(x- v) 2 ] d[F(v)- Fa'(x)v]

where <5 is chosen so small that for a given e > 0,

Ix

x+h

I dF(u)- Fa'(x)u I ~ e I hI

for

I hI

~ <5,

(9)

IV.

154

CONVERGENCE AND SUMMABILITY THEOREMS

which is possible in view of Lemma 4.9.1. I

I ~ (n.l.)- I

12 1 ~ (n.t.)- 1

1/A:Slul

1

1/A:::Oiul

u- 2 1 d[F(u

u-2 d[

+ x)- Fa'(x)(u + x)] I

Iu I d[F(u + x)- Fa'(x)(u + x)] I] 0

Iu Id[F(v + x)- Fa'(x)(v + x)] !leo

= (n.l.u 2 )- 1

+ (n.l.u

0

2 )-1

I: I

VA

d[F(v

+ x)- Fa'(x)(v + x)] 11:A

+ 2(nJ.)-1 I1/A::iiilul u-3 du Iuo I d[F(v + x)- Fa'(x)(v + x)] I· From (9), I

12 1 ~ Ce

+ (2efn.l.) I1/A:iilul u-

2

du

(10)

for some constant C, which may depend on d, and

(11) Finally

I /2!

~ (.1./n)

I

lul:iil/A

I d[F(u + x)- Fa'(x)(u + x)] I < 2efn

(12)

by (9). Equations (10)-(12) show the theorem. We finally remark that we may replace (C, 1)-summability in Theorem 4.9.1 by A-summability, that is, in place of uT(x), we may take y

>0,

and let y - 0+, to obtain a similar conclusion. We will not go into details.

4.10. Fourier Series and Approximate Fourier Series of a Fourier-Stieltjes Transform Let F(x) E M and its Fourier-Stieltjes transform be f(t). Consider the restriction of f(t) to (- T, T) and, if necessary, make the periodic extension of it. We study the convergence of the Fourier series of the function obtained in this way. We call such a Fourier series the Fourier series of f(t) over (-T,T).

4.10.

FOURIER SERIES OF A FOURIER-STIELTJES TRANSFORM

155

So the Fourier coefficients are .

a" = T- 1 I:T f(t) cos(nntfT) dt,

n = 0, 1, 2, ... ,

(1)

b,. = T-1 I:T f(t) sin(rmt/T) dt,

n = 1,2, ... ,

(2)

n = 0, ±1, ±2, ....

(3)

or

c,.

=

(2T)- 1 IT f(t)e<-nnitltT dt, -T

For simplicity, we suppose T = :n;. There is no loss of generality. The following theorem is due to Dugue [3, 4].

Theorem 4.10.1. The Fourier series of the Fourier-Stieltjes transform f(t) over ( -:n;, :n;) converges to f(t) when -:n; < t < :n;, and to l[f(:n;) + /( -:n;)] when t = :n; or -:n;. Proof. Suppose series of f(t) is

-:n;

sn(t) = :n;- 1


:n;.

Then the partial sum sn(t) of the Fourier

J:" f(u)Dn(u- t) du

= :n;-1 I:J(u)[sin(n + l)(u- t)/2 sin t(u- t)] du

(4)

[by Section 4.1, Eq. (5)]. Hence

sn(t)- f(t) = :n;- 1 I: [J(u)- f(t)]Dn(u- t) du = :n;-1 I: D,(u- t) du =

I:

I:

[eiux- eitx] dF(x)

eitx dF(x) :n;-1 I: Dn(u- t)[ei
(5)

Now it is easy to see that the inner integral is uniformly bounded, since

I: [sin J.(u-x)/2 sin t(u-x)]eiux du- I: [sin J.(u-x)f(u-x)]eiux du is seen to be uniformly bounded and the second integral is also uniformly bounded from Lemma 2.6.4. Further, the inner integral converges to

ei(u-t):a: - 1 lu-t'

IV.

156

CONVERGENCE AND SUMMABILITY THEOREMS

which is 0 from the convergence criteria for Fourier series (e.g., Corollary 4.1.1). Therefore, sn(t) converges to f(t). Next we suppose that t = :n:. Then as before

I:

J:,. Dn(u- :n;)eiuz du = I: eizn dF(x) J:,.D (u- :n;)ei
sn(:n;) =

dF(x) :n;- 1

:n;-1

11

(6)

Notice that the inner integral is written by

I~ Dn(v)e-ivz dv and

Dn(v) = sin(n has singularities at v

I:

(7)

+ !)v/2 sin !v

= 0 and v = 2:n:. Now we break (7) into two parts:

D11 (v)e-ivz dv

+ J~ Dn(v)e-ivz dv

=I: D (v)e-ivz dv +I: D,.(2:n:11

=

I:

Dn(v)e-ivz dv

+ e-i2nz

I:

w)e-i< 2n-wlz

dw

Dn(w)eiwz dw.

Hence from (7),

s11 (:n:) =

I" Dn(v)e-ivz dv + :n;-1 Ioo e-izn dF(x) I" Dn(v)eivz dv.

:n;- 1

Ioo

-oo

eizn dF(x)

0

-oo

Now :n;-1

0

(8)

I" D (v)e-ivz dv o

n

is the partial sum of the Fourier series at u = 0 of a periodic function g(u) = 0 for -:n; < u < 0, and = e-iuz for 0 < u < :n;, with period 2:n:. Hence by Theorem 4.1.3, it converges to ![g(O-) + g(O+)] which is t· In the same way the inner integral of the second integral of (8) also converges to t· This shows that

sn(:n;)-+ H/(:n:) The same result holds for t

+ /( -:n;)] ·

= -:n; and the proof is complete.

4.10.

FOURIER SERIES OF A FOURIER-STIELTJES TRANSFORM

157

In some probabilistic problems (see Papoulis [1 ]), the following series is important for the approximation of the Fourier-Stieltjes transform. As before,

f(t)

r:

=

eitx dF(x),

(9)

F(x) EM. Now define, for any T, Fn = F11 (T) = f

21n+llntT dF(x), 2nn/T

n

= 0, ±1, ...

(10)

and define the trigonometric series 00

f(t, T)

L

=

Fne2nnit!T.

(11)

n--oo

Actually this series is absolutely and uniformly convergent for -oo < t < oo since Fn ~ 0 and L:O Fn = J:X, dF(x) < oo. Therefore f(t, T) is well-defined and is uniformly continuous over (-oo, oo). Thus the series ( 11) is the Fourier series off( t, T) (by Theorem 2.2.1 ), where f(t, T) is a periodic function of period T. The Fourier series (11) is called the approximate Fourier series of f(t). We will now show that if T is large, then the series ( 11) really approximatesf(t). More precisely we have:

Theorem 4.10.2. For every t and T such that T > 21 t I, we have IJ(t)- f(t, T) I

~

21 sin(:rr:t/T) I [F(oo)- F( -oo)].

(12)

From this theorem we say that f(t, T) converges to f(t) as T-oo, uniformly for every finite interval oft.

Proof. lf(t)- f(t, T)

I= =

IL oo

[

n=-oo

L 00

J2(n+l)n/T

2nn!T

n=-oo ~

L 00

n--oo

f21n+llntT eitz dF(x)- f21n+1)ntT e2n:nit!TdF(x) ] 2nn/T 2nn/T

[

11 _

J2(n+l)n/T

2nn/T 2(n+l)n/T

e<2nntT-zlit 1dF(x)

11- e<2nn!T-zlit 12 dF(x) ] 1/2 ] 1/2

dF(x) 2nn/T (by the Schwarz inequality), X

[ J

I

IV.

158 lf(t)- f(t, T)

Jll(n+l)n/T

oo I~ [ L

n--co

I 1 - e
2""'T

Jll(n+l)n/T oo L n--co 2""'T

[

X

CONVERGENCE AND SUMMABILITY THEOREMS

dF(x)

] 112

(by the Cauchy-Schwarz inequality) J2(n+lln/T L n--co 2""'T 00

=

[

2

(1- cos[(2nn/T)- x]t) dF(x)

] 1/2

X [F(oo)- F(-oo))1'2.

Since

1- cos[(2nnfT)- x]t

~

2 sin 2 (ntfT) t:

10r

{T>2ltl, (2nnfT) ~ x

~

[2(n

+ l)n/T],

the last expression is lf(t)-f(t, T)

I~ 2

00 [

nLsin 2(ntfT)

J2(n+lln/T

""'T

2

dF(x)

] 1/2

X [F(oo)- F(-oo)]112

=

21 sin(ntfT) I [F(oo)- F(-oo)].

4.11. Some Examples, Using Fourier Transforms 1.

EXAMPLE

Consider the function, for A.> 0 (Fig. 4.5), 0, S(x, A.)= {

l, 1,

• I

I xi> X

A.,

=±A,

(1)

I xi< A.

I

+

I

I

I

FIGURE

4.5.

Then the Fourier transform of S(x, A.) is (2n)- 1' 2

J:

S(x, A.)e-itx dx = (2/n)I' 2 sin A.tft.

(2)

4.11.

159

SOME EXAMPLES, USING FOURIER TRANSFORMS

Hence by, the inversion formula (Theorem 4.3.4), x- 1 lim T-+oo

JT

(sin Atjt)eizt dt = S(x, A),

(3)

-T

which is no more than Lemma 2.6.4. If we write x = 0 in (3), we have Lemma 2.6.3. Let A1 , ••• , An be any set of positive numbers. Then the Fourier transform of the convolution S 1(x; A1, A2 ) of S(x, A1) and S(x, A2 ) is given by (from Theorem 2.12.2)

.yz;;- (2/.n)(sin A t/t)(sin A tft). 1

2

Therefore, by the inversion formula (Theorem 4.3.4) (2/.n)

J:

(sin A1t/t)(sin A2t/t)eizt = S(x; A10 A2),

(4)

where the integral on the left is absolutely convergent. It is seen, by a direct observation or from the fact that the set of discontinuities of a convolution of two step-functions is the vector sum of the sets of discontinuities of these step-functions, that

in other words, (5)

Repeating the argument, we see that for

n = 1, 2, ... , where if n = 1, (See Bochner [5].)

+ ··· + Ak>

(6)

replaced by limT-+oo

J:T.

Ix I >

J:., should be

A1

EXAMPLE 2. We perform the direct computation of S(x; A1 , A2 ) which is the convolution of S(x, A1) and S(x, A2). We then see that if A2 ~ A1 > 0 (Fig. 4.6), x ~ -A 1 - A2 , 0, - A1 - Aa ~ X ~ - A2 A1 , X+ A1 A2, S(x; A1, Aa) = ZA1, (7) - Aa A1 ~ X ~ A2 - A1 , Aa - A1 ~ X ~ A1 A1 , -x A1 Aa, 0, A1 Aa ~ x.

+

+ +

+ +

+

+

IV.

160

(-).2+

CONVERGENCE AND SUMMABILITY THEOREMS

>..,2>.,)

/ FIGURE

4.6.

In particular (see Fig. 4.7), n-1

J:

0, (sin 2 ).tjt2 )eizt dt = { ix + )., ). - ix,

I X I ~ 2)., -2).

<X~

0,

(8)

O~x
More particularly, (9)

which is, of course, known. (0,~)

A

-2~

FIGURE

4.7.

2~

Also we remark that n-1

J"" -oo

(sin afa), { (sin tft)[sin ).(t- a)f(t- a)] dt = (sin a).ja), -(sin afa),

).

~

I ;.I

1, ~ 1,

).

-1.

~

(10)

This is obtained from the application of Theorem 2.12.2. EXAMPLE 3. We now give an example of a different type. Suppose that a function f(x) is given, and that

f'"'(x)

E

V(-oo, oo),

k = 0, 1, 2.

Let /(t) be the Fourier transform of f(x). Then the Fourier transform of f"(x) is given by -tzj(t), by the remark at the end of Section 2.9 (seep. 69). Suppose further that

(1

+ x )f(x) E V(-oo, oo). 2

Then /"(t) [the second derivative of /(t)] is the Fourier transform of

4.11.

161

SOME EXAMPLES, USING FOURIER TRANSFORMS

-x2f(x) (the last remark in Section 2.9). We can say that under the above conditions, the Fourier transform of the function

is given by

So notationally, (11)

Now consider the function Hn(x)

=

dne-z• (-1)nez• dxn , -oo <X< oo, n

=

0, 1, 2,.. ..

(12)

This is seen to be a polynomial of the nth degree, and is called the Hermite polynomial, n = 1, 2, .... We may see that Hn(x) satisfies the differential equation

Z

(13)

-2xy+2ny=0

because, by (12), From this, from which and Therefore,

+

H;:+l(x)- 2xHn+l(x) 2(n = -H;:'(x) 4xH;:(x)

+

Hence, if we write

+ 1)Hn+t(x) + (2 - 4x

2 -

2n)Hn'(x)

+ 4nxHn(x).

IV.

162

CONVERGENCE AND SUMMABILITY THEOREMS

then we have from the above equation that (14) By definition H 0 (x) = 1, and hence W0 (x) = 0,

-oo

<x<

oo.

Then from (14), W1 (x) = 0, and generally Wn(x) = 0 or Hn(x) satisfies (13). Now since Hn(x) is a polynomial, which is seen from (12) immediately, we find, putting Hn(x) = anxn + · · · + a0 and inserting y = Hn(x) in (13), that, except for a constant factor,

Hn(x)

=

2n[ xn- n(n :;-1) xn-a

+ n(n -1)(n4~ 2)(n- 3) xn-c

_ n(n -1)(n- 2)(n- 3)(n- 4)(n- 5) xn-a 4·8·12

+ ... ] (15 ) .

Write

n = 0, 1, 2, ....

(16)

These are called the Hermite functions. These functions cf>n(x) satisfy the differential equation

(17) This can be written as

Hence the Fourier transform ~n(x) of cf>n(x) is the Fourier transform of

1 ( d2 ) - 2n + 1 dJc2 - x2 cf>n(x). From (11) it is

4.11.

163

SOME EXAMPLES, USING FOURIER TRANSFORMS

In other words, again ~n(t) satisfies

d2

dt 2 ~n(t)- t 2~n(t)

= -(2n + 1)~n(t).

(18)

Now

J~"" xn exp(-x2 /2) e-itz dx = J~"" xn exp[-(x + it) 2 /2] dx =

exp( -t2/2)

J

oo+it -oo+it (z-

it)n exp(-z 2 /2) dz,

where the last integral is taken over the line u - it, - oo < u < oo, parallel to the real axis. Since the integrand is an entire function, and using the Gauchy integral formula and the fact that

J

A +it A

(z- it)n exp(-zn /2) dz

-+

as

0

jAj-+oo

for a fixed t, we see that

J~"" xn exp(-x2 /2) exp(-itx) dx

=

exp(-t2 /2) J~"" (x- it)n exp(-x 2 /2) dx

=

(-t 2 /2) Qn(t),

(19)

where Qn(t) is a polynomial of the nth degree. Hence the Fourier transform of c/Jn(x) is of the form exp( -t2 /2) Pn(t), where Pn(t) is a polynomial of the nth degree. That is,

Now y

= exp(t2/2) ~n(t)

satisfies Eq. (13) with tin place of x, from which we see that ~n(t) exp(t 2/2) is equal to Hn(t) up to constant factors. This shows that -00

< t < oo.

(20)

[If we use the fact that the differential equation (13) has a unique analytic solution except for a constant factor, then we do not need the above discussion to get (20).] The coefficient of tn in Qn(t) in (19) is

164

IV.

CONVERGENCE AND SUMMABILITY THEOREMS

Hence we have Cn = ( -i)n. That is, the Fourier transform of c/Jn(x) is given by ( -i)"~n(t). (See Wiener [7].) NOTES 1.

In connection with the Dirichlet kernel, we mention that

e.. =

I" [I

sin(n

+ !)x l/2 sin !x] dx = :n:-1 log n + 0(~.

~

0

(See Bary [1], p. 108 and Zygmund [1], p. 172.)

:n:- 1!?..

is called the Lebesgue constant.

2. The single condition that the function be continuous at a point x is not sufficient for the convergence at x of the Fourier series. Such an example was given by Fejer. (See Bary [1], Zygmund [1, 2].) For the single or double Fourier integral theorems, the single continuity condition is not sufficient even if the function is integrable over (-oo, oo). (See Titchmarsh [2].) 3. In the proof of Fourier's single integral theorem (Theorem 4.3.1) the Dirichlet integral, the integral in Section 4.3, Eq. (8) or (9) was broken into two parts f 1ul<..4. and flui>A. In the first integral, the Riemann-Lebesgue lemma (Theorem 2.1.1) was used to show it to converge to zero as i.--. ±oo. In place of the Riemann-Lebesgue lemma, we may use the results in notes 1-3, and notes in Chapter II. For instance we have the theorem: If for some A > 0, u-1

I:

/(v) dv,

u >A

and

u-1

I

-A

u

f(v) dv,

u < -A,

are of bounded variation over [A, oo) and ( -oo, -A], respectively, then the statements (i) and (ii) of Theorem 4.3.1 hold. See the references quoted in Notes 1-3, and Notes in Chapter II. 4. In Theorem 4.2.1, (5) holds for F(x) e V( -:n:, :n:). Also (1) in Theorem 4.4.1 (inversion formula) remains true for F(x) e V(- oo, oo ). 5.

In Theorem 4.4.3, if F(x) eM, then (6) takes the form F(x)

=

![F(oo)

+ F( -oo)]

- :n:- 1

f-+oo 0

[Im(f(t)e-~ 1 "')/t]

dt.

Theorem 4.4.3 was used in the paper by Zolotarev [1]. 6. In the inversion formula (1) in Theorem 4.4.1 for a Fourier-Stieltjes transform, the limit of the symmetric integral limr4 oo (2:n:)- 1 f~r cannot be replaced by a general improper integral (2:n:)- 1 f~oo. In order that this be possible, it is necessary and sufficient that

I:

exist for some e

{[G(u, x) - G(u, 0)]/u} du

> 0, where G(u, x) =

F(u

+ x) -

F( -u

+ x)

(Kawata [8]).

NOTES

165

7. Toeplitz [1] showed Theorem 4.5.1. Silverman also showed the same result (Silverman [1]). Conditions (4)-(6) are also necessary for the convergence of u,. for any s,. that converges. This was shown by Kojima [1] and Schur [1]. The theorem was also given for the case in which a,.k = a,.k(t) depends on a real parameter. (See Dienes [1].)

8. Theorem 4.6.1 (Fejer's theorem) was proved by Fejer [1].

9. Vander Vaart has shown Theorem 4.9.1 in the multidimensional case, with a more general summability kernel (van der Vaart [1]). Theorem 5.4.2 in the next chapter is the generalization of Theorem 4.9.1, which is substantially van der Vaart's result in the one-dimensional case. For other determinations of the absolutely continuous component of the distribution function, see Loeve [1], p. 188.

10. We add here recently obtained significant results on the almost everywhere convergence of Fourier series. Carlson [1] has shown that the Fourier series of any function belonging to L 1 ( -:n;, :n;) is convergent almost everywhere. Adopting his method, R. A. Hunt [1] has proved that the Fourier series of any function belonging to L"( -:n;, :n;), 1 < p < oo, converges almost everywhere.

v General Convergence Theorems

5.1. Nature of the Problems The convergence and (C, 1)-summability for the Fourier series of a periodic function are reduced to the convergence of the integral

J:,. Kn(x- u)f(u) du as n

--+-

(1)

oo, where Kn(u) = :n-1 [sin(n

+ l)u/2 sin lu]

(2)

for the convergence and (3) for the (C, 1)-summability (Sections 4.1 and 4.6). For the corresponding problems for Fourier transforms we have the problem of convergence 166

5.2.

SOME GENERAL CONVERGENCE THEOREMS

of the integral S 1 (x, ,l) =

J:

I

K(x- u, .l)f(u) du,

167

(4)

as ,l -+ oo, where K(u, ,l) = n-1 (sin ,lufu)

(5)

or (6)

For A-summability for Fourier series and the corresponding problem for the Fourier transform, we also deal with (1) or (4) with Poisson kernels K. In order to unify or generalize these theories, it is natural to discuss the convergence of (4) with a general kernel K(u, .l), or the convergence of the integral

S 2 (x, .l) =

J:

K(x, u, .l)f(u) du

(7)

with a general K(x, u, ,l). Our aim now is to do this. There are other types of theory also as to convergence, such as convergence in norms in some spaces. Here we will be concerned with ordinary convergence, i.e., pointwise convergence, and aim at finding results which contain directly as special cases the theorems we have already obtained, and give some applications. The integral (7) is called the singular integral. The theorems which include the convergence of (4) or (7) as ;, -+ oo are called the general convergence theorems and are sometimes called the general summability theorems in reflection of the generalization of (C, 1)summability or A-summability. In connection with (4) or (7), we also consider the integral of the form T(x, ,l)

=

J:

K(x, u, ,l) dF(u),

the convergence problem of which has important applications in probability theory and mathematical statistics.

5.2. Some General Convergence Theorems I We shall begin with the following theorem due to Titchmarsh [3], which is a direct generalization of Theorem 4.6.1 or Theorem 4.6.2.

V.

168

GENERAL CONVERGENCE THEOREMS

Theorem 5.2.1. Suppose that for some a > 0 K(x, u, J.) = O(J.) = O(A" I X-u

11+")-1

for

I x - u I ~ ;.- 1 ,

(1)

for

Ix

(2)

-

u

I ~ ;.- 1 ,

for large ;. > 0 and lim A-+oo

If

f(x)f(1

Joo K(x, u, J.) du = -oo

1.

(3)

+ I x II+") E £1(- oo, oo)

(4)

and

J: I f(x + u)- f(x) I du = ash

o(h)

(5)

f(x).

(6)

-o, then lim A__..oo

Joo

f(u)K(x, u, J.) du

=

-oo

From condition (2), for each x, K(x, u, J.) is, as a function of u, an element of £1(- oo, oo ). Hence condition (3) makes sense. Also the integral in (6) does exist because for each x and each ;. > 0, K(x, u, J.) is bounded for every finite interval of u, which is seen from (1) and (2), and (2) and (4) assure the integrability of K(x, u, J.)f(u) over the neighborhoods of ±oo, as a function of u.

Proof.

In view of (3 ), it is sufficient to show that

!~~

J:

[f(u)- f(x)]K(x, u, J.) du = 0,

!~~

J:

or

c/J(u)K(x, x

+ u, J.) du =

0,

(7)

where

c/J(u)

=

c/Jz(u)

= f(x + u) -

f(u).

Write

J:

c/J(u)K(x, x

+ u, J.) du = J~ c/J(u)K(x, x + u, J.) du + J:oo c/J(u)K(x, x + u, J.) du =I+],

(8)

5.2.

SOME GENERAL CONVERGENCE THEOREMS

I

169

say. We are going to show that

I-+0

A.-+ oo.

as

(9)

Now we break I up into two parts:

I=

J

l/A

0

+ foo1/A = I1 + I2.

(10)

From (1)

II1 I ~

fmI lf>(u) II K(x, x + u, A.) I du ~ C A. Jm I lf>(u) I du 1

0

0

for some constant C 1 • Hence by (5),

A.-+ oo.

as

(11)

Using (2),

I 2 1~

Joo1/A

llf>(u)

II

K(x, x

+ u, A.) I du

~ C2 A.-"'J 1/A [llf>(u) lfu1+"'] du 00

(12) for some constant C 2 independent of A.. Write X(t)

=

Because of (5), there is an TJ that

J: I

lf>(u) I du.

> 0 for an arbitrarily given e > 0, such

X(u) < eu

for

0 ~ u ~ TJ·

(13)

We again split the integral I 2 in (10):

A.-<><

Joow

(llf>(u) lfu1+"') du =A.-"

say. Using integration by parts,

[by (12)]

J'lm

+A.-"

Joo = 'I

Iu

+ I22,

V.

170

GENERAL CONVERGENCE THEOREMS

Therefore, lim sup / 21 A-+oo

e(a

=

+

1)a- 1 •

(14)

Obviously, lim sup 122 = 0.

(15)

A-+co

Equations (11), (12), (14), and (15) show that I--+ 0. The proof that is carried out in the same way. This completes the proof.

J --+ 0

Corollary 5.2.1. Suppose the conditions on K(x, u, A.) in Theorem 5.2.1 are satisfied. If f(x)/(1 + I x 1 1 ) E V( -=, =) for some a> 0, 11 -

then (6) holds almost everywhere. Suppose more specifically that

K(x, u, A.)= A.K[A.(u- x)]

(16)

as in many cases it does; for instance, as in the case of the Fejer kernel. Then Theorem 5.2.1 turns out to be:

Theorem 5.2.2. Suppose that, for some constants C 1 and C 2 ,

I K(u) I ~ Cl I Ktu) I ~ (C2/I u 11+

11 )

I u I < 1, for I u I ~ 1, for

(17) (18)

for some a> 0. Ifj(x) satisfies (4) and (5), then

}~A. J~oof(u)K[A.(u- x)] du = f(x)

J:

K(u) du.

(19)

If

J~oo K(u) du "# 0, this is just a special case of the preceding theorem. Even if

J:

K(u) du = 0,

(19) still holds. This is easily seen if we take a look at the proof of Theorem 5.2.1.

Corollary 5.2.2. If K(u) satisfies (17), (18), andf(x) satisfies (4), then (19) holds almost everywhere.

5.2.

SOME GENERAL CONVERGENCE THEOREMS

I

171

We remark that if a = 0, then the theorems cease to be true, because the Dirichlet kernel satisfies (17) and (18) with a = 0, but for the conclusion (19) we need, for instance in Theorem 4.3.1 (Fourier's single integral theorem), some other conditions on f(x) [Section 4.3, Eq. (7) or condition (ii) in Theorem 4.3.1, are needed] (see Chapter 4, Note 2). It is easily seen that Theorem 5.2.2 implies the theorem for the Fejer kernel (Theorem 4.8.1). We see that (19) is equivalent to

In this form, the problem seems very simple. Actually the direct application of the dominated convergence theorem gives us the result:

If f(x) is bounded over (-oo, oo) and is continuous at x and K(x)

e Ll(- oo, oo ), then (20) holds. However, the condition that f(x) is bounded over (-oo, oo) is very much restricted and we are concerned with the question of when (20) holds, mostly in the case in which some sort of integrability condition on f(x) is supposed. We shall show another theorem.

Theorem 5.2.3. Suppose that f(u)/(1

+ I u I) ELl( -oo, oo),

(21)

and K(u) ELl( -oo, oo).

(i)

(22)

If

I uK(u) I ~

(23)

C

for some constant C and f(u) is continuous at x, then (19) holds. (1/q) = 1, and (ii) If p > 1, q > 1, (1/p)

+

J:

J: I

u

oo,

(24)

[lf(u)

IP/(1 +I u I)] du < oo,

(25)

lq-t I K(u) lq du <

and f(u) is continuous at x, then (19) holds.

172

V.

GENERAL CONVERGENCE THEOREMS

Statement (i) is thought of as the case where p = 1, q = oo in (ii). Statement (i) was given by Bochner [5] and (ii) by Bochner and Izumi [1]. Proof. We first prove (ii). We may, without loss of generality, suppose x = 0. We are going to show that

lim A. Joo J(u)K(A.u) du = J(O) Joo K(u) du.

A-+oo

0

(26)

0

(For the part foo, the same argument as that which will be used in proving (26) is applicable.) Since, by the Holder inequality, for TJ > 0,

IA f~ J(u)K(A.u) du I~A[ f~
J:

= [

< oo,

(27)

the integral on the left-hand side of (26) exists. Write 4>(u) = f(u) - f(O). Then we have to show that lim Joo 4>(u)K(A.u) du = 0. A-+oo

From (27), A.

J;' f(u)K(A.u) du-+ 0

A. Joo K(A.u) du =

(28)

0

as A.-+ oo, and

Joo K(v) dv-+ 0 A'l

'I

for each fixed TJ > 0. Hence J;' 4>(u)K(A.u) du-+ 0 as A.-+ oo. Now, choose TJ > 0 so small that 14>(u) I < e for 0 ~ u ~ TJ, for an arbitrarily given e > 0. Then

IA. J: 4>(u)K(A.u) du I~ eA. J: I K(A.u) I du ~ e J~ I K(v) I dv. This completes the proof of (28). We now turn to the proof of (i). Using (23), we see that

I J: A

4>(u)K(A.u) du

I~ cl J: [14>(u)

lfu] du,

which is made as small as desired if A is taken large enough. For A.

J:

4>(u)K(A.u) du,

(29)

5.2.

SOME GENERAL CONVERGENCE THEOREMS

I

173

we have (29) as before. It remains to show that lim A .1.-+oo

r 'I

~(u)K(Au) du = 0.

(30)

For any e > 0, there is a bounded function g(u) such that

J: I [~(u)fu]Let I g(u)

g(u) I du < e.

I ~ C 2 • We then see that

IA J: ~(u)K(Au) du I ~ A J: I [~(u)/u]- g(u) I u I K(Au) I du

J: ~ cl J: [~(u)/u]+ J: I

+ A I g(u) I u I K(Au) I du 1

g(u) 1 du

g(u)K(Au) I du

AA

~ C 1 e + C 2A

Ju. I K(v) I dv. A'l

Hence

which is sufficient to conclude (30). This finishes the proof. This theorem also implies the (C, 1)-summability theorem for Fourier transforms or Fourier series. But condition (22) on K is not satisfied by the Dirichlet kernel. The following results are related to Theorems 5.2.1 and 5.2.3.

Theorem 5.2.4. If /(u)/(1 +I u IP) then

where

E

£1(-oo, oo),

p > 1,

V.

174

GENERAL CONVERGENCE THEOREMS

provided that

J:

lf(x

+ u) -f(x) I du =

o(h)

as h -o.

Theorem 5.2.5. Ifj(u) exp -cu2 E Ll(-oo, oo) for some c > 0, then

~~[(2n)1t2<5]-1 J:J(u) exp[-(u- x) 2f2<5 2] du = f(x)

(32)

provided that

J:

lf(x

+ u)- f(x) I du =

as h---+ 0.

o(h)

Theorem 5.2.4 is a direct consequence of Theorem 5.2.1. Theorem 5.2.5 is also a consequence of Theorem 5.2.1 or Theorem 5.2.3 if f(u)/(1 + I u 1)"'+1 ELl( -oo, oo) for some a ~ 0. However, we can see that, under the conditions in Theorem 5.2.5, (32) is true if we reproduce the proof of Theorem 5.2.3 (i). Here we have to use

(Zn)-1' 2 J : exp(-u 2/2) du = 1.

(33)

The integral on the left-hand side of (32) is called the Weierstrass singular integral.

5.3. Some General Convergence Theorems II In this section, we shall give a general convergence theorem which implies the Fourier single integral theorem.

Theorem 5.3.1. Suppose that for a fixed x, the improper integral

J:~oo K(x, u, A) du exists for A > 0. Suppose further that

I K(x, u, A)

I.~

c1

I X - u l-1

for

Ix - u I >

'TJ

> 0,

(1)

where 'TJ is any positive number and C 1 = C1(TJ) is a constant independent of A> 0, and -+oo lim f K(x, u, A) du

A-+co

lim A-+oo

:z

f

-+oo

fl

K(x, u, A) du

= =

lim

-+oo

A

0,

f:z

K(x, u, A) du

= t,

(2)

-+-oo

(3)

5.3.

SOME GENERAL CONVERGENCE THEOREMS

II

175

for each {.J > x,

J'"

K(x, u, .f.) du = 0,

(4)

Jfl (u- x)K(x, u, .f.) du = 0,

(5)

lim

A-+oo

-+-oo

for each a < x, lim A-+oo

11

for every pair a, {.J such that x < a < {.J or a < {.J < x, and (6) for every a, {.J, C 2 being a constant independent of a, {.J, and .f.. Suppose thatf(u)/(1 + I u I) E £1( -oo, oo) and thatf(u) is of bounded variation in a neighborhood of x. Then

!~~ J~oo K(x, u, .f.)J(u) du

= l[f(x

+ 0) + f(x- 0)].

(7)

We first note that the integral on the left-hand side of (7) exists (as a Lebesgue integral). In fact, by (1 ),

Proof.

f

lui>A.

I K(x, u, .f.)J(u) I du ~

cl

f

IUI>A.

(IJ(u)

l/1 u-x I> du,

A> I X I.

which converges to zero as A - oo. We shall prove that

1~[ J~ K(x, u, .f.)J(u) du- f(x + 0) J:oo K(x, u, .f.) du]

= 0.

(8)

In the same way we will be able to show that

1~[

J:oo K(x, u, .f.)J(u) du- f(x- 0) f~--oo K(x, u, .f.) du] =

0. (9)

Proof of (8) and (9) will give us the required proof of the theorem. Now we may suppose without loss of generality that J(u) is nondecreasing in a right neighborhood of x. For any given e > 0, choose TJ so that

lf(u)- f(x

+ 0) I <

e

for

x ~u ~x

+ TJ·

(10)

V.

176

GENERAL CONVERGENCE THEOREMS

The expression in brackets on the left-hand side of (8) is written as I=

=

I:..., K(x, u, A)[j(u)- f(x + 0)] du +

I:

Il

+ I2-Ia.

K(x, u, A)j(u) du- f(x

+ 0)

I:

K(x, u, A) du (11)

say. From (3)

II 3 I - 0

as

(12)

A - oo.

Using the second mean value theorem, we have I

I 1 1 ~ [J(x

for some C, 0

Z-Hl + 1J)- f(x + 0)] Iz+C K(x, u, A) du

< C< "'· By (6) and (10),

I I1 I ~ C2e.

(13)

In order to handle I 2 , we split it into two parts: I2

=

z+.A K(x, u, A)f(u) du + Ioo IZ-Hl

~.A

say, where A is taken so large that A

K(x, u, A)f(u) du

>

=

I 21

+ I 22 ,

'f/, and

Ioo [I f(u) If(u- x)] du < e.

(14)

~.A

Then

I I22l ~ Ioo [I f(u) If(u- x)](u- x) I K(x, u, A) I du ~.A

[by (1)] and

I I221

~ cl

Iooz+.A [I f(u) If(u- x)] dx < Cle. step-function g(u) in [x + 1J, x +A],

Next choose a number of discontinuities, in such a way that

I::

I [f(u)f(u- x)]- g(u) I du <

(15)

with a finite

e

(16)

for fixed x, 1J, A. We then have I 21 =

z+.A (u- x)K(x, u, A)[[J(u)f(u- x)]- g(u)] du Iz-hl +

I::

(u- x)K(x, u, A)g(u) du.

(17)

5.3.

SOME GENERAL CONVERGENCE THEOREMS

II

177

Equations (1) and (16) give us that the first integral is, in absolute value, not greater than

c1

f


I [J(u)f(u- x)]- g(u) I du <

C1e.

The second integral on the right-hand side of (17) converges to zero as A. ----. oo because it is a finite linear combination of

f

flt

(u- x)K(x, u, A.) du

tzt

(i = 1, ... , m for some m)

and then converges to zero as A.----. oo because of (5). Hence for A. A.0 being a suitably large number,

~

A.0 ,

IJ:: (u- x)K(x, u, A.) du I< C e 3

for some constant C 3 • Altogether we see that (18)

From (15) and (18),

for A. ~ A. 0 , which shows, with (12) and (13), that (8) holds. This completes the proof.

Theorem 5.3.2. Suppose that

J::-':. K(x, u, A.) du

exists and

I K(x, u, A.) I ~ c1 I X - u l-1

(19)

for every pair x, u, C1 being a constant independent of x, u, A., and all other conditions (2)-(6) of Theorem 5.3.1 are satisfied. Suppose that f(u)f(1 + I u I) E Ll(-oo, oo) and that

f

lui<"

I [J(u + x)- f(x)]fu I du <

for some 15 > 0. Then

!~

J:

K(x, u, A.)f(u) du

= f(x).

oo,

(20)

(21)

The condition (19) is stronger than (1). The proof remains the same as in the proof of Theorem 5.3.1, replacing

V.

178

f(x

GENERAL CONVERGENCE THEOREMS

+ 0) by f(x), except in the treatment of 1

1•

I 11 I

[with f(x) in place of f(x ;;;::;C1

J

Z+r/

z

+ 0)]

[lf(u)-f(x)lf(u-x)]dx,

which is made as small as we wish, (20) being used. The proof is now complete. If K(x, u, A.)= A.K(A.(u- x)) as in Section 5.2, Eq. (16), Theorem 5.3.1, with slight specialization, reduces to a considerably simpler form.

Theorem 5.3.3. Suppose that K(u) is an even function such that J:"" K(u) du exists, and

IJ: K(u) du I~ c1

for every

fJ > 0,

(22)

I uK(u) I ; ;: ; c2.

(23)

where c1 and c2 are constants. Ifj(u)/(1 + I u I) E £1( -oo, oo) andf(u) is of bounded variation in a neighborhood of x, then

~i_:, A. J~oof(u)K(A.(u- x)) du = i[f(x + 0) + f(x- 0)]

J::

K(u) du. (24)

We remark that: If the condition that/( u) is of bounded variation in a neighborhood of x is replaced by the finiteness of

f

lul
lf(x+u)-f(x)lfluldu

we have (24) withf(x) in place of i[J(x + 0) hand side of (24 ).

for some

«5

> 0,

+ f(x- 0)] on the right-

Obviously the Dirichlet kernel (sin u)fu satisfies the conditions on K(u). The Fourier single integral theorem is a particular case of Theorem 5.3.3.

5.4. General Convergence Theorems for the Stieltjes Integral We shall study the convergence of

A.

J: K[A.(u- x)] dF(u)

(1)

when A.--+- oo and F(u) EM. If F(u) has only the point spectrum

5.4.

CONVERGENCE THEOREMS FOR THE STIELTJES INTEGRAL

179

S(F) = P(F) = {x11 , n = 1, 2, ... }, then (1) becomes 00

A L K[A(Xn- x)]Pn•

(2)

n-1

+

where Pn = F(xn 0)- F(xn- 0). The X11 of course are not always arranged in order of magnitude. If X 11 is once arranged in some way, we fix it and consider the series (2). X 11 may have finite limiting points. Still it is true that for a sufficiently large N 1 = N 1 (e 1 ), (3)

Now suppose that (4)

lim uK(u) = 0, u-+±oo

which of course implies

I uK(u)

I~

c,

-00

< u < oo,

for some constant C. Now let x be a continuity point of F(u). Then x =I= from (4), AK[A(xn- x)]--+ 0 as A--+ oo for each n. Hence for any fixed N,

) AK[A(Xn- x)]pn--+ 0

n~

as

(5)

X 11

A--+ oo.

,

for each nand

(6)

Now suppose further that 00

L (Pn/1 Xn n-1

X I) <

00 •

(7)

Needless to say, we may equivalently assume that

L

lx,.-xl
(Pn/1 Xn - X I) <

00

for any fixed 15 > 0. Then

I An>N L K[A(Xn- x)]Pn I ~ n~ ) AI X

11 -

~

X I K[A(Xn- x)](Pnfl X"- X I)

C ) fi

[by (5) and (7)] if N has been chosen large enough. We have shown the lemma:

180

V.

GENERAL CONVERGENCE THEOREMS

Lemma 5.4.1. Suppose that F(u) EM has only the point spectrum {xn} [F(u) is purely discontinuous]. If K(u) is a function over (-oo, oo) satisfying (4) and at a continuity point of F(u),

L
X I) < oo,

(8)

where Pn is the amount of jump of F(u) at Xn, then lim A. A-+oo

Joo

K[A.(u- x)] dF(u)

= 0.

(9)

-oo

Now write for F(u) EM, (10)

where Fa, Fd, and Fa are, respectively, the absolutely continuous, purely discontinuous, and continuous singular components. If F,(u) = 0, then the combination of Lemma 5.4.1 and Theorem 5.2.3 gives us the following theorem.

Theorem 5.4.1. Suppose that F(u) EM has no continuous singular component. Suppose that K(u) is a function satisfying lim uK(u) = 0

(11)

u-+±oo and K(u)

E

V(-oo, oo).

(12)

Let x be a point at which Fa'(x) exists and (7) holds, where Pn is the saltus of Fd(u) at its discontinuity Xn· Then

!~ J: K[A.(u- x)] dF(u) = Fa'(x) J~oo K(u) du.

(13)

This was given by Murthy [1 ]. If we consider the analog of ·Theorem 5.2.2, we have the following theorem:

Theorem 5.4.2. Suppose that

I K(u)

I~

I K(u) I ~

C1 C 2/l u

11+"'

for

I u I ~ 1,

(14)

for

I u I > 1,

(15)

5.5.

181

WIENER'S FORMULA

for some a > 0 and for some constants C 1 and C 2 • If F(u) eM and at x, F,.'(x) and F 8 '(x) exist [F8 '(x) = 0], then (13) holds. It is to be noted that condition (7) is not required. The conclusion of this theorem may be said to be that (13) holds for almost every x. Proof.

The proof is similar to the one for Theorem 4.9.1.

A I~oo K[1(u- x)] dF(u)- F,.'(x) I~oo K(u) du

=I: 1K[1(u- x)] d[F(u)- F,.'(x)u] =

I~oo 1K(1u) d[F(u

+ x)- F,.'(x)(u + x)]

=

I

= /1 + I.,

lui <1/.1.

+ I1Uiil:1/A

say. Because of (12),

I l1 I ~ ;.c1 I lul
C 1e,

where e > 0 is arbitrary and;. is large enough, as in Section 4.9, Eq. (12).

I 18 I ~ (Cz/A") •fIUiil:l/.1.

(1/ul+") I d[F(u

+ x)- F,.'(x)(u + x)] I,

which is seen to be made as small as we want, just as in the handling of / 2 in the proof of Theorem 4.9.1 [Section 4.9, Eq. (11)]. Theorem 5.4.2 is a generalization of Theorem 4.9.1.

5.5. Wiener's Formula Again consider the singular integral A I~00 j(u)K[1(u- x)] du or, equivalently,

I:J[x + (~/1)]K(~) d~.

(1)

(2)

Now differently from before, we let;.---+ 0. Ifj(x)---+ c for x---+ oo, then it is expected that (2) converges to

C

I: K(~)d~

182

V.

GENERAL CONVERGENCE THEOREMS

and actually it does under some conditions on f and K. However, if f(x) --+- c does not hold, still the limit exists and is an average ofj(x) at oo under suitable conditions. We are letting A--+- 0+, so it is quite natural to talk about, for instance,

I~ f(u/A)K(u) du

(3)

instead of (1) or (2). The following theorem is due to Bochner [5] and Bochner and Hardy [1].

Theorem 5.5.1. Suppose that K(x) is absolutely continuous over every finite interval in (0, oo) and satisfies for

x

> 1

(4)

for some constant C 1 • Suppose thatf(x) is a function such that lim T- 1 IT f(x) dx

T-+oo

= 1(/)

(5)

0

exists and (6)

for some constant C 2 • Then lim

.t-+o+

Ioo f(x/A)K(x) dx = 1(/) Ioo K(x) dx. o

(7)

o

Equation (7) means that the integral on the left of (7) exists for A > 0 and its limit when A --+- 0+ also exists and is equal to the right of (7). We begin with the proof of the existence of the integral on the left. It would be convenient for this to state the following lemma.

Lemma 5.5.1. If (6) holds, then

I:

Proof.

[1/(x) l/x 2 ] dx

Write

(J)(x) =

[I f(x) l/x2 ] dx = [(J)(x)fx 2 ]

~

(8)

I: I

Then using integration by parts,

I:

~ 2C2 /A.

/(u) I du.

1: + 2 I: [I (J)(x) l/x

[(J)(B)/Bz]

+ 2C

2

I:

x-z dx

3]

dx

5.5.

183

WIENER'S FORMULA

since, by (6), (l)(x)

I: [I

~

CiJC,

f(x) l/x2] dx ~ (C2/B)

+ (2C2/A)- (2C2/B)

~ 2C2fA.

This shows (8).

Proof of Theorem 5.5.1. Let cf>(x) = f(x)- .L(f). Then

.L(cf>) = 0 and T- 1

I: I

cf>(x) I dx

~ c2 + I.L{f) I =Ca.

(9)

(10)

say. What we have to show is that lim A-+o+

Iooo cf>(xjA)K(x) dx = 0.

(11)

The existence of the integral in this equation is easy to show from (4) and the lemma, which gives us (12)

for any A > 0. Now we are going to show (11). For an arbitrarily given e > 0, choose A > 1 so large that (13)

We break up the integral in (11):

I~ cf>(xjA)K(x) dx =I:+ I:= 11 + 12 , say. Then

I I2l

=

I;. I:,A cf>(u)K().u) du I

~ ).-1

Ioo

AlA

[I cf>(u) l/u 2](Au) 21K(Au) I du,

which is, by (4), (12), and (13),

I /2l ~

(C1/A)

Ioo

AIJ.

[I cf>(u) l/u2] du ~ 2C1Ca/A ~e.

V.

184

GENERAL CONVERGENCE THEOREMS

Hence (14) Now write


=

f 1/J(uj).) du = x(Ajx) JZ/A t/J(u) du, x

0

0

from which we obtain
~

() > 0

(15) Also from (10),

I
Now choose () in such a way that ()

C3 ()

J: I

~

C;;c.

(16)

< A and

K'(x) I dx

~

(17)

E.

Then choose 'YJ to be the smaller of AE/C1 and E[f: xI K'(x) I dx]- 1 • We now consider / 1 • Write

11 =

J: 1/J(xj).)K(x) dx

=
J:


J:


=
say. By (4) and (15),

I
~E.

Using (16) and (17),

I lu I ~ c3

J:

X

I K'(x) I dx ~ Ca()

Using (15),

Hence, altogether, This with (14) proves (11).

J: I

K'(x) I dx

~

E.

5.5.

185

WIENER'S FORMULA

Theorem 5.5.1 may also be stated in the following form.

Theorem 5.5.2. If K(x) is an even function defined over (- oo, oo) satisfying the conditions in Theorem 5.5.1 for x > 0, and ifj(x) is such that (2T)- 1

(T lf(x) I dx ~ Ca,

T> 0,

(18)

on (- oo, oo) for some constant C 3 and

~i~ (2T)- 1

(T f(x) dx = .L{f)

(19)

exists, then

}~~ J~oof(xfJ.)K(x) dx = .L{f) J~oo K(x) dx.

(20)

It is easy to show this if we split the integral on the left-hand side of (20) into J;;" + f-.oo and apply Theorem 5.5.1 in each integral. A special case of this theorem is the formula (21)

with the conditions (18) and (19). The formula (20) is called the Wiener formula. Wiener [3] first showed the particular case (21). Without any change in the proof of Theorem 5.5.1 we can state the following:

Theorem 5.5.3. Let K(x) satisfy all the conditions of K(x) in Theorem 5.5.1. Let F(x) be a function of bounded variation over every finite interval in [0, oo) such that x-•

f: I

dF(x)

I ~ cz

for

1

~

x

< oo

(22)

and

F(x)fx-+ m

as

x-+ oo.

(23)

Then lim ). Joo K(J.x) dF(x) A~+

0

= m Joo K(x) dx. o

(24)

V.

186

GENERAL CONVERGENCE THEOREMS

A particular case of this theorem is the following :

Corollary 5.5.1. If {sn} is a sequence such that as

n-.oo

(25)

and (26)

for some constant C, then with the same conditions on K(x) as in Theorem 5.5.1, we have lim h h-+o+

f snK(hn)

n-1

=

s Joo K(x) dx. o

(27)

5.6. Applications of General Convergence Theorems to the Estimates of a Distribution Function Now we should like to give an application of general convergence theorems to the problem of estimating the density function of a distribution. Let F(x) be a distribution function and let (1)

be a sample of size n of this distribution function. In other words, (1) is a set of independent random variables with identical distribution functions F(x). Knowing (1) for some co = co 0 , we want to estimate F(x). Suppose we face the problem of estimating some parameter () which is involved in the unknown distribution function F(x) or which is some quantity that has something to do with F(x). We usually construct some function of (1) such as

which is generally called a statistic, or more specifically an estimate as a statistic for the purpose of estimating 0. If, by an observation or an experiment, the realized values of the sample (1) are obtained, or equivalently,

5.6.

187

ESTIMATES OF A DISTRIBUTION FUNCTION

the values of ( 1) for some w value

=

w0

are obtained, then the corresponding

is taken to be an estimate of 0. In order to construct the estimate Tn, some conditions are usually required. If

W Tn(Xt•· . . , Xn)

= 0,

(2)

then Tn is called an unbiased estimate. If, in place of (2), we have lim W Tn(Xl, . .. , Xn)

= 0,

(3)

n->-oo

then Tn is called an asymptotically unbiased estimate. Furthermore, if lim var Tn(X1 , ••• , Xn)

= 0,

(4)

n->-oo

then it is called a consistent estimate. There are other ideas concerning estimation but we shall not discuss them here because they are beyond the scope of this book. Now our problem is to estimate F(x) itself or the probability density, p(x) being supposed to exist. We thus take 0 = F(x) or p(x) for each x. If n is very large and a very large number of sample values is available, usually we draw the so-called histogram from which we guess what the probability density (it is often called a relative frequency curve in statistics) looks like. Suppose n is not sufficiently large to permit the construction of a histogram. In this case we proceed in the following way. We are given (1 ). For each w, we rearrange (1) in increasing order of magnitude as (5) Then XU,, j = 1, ... , n is again a random variable (that is, a measurable function). Now define a random variable Fn(x, w), depending on a real variable x (namely, for each x, Fn(x, w) is a random variable), by F.(x, w)

~ { ~fn

if x ;:;;; XAU(w), if Xnk (w) < x ;:;;; if Xknl(w) <X.

X~k+ll(w),

1 ;:;;; k ;:;;; n- 1, (6)

In other words, nFn(x, w) is the number of values XAkl( w) less than x. This random variable depending on x is called the empirical distribution

V.

188

GENERAL CONVERGENCE THEOREMS

function, based on a sample of size n from F(x). Now F(x) is supposed to be left-continuous. For each x, Fn(x, w) provides an estimate of F(x) [Fn(x, w) is a statistic]. This estimate is unbiased for each x, namely (see Fig. 5.1), W Fn(x, w) = F(x).

(7)

r--

1

4/nl I

I

4/nl

iI

4/nl

)(.(3) n

FIGURE

5.1.

As a matter of fact, we may easily find the distribution function of the random variable Fn(x, w) for each x. For this, we introduce the random variable if X~:(w) < x, Y~:(x, w) = { if X~:(w) ~ x.

b

Then it is easy to see that n

Fn(x, w) = n- 1 L Yk(x, w ).

(8)

k-l

Now

and

Y~:(x,

L~- 1 Y~:(x,

P[Y~:(x,

w) = 0] =

P[X~:(w) ~

P[Y~:(x,

w) = 1] =

P[X~:(w)

x] = 1 - F(x)

< x] = F(x)

w) is a sequence of independent variables. Therefore, w) depends on the binomial distribution function

P[nFn(x, w)

=

k]

= ( ~ )Fk(x)[(1- F(x)]n-k,

0 ~k

< n.

Hence

n W Fn(x, w) = nF(x), which shows (2) with Tn(x 1,. •• , Xn) = Fn(x, w). Furthermore

n2 var Fn(x, w) = nF(x)[1 - F(x)],

(9)

or var F,(x, w)

=

n- 1F(x)[1 - F(x)].

(10)

Now we are interested in how to make an estimate of the probability density when F(x) is absolutely continuous. For this purpose, the following estimate is constructed:

Pn,A(x, w) =A J~oo K[J.(x- u)] dF,(u, w).

(11)

For each w, F11 (u, w) is a nondecreasing function of u. This was proposed by Parzen [1, 2]. Suppose that K(x) is a Borel-measurable bounded function. Then, denoting the probability measure in our probability space Q by P,

From Lemma 3.10.1, or more exactly, Section 3.10, Eq. (4), the righthand side is

A

J: K[J.(x- u)] d[ LF (u, w) dP] =A J~oo K[J.(x-u)] d8"F (u, w) 11

11

= A J~oo K[J.(x- u)] dF(u) (12) by (9). If F(x) is absolutely continuous with probability density function p(x), then (11) becomes

A J~oo K[J.(x- u)]p(u) du.

(13)

Hence directly from Theorem 5.2.3, we may claim that (13) converges to p(x) at its continuity point x if the same conditions as in Theorem 5.2.3 are imposed on K(x) here and K(x) dx = 1 is assumed in addition. In Parzen's theory of estimation, ;. is taken to be a number ;., depending on n, the sample size of (1 ), and in place of (11) we consider the estimate

f::'oo

P 11 (X, w) = A 11 J~oo K[A 11 (x-u)] dF11 (U, w).

(14)

We now remark that P 11 (x, w) can also be written as

Pn(x, w) = (J.nfn)

n

L K[A j=l

11

(X- Xj)].

(15)

V.

190

GENERAL CONVERGENCE THEOREMS

For (14), it is obvious that (") l:n K[J.,.(x- X,.')].

P,.(x, w) = (J.,.fn)

(16)

J-1

But for each w, (X~11 , ••• , X~21 , ••• , X~" 1 ) is a rearrangement of (X~> X 2 , ... , X,.). Of course this rearrangement may differ from another in accord with different values of w. However, L K[J.,.(x - XU1)] does not change by rearrangement of {XU1} for each w. Hence the right sides of (15) and (16) are the same. Then the statement following (13) may be stated as:

Theorem 5.6.1. Let F,.(x, w) defined by (6) be the empirical distribution function based on a sample of size n from the distribution F(x). Suppose F(x) to be absolutely continuous and have the density function p(x).

Suppose that K(x) is a Borel function of Ll(- oo, oo ),

J~oo F(x) dx = 1, and

I xK(x)

I~

c

(17) (18)

for some constant C. Then at the continuity point x of p(x), the estimate (14) or (15) is an asymptotically unbiased estimate of p(x), where;.,. is a sequence such that ).,. - oo. If we apply Theorem 5.4.2, we have the following:

Theorem 5.6.2. Let F,.(x, w) be the empirical distribution function based on a sample of size n from F(x). Let Fa(x) be the absolutely continuous component of F(x). Suppose that K(x) is a Borel function belonging to Ll(-oo, oo) with condition (17) and satisfying

I xI ~ 1, for I xI > 1, for

(19) (20)

for some a > 0 and for some constants C 1 and C 2 • Then for almost every x, (14) or (15) is an asymptotically unbiased estimate of Fa'(x). The analog of Theorem 5.4.1, which is due to Murthy [1], is also valid.

5.6.

191

ESTIMATES OF A DISTRIBUTION FUNCTION

Next we consider the variance of Pn(x, w ). Using (15), var Pn(x, w) = (A/·/n 2 ) var

n

L K[An(x- X;)]. j-1

Since the K[An(x- X;)], j = 1, 2, ... , n are independent random variables, var Pn(x, w) = (An 2 /n 2 )

n

L var K[An(x- X;)].

i=I

Now var K[An(x- Xi)]= W K 2 [An(x- X;)]- {W K[An(x- X;)] }I =

J:

K 2 [An(x- u)] dF(u)

- {J:

K[An(x- u)] dF(u)r

Hence var Pn(x, w) = (An 2 /n)

Joo -oo

-

(1/n) {An

K 2 [An(x- u)] dF(u)

I:

K[An(x- u)] dF(u)r

If F(x) is absolutely conti~uous with density function p(x), (n/A 11 ) var Pn(x, w) =An J~oo K 2 [An(X- u)]p(x) dx

- (1/An) {An J~oo K[An(x- u)]p(x)

dxr (21)

Suppose that the conditions on K(x) in Theorem 5.6.1 hold and that An-+ oo, then from Theorem 5.2.3 the last member converges to zero at the continuity point of p(x). Theorem 5.2.3 applies also to the first integral of the right-hand side of (21). We then get the following:

Theorem 5.6.3. Under the same conditions as those in Theorem 5.6.1,

!~ (n/An) var Pn(x, w)

= p(x)

J:

K 2 (x)

dx.

Note that

because of (18). This theorem implies the following theorem.

(22)

V.

192

GENERAL CONVERGENCE THEOREMS

Theorem 5.6.4. Under the conditions in Theorem 5.6.1, P,(x, w) is a consistent estimate of the probability density at its continuity point, if ().nfn)--+ 0 and A11 --+ oo. This is obvious from (22). NOTES 1.

Let
IT

[1 -

(I u 1/T)]"'
for

s

a>

-1,

-T

then

is said to be (C, a)-summable to s as T--.. oo and we write

(C, a)

I

-+00

-+-oo
Using this definition, we may show that if f(x) transform, then

I

E

=

s.

Ll(- oo, oo) and /(t) is its Fourier

-+00

(2n)-1ii(C, a)

eizlf(t) dt = f(x) +-oo

almost everywhere when a > 0. Actually, we may show this using Theorem 5.2.1, since (2n) -1/1

I:T [1 -

(I t 1/).)]"'eizlj(t) dt

is seen to be (2n)- 1 I~oo K(x, u, T)/(u) du,

where K(x, u, T) = n:- 1 I~ [1 - (t/T)]"' cos t(x - u) dt,

which is shown to satisfy Section 5.2, Eqs. (1) and (2). (See Titchmarsh [2].) 2. Wiener [3] gave formula (21) of Section 5.5. Also see Wiener [5] and Bochner and Hardy [1]. 3.

Jacob [1] has shown that, if for some a (0

~a

< 1),

for

lim (1/T1 -'") T-+oo

IT 0

f(t) dt

=

T > 0, A

(1)

(2)

193

NOTES

exists, then lim (e1 -"'/y.,n)

Joo f(t)(sin etfet)

where y.,

=

2

dt = A,

(3)

0

a-+O+

2"'- 1 (1 - a)/[F(2

+ a) cos(na/2)].

Furthermore, if /(t) ~ -K, K being a constant, then (3) implies (2). See also Littauer [1] and Wiener [5, 6]. For the generalization of Jacob's result in the case where a general kernel comes in, see Takahashi [1].

4. Izumi [3] replaced condition (4) on K(x) in Theorem 5.5.1 by the condition that xK(x) is of bounded variation over (0, oo). He also gave some generalizations of this theorem and other sufficient conditions for the conclusion of Theorem 5.5.1.

5. The case p = q = 2 in Theorem 5.2.3 (ii) was given by Izumi [2]. = sin 2 xfx• holds, and if s,. is given as :E~ ak, then :E;"' a,. is called Riemann-summable to s or (R, 2)-summable to s. The corollary in Section 5.5 says that a series that is (C, 1 )-summable to s is (R, 2)-summable to the same s. For this, see Hardy [2], Chapter XII.

6. If (27) in Section 5.5, with K(x)

7.

As a partial generalization of Theorem 5.3.3, we may show: Suppose that/(x) has a continuous rth derivative in a neighborhood of x, r :;;:: 0, and f(x) e V( -oo, oo). Suppose that x 2 K(x) e V( -oo, oo), K(x) e £1( b), > 0, and K(x) = O(x- 1 ) for large J x J. Then, as). ~ oo,

-o,

). f~oo f(u)K[).(u -

x)] du

=

f(x)

o

f~oo K(u) du + [/'(x)/1 !).] f~oo 11K(u) du

+ · · · + [/ 1' 1(x)fr!)!]

J:

11'K(11) du

+ o().-').

This is a completion of a result of Aonuma [1].

8.

Concerning the empirical distribution functions m Section 5.6, see, for example, Chung [1], pp. 123-124, where the proof of the following basic fact is found. F,.(x, w)

~

F(x)

with probability one (for almost all w ).

9.

K(x) in Theorems 5.6.1 and 5.6.2 is called a window in time series analysis. Other applications of the generd convergence theorems can be found, for example, in the work of Parzen [2] or in his original papers listed in his reference [2].

VI L2 - Theory of Fourier Series and Fourier Transforms

6.1. Fourier Series in an Inner Product Space We are now in a position to study the Fourier series of a function belonging to L 2 ( -n, n). However, so far as the L 2-theory is concerned, it is a special case of a more general theory of orthogonal Fourier series. A sequence {1/>n(x), n = 1, 2, ... } offunctions 1/>n(x) e £2(a, b) is called orthogonal if

m =f=. n.

(1)

The interval (a, b) may be a finite or an infinite interval. Moreover, if

m=n,

{1/>n(x)} is called a normal orthogonal sequence, or an orthonormal sequence. 194

6.1.

195

FOURIER SERIES IN AN INNER PRODUCT SPACE

Letf(x) E L 2 (a, b) and let sequence on (a, b). Define Cn =

{~n(x),

n

= 1, 2, ... } be a fixed orthonormal

J:f(x)~n(x) dx,

(2)

which are called the Fourier coefficients of f(x) with respect to Form a formal series

{~n(x) }.

(3)

This is called the Fourier series of f(x) with respect to thogonal series of f(x). We usually write

{~n(x)}

or an or-

00

f(x)

"'L Cn~n(x) n-1

(4)

as in the theory of ordinary Fourier series. But the theory of the series given in (4) is again a particular case of the more general theory of Fourier series in an inner product space and, as a matter of fact, most of the theorems in the L 2-theory of Fourier series hold for the general Fourier series. Therefore, we shall talk about the Fourier series in an inner product space. Let X be an inner product space, that is, be a linear space with the inner product (x 1 , x 2 ) for any pair of elements x 1 , x 2 of X. The inner product (x 1 , x 2 ) is a complex number with the properties given in Section 1.16 (i)-(v). If we write II x 11 2 = (x, x), x EX, II x II becomes a norm [Section 1.16]. Hence when we speak of an inner product space, we shall always mean the normed inner product space. We know from Section 1.16 that the Schwarz inequality

always holds for any x 1 , x 2 E X. The equality sign is valid if and only if x 1 and x 2 are linearly dependent, that is, x 1 lx 2 = 0 for some complex number A., where 0 is the null element. Throughout this section, X is a normed inner product space. If (x 1 , x 2 ) = 0, x 1 , x 2 E X, then x 1 and x 2 are called orthogonal. When

+

the element lx2 is called the projection of x 1 on x 2 • In fact,

196

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

A set S of elements of X is called orthonormal if, for any x, y (x,y)

=

~:

{

E

S,

-:f=. y, x=y.

X

Now let (5) be a finite or a countably infinite sequence of elements of X, which forms an orthonormal set. Let x be any element of X. (x, Cn)i;, is the projection of x on Cn since II /; 11 II = 1. This is also called the Fourier coefficient of x with respect to (5). The formal series 00

L (x,

X""

Cn)C,.

(6)

n.-1

is called the Fourier series of x with respect to {/;11 } .

Theorem 6.1.1. Let /; 1 ,

/; 2 , •••

be an orthonormal sequence and let x

be any element of X. Then (7) for any set a 1 , ••• , a11 of complex numbers, where n is an arbitrary positive integer.

Proof.

n

=

(x, x)-

n

L ak(l;k> x)- L iik(x,

k-1

n

+ k=1 L (Ck> x)(x,

Ct) -

k=1 n

L (Ck> x)(x,

+ k=1 L

+ k-1 L I ak 1

2

Ct)

k=1

II

= (x, x)

n

Ck)

II

I

ak- (x, C~:)

12 -

L I (x,

l;k)

12 •

(8)

k-1

Hence II x - 2;;= 1 akl;k II attains its minimum when ak is given by (x, Ct), k = 1, 2, ... , n. This shows (7).

6.1.

FOURIER SERIES IN AN INNER PRODUCT SPACE

197

From (8) we have the following theorem.

Theorem 6.1.2. 00

L I (x, ck> 12 ~II xll 2• k=1

(9)

where x is any element of X and { Cn} is an orthonormal sequence. The inequality (9) is called the Bessel inequality. We now show the following theorem, which is due to Riesz [1].

Theorem 6.1.3. Let X be a complete normed inner product space and let {t'to k = 1, 2, ... } be any sequence of complex numbers such that (10)

Let {Ck, k = 1, 2, ... } be a given orthonormal sequence of X. Then there is an x E X such that

k = 1, 2, ... ,

(11)

and as

Proof.

Consider

n---+ oo.

(12)

n

s,

=

L ckck.

(13)

k-1

Then it is easily seen that, for m < n,

which converges to zero as n ---+ oo, m ---+ oo. By the completeness of the space X, we see that there is an element x such that

II X -

Sn

11---+0

as

n---+ oo.

Now for n > k,

= (x, Ck)-

n

L cj( cj, Ck) =

i-1

(x, Ck)- ck.

(14)

198

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

Hence

I (x,

C~:)

by (14). Hence c,.,

- c,., I = I (x - Sn, c,.,) I ~ II X - Sn II II c,., II =II x - sn II- 0 as n- oo =

(x,

C~:)

and (14) is just (12).

Now we introduce the terminology, the completeness of a sequence of elements of a normed inner product space X. A sequence {xn, n = 1, 2, ... }, xn eX is called complete if (x, Xn)

= 0,

n = 1, 2, ... , x e X,

implies x = 0. Here it is hoped that readers will not confuse the completeness of a space and the completeness of a sequence. For example, let X= £2( -n, n), which is a complete normed inner product space with inner product (x, y)

=

J:,. x(t)y(t) dt,

x

=

x(t),

y

=

y(t).

The sequence {xn, n=O, ±1, ±2, ... }, where x,. = xn(t)= einte £2( -n, n) is complete in £1( -n, n) (Theorem 2.10.1 ). Hence eint is complete also in £2( -n, n) since £2( -n, n) c £1( -n, n).

Theorem 6.1.4. Let X be a complete normed inner product space. If

{C,., n = 1, 2, ... } is a complete orthonormal sequence of elements of X, then the Fourier series L::":1 c,.C,., of any element x eX, c,. = (x, Cn), converges to x: as n- oo,

(15)

and the equality 00

II X II holds.

=

L I c,. 1 n-1 2

(16)

The equality (15) is called the Parseval relation or Parseval equality, which is a refinement of the Bessel inequality for a complete orthonormal sequence {C,.}. Proof.

Lets,. be the partial sum of the Fourier series of x: (17)

6.1.

199

FOURIER SERIES IN AN INNER PRODUCT SPACE

From Theorem 6.1.2 and Theorem 6.1.3, there is an element z e X, such that (18) lim II sn - z II = o n-+oo

and k = 1, 2, .... Since ck is the Fourier coefficient of x, k

= 1, 2, ....

Hence from the completeness of the sequence {C,.}, x turns out to be (15). Now from (8), with ck in place of ako

= z. Hence (18)

(19)

which gives us (16) because of (15). This completes the proof. We shall remark that the validity of the Parseval relation (16) is equivalent to the validity of 00

(x, y)

=

L (x, k=l

Ck)( y, Ck)·

(20)

If x = y, (20) reduces to (15). Equation (20) follows from (15) and (19):

I(x,y)- ,~1 (x, Ck)(y, C,.) I

=I

(x-

t (x, Ck)Ck,y- t (y, Ck)Ck) I

k-l

k-1

Let us consider again ordinary Fourier series. Since L 2 ( -n, n) is a complete, normed inner product space and the sequences {

sin x cos 2x sin 2x } 1 cos x (2n )112' --;tii"2' --;tii"2' ~' ~' ...

and {

einz

2n ,

n

= 0, ±1, ...

}

200

VI. L 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

are orthonormal sequences that are complete in L 2 ( -n, n), we have the following theorems directly from Theorem 6.1. 4:

Theorem 6.1.5. Let f(x)

E

V( -n, n) and its Fourier series be 00

f(x) ,_, (a0 /2)

+ L (an cos nx + bn sin nx).

(21)

n-1

Let sn(x) be its partial sum. Then

II sn(x)- f(x) ll2- 0

(22)

as n - oo, and the Parseval relation

holds. For the complex Fourier series, we have (22) with n

= ".(_,

sn

ck eikz •

k--n

ck being the Fourier coefficient of f(x) and

(2n)-l

f

1

J(x)

12

dx

=

-n

~

1

c,.

12 •

(24)

n=-oo

We also have from (20) the result that: If f(x) and g(x) are of L 2 ( -n, n) and their Fourier coefficients are, respectively, an, bn, and an', bn'• and in complex form, en and en'• then (25) (26)

Theorem 6.1.3 of Riesz turns out to be:

Theorem 6.1.6. If a 0 , a 1 , ••• , b1 , b2 ,. • • is a sequence for which

then there is a function f(x)

E

V( -n, n) such that the given sequence is

6.2.

FOURIER TRANSFORM OF A FUNCTION IN

£ 2( - oo, oo)

the sequence of Fourier coefficients of f(x). If {c,., n sequence such that

201

= 0, ± 1, ... } is a

00

L: I c,.l2 <

oo,

-oo

then there is a function f(x) E £2( -n, n) such that the sequence is the sequence of the complex Fourier coefficients of f(x). In both cases, f(x) is uniquely determined except on a set of measure zero. The last statement follows from the completeness of the trigonometric sequence.

6.2. Fourier Transform of a Function in L 2 (-oo, oo) We defined the Fourier transform of a function belonging to U( -oo, oo) to be

(2n)-1i2

J: e-ixlj(x) dx;

but, if we assume thatf(x) is in £2( -oo, oo) and is not in U( -oo, oo), then this definition does not work as it stands. So we shall begin with the problem of how to define it. This was achieved by Plancherel [1] and the subsequent theorems, 6.2.1 and 6.2.2 are called Plancherel's theorems. We recall that if f(x) E £2(- oo, oo ), then we may write (1)

Theorem 6.2."1. Let f(x) E £2( -oo, oo). Then there 1s a function /(t) E £2( -oo, oo) such that

II (2n)-112 J:a f(x)rixt dx -

/(t) 112 - 0,

(2)

as a- oo. There are several ways of proving this theorem but we give here a classical proof. (See Titchmarsh [2].) For other proofs also see Bochner and Chandrasekharan [1], Wiener [6], or Rudin [1].

Proof.

Set for for

I xI ~a, I xI >a,

(3)

202

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

and Ja(t) = (Zn)-112

J: fa(x)e-izt dx.

(4)

Since/4 (x) E L 1(-oo, oo) .'1 L 2(-oo, oo), (4) is well defined. Now we compute the following integral. Let 15 > 0 be an arbitrary number. Since / 4 (t) is bounded, the integral exists.

J: exp[-ll52t2] 1/a(t) 12 dt = (2n)-1 J: exp(-l15 2t 2) dt J: exp(-ixt)/ (x) dx •J: exp(iyt)/ (y) dy = (2n)-1 J: fa(x) dx J: fa(Y) dy J: exp[ -l15 2t 2+it(y-x)] dt.

I( a, 15) =

4

4

The interchange of integral signs is permitted, since the integrals are absolutely convergent. The inner integral is shown to be

which follows from

J: exp[-l(u -

it) 2] du

=

J: exp( -lx2) dx = (2n)112

(by complex integration). Hence

l(a, 15) = (Zn)-11215-1

J: f (x) dx J: fa(Y) exp[-l(x- y)2/l52] dy,

which is, by the Schwarz inequality for double integrals, I(a, 15)

~

(Zn)-11215-1

[f: J: 1/

4

(x) 12 exp[- Hx

-

y) 2fi52J dx dy

. [f~oo f~oo lfa(y) 12exp[ -l(x- y) 2/l52] dx dy =

(Zn)-11215-1

= (Zn)-11215-1 =

J:

lfa(x)

(5)

4

J: J: I dx.

2

fa(x) 12 exp[- l(x- y)2fl52] dx dy

J~oo I fa(x) 12 dx 12

r

J:

exp( -lu2fl52) du

r 2

6.2.

FOURIER TRANSFORM OF A FUNCTION IN

£ 2( - oo, oo)

203

Hence we have obtained

Since exp( -l <5 2t 2) --+ 1 monotonically when <5 decreases to 0, the monotone convergence theorem gives us

ll]a(t) ll2 ~ 11/a ll2o which shows that fa(t) also be written as

l(a, <5)

E

L 2 ( -oo, oo) for each

a

> 0. Equation (5) can

= (Zn)-1' 2<5-1 I~oofa(x) dx I~oofa(X + y) exp(-ly2/<5 2) dy (2:n:)-112<5-1 I~oo exp( -ly2f<52) dy I~oo fa(x)Ja(x

=

+ y) dx.

(The interchange of integrations is permissible.)

where

¢'(y) is bounded and continuous over -oo < y < oo. This is seen from I

¢'{y) 12 ~ I~oo lfa(x)

12

dx

I:

lfa(X

+ y) l

2

dX

=Ilia II~~ 11/lll. I ¢'{y + h) -

¢'(y)

12 ~ I~oo Ifa(x) 12 dx

I:

lfa(x+y+h)- fa(x+y) 12 dx

= 11/a II~ I~oo lfa(X +h)- /a(x) 12 dx--+ 0 as

h--+ 0.

Therefore from Theorem 5.2.5,

which is

11/a II~.

204

VI. £ 2-THEORY

Thus letting t5

---+

OF FOURIER SERIES AND FOURIER TRANSFORMS

0+, we have, from (6) that (7)

If we take fa(x) - fb(x) in place of fa(x), then from (7) we should have

Hence if b ---+ oo, a

---+

oo, then

Because of the completeness of L 2 ( -oo, oo), there is a function /(t) e £2(- oo, oo) such that as

a---+ oo.

(8)

This proves the theorem. The function /(t) in (2) is called the Fourier transform of f(x)

e £2( -oo, oo) or the Fourier transform in £2 of f(x). Needless to say, /(t) is defined for almost all values oft. Sometimes we describe (8) as l.i.m./a(t) a__.,oo

=

l.i.m. (2n)- 1' 2 a"""'"'oo

Ja

f(x)e-ixt dx

= /(t).

(9)

-a

Here, l.i.m. is the abbreviation of "limit in the mean," which is the same as the limit in norm (£2). The following remark is useful. REMARK.

Hf(x) e L 2 (-oo, oo) and it happens that lim (Zn)- 112 a-+oo

r

f(x) e-ixt dx

-a

converges to / 1 (t) almost everywhere (pointwise), then / 1 (t) is equal to the Fourier transform /(t) in L 2 ( -oo, oo) of f(x) almost everywhere. Therefore, we also see that if f(x) e £1( -oo, oo) n £2 ( -oo, oo), then the Fourier transforms off(x), since f(x) e £1( -oo, oo) and since f(x) e £2( -oo, oo), are equal to each other almost everywhere. Now since 11/a 11 2 -+II/ 11 2 if (8) holds, and II fa 11 2 -+I If 11 2 , (7) implies the following theorem.

6.2.

£ 2( - oo, oo)

FOURIER TRANSFORM OF A FUNCTION IN

Theor.!m 6.2.2. If/(t) is the Fourier transform of f(x)

205 E

£2(- oo, oo),

then (10) Equation (10) is called the Parseoz,·al relation for Fourier transforms. We remark concerning the definition of the Fourier transform, that Eq. (2) can be replaced by II

(2n)-1

2

J:J(x)ri.rt dx- /(t)

11 2 ---+

0,

(11)

as b ---+ oo, a ---+ - oo. The proof remains the same. In connection with the notation l.i.m., we mention the following relation. If l.i.m. Fa(x) a-•oo

exists, then for every G(x)

J: l~i_:~·

E

£2(-oo, oo),

Fa(x)G(x) dx =

!~r::,

J:

Fa(x)G(x) dx.

(12)

In this case Fa(x) is said to converge weakly to F(x). Equation (12) is shown in the following way. If we write

F(x)

l.i.m. Fa(:-c) = F(x), a-+oo

E

£2( -oo, oo),

then I

J: =I J~oo

Fa(x)G(x) dx -

f~oo F(x)G(x) dx

fFa(x)- F(x)]G(x) dx

r

r ~II Fa- F 11211 G 112,

which shows (12). Using (12), we shall show the following relations which are also called the Parseval relations. Ifj(x) E £2(-oo, oo),g(x) E £2(-oo, oo)andtheirFouriertransforms in L 2 are, respectively, /(t) and g(t), then

J~oof(x)g(x) dx = J~oo/(x)g(x) dx,

(13)

J:J(x)g(x) dx

(14)

=

J:f
VI. £ 2-THEORY

206

OF FOURIER SERIES AND FOURIER TRANSFORMS

Using the definition fa(t) in (4) and ga(t) corresponding to g(x), we have, using (12),

J:

f(x)g(x) dx = =

J:

f(x) l~i~. la(x) d:c

!~ J~oof(x)ga(x) dx

= !~~ J~00 f(x) dx (Zn)-112 J:a g(u)e-izu du =

!~ !~~ J~b f(x) dx (Zn)-112 J:a g(u)e-ixu du

=

!~ !~~ J:a g(u) du (Zn)-11 2 (b f(x)e-izu dx

= lim J" g(u) du l.i.m.(Zn)-11 2 Jb f(x)e-i= dx a-+oo -a

=

!~

J:

b-+oo

g(u)/(u) du

=

-b

J~oo g(u)/(u) du,

which shows (13). In the same way we can show (14). A way analogous to the proof of Section 6.1, Eq. (20) is also applicable. Equations (13) and (14) are also derived from (10), when it is applied tof(x) + g(x), whose Fourier transform is/(t) + g(t) and tof(x) + ig(x). Equations (13) and (14) are therefore equivalent to (10). It is also remarked that the transformation from f(t) to /(t) is linear, which is straightforward. Now we show the following theorem.

Theorem 6.2.3. Let/(t) be the Fourier transform off(x) e £2( -oo, oo). Then f(x) = l.i.m.(2n)-112 Ja f(t)eizt dt. (15) a-+oo -a This gives us the inversion formula for the Fourier transform of the 2 L -function. The integral of the form in (15), namely, l.i.m.(Zn)-112 a-+oo

r -a

1/J(t)eizl dt,

(16)

where ¢(t) e L 2(-oo, oo), is called the inverse Fourier transform, i.i which eizt appears, instead of e-izt in the definition of the Fourier transform [in (2)].

6.2.

FOURIER TRANSFORM OF A FUNCTION IN

£ 2(-oo, oo)

207

Proof. /(t) e L 2(-oo, oo) as was shown in Theorem 6.2.1. Hence by the same theorem,/(-t) has the Fourier transforrn/(x) in £2(-oo, oo), since

J:a /( -t)e-itz dt

=

J:a /(t)eizt dt,

l.i.m.(Zn)-1' 2 Ja j(t)eizt dt = /(x). a-+oo

-a

Now for any E,

r

(2n)-1' 2 /(t)eitx dt fe.f<x) dx = fe dx l.i.m. a-+oo -a 0

0

= lim

a-+oo

r

dx (2n)-1' 2

0

= (2n)-1' 2 lim a~oo

r

-a

r-a

}'(t)eitx dt

f(t)[(eitE- 1)fit] dt.

The last integral is absolutely convergent when a---+ oo, for /(t) e £2(-oo, oo) and (eite- 1)f(it) e L 2(-oo, oo). Therefore, we may write the above equation in the form

J:

/(x) dx = (2n)-1' 2 J~oo j(t)[(eitf - 1 )fit] dt.

(17)

Now consider the function

g(x) = {

~

for x between 0 and for other x.

E,

Then g(x) e L 2( -oo, oo) and its Fourier transform is ±(Zn)-112

J:

e-itzdx= ±[(e-itf-1)f(2n)1!2(-it)],

where plus or minus is assumed according as E > 0 or E < 0 (because of the statement below Remark on p. 204). Therefore from (14), ±

r 0

j(x) dx =

foo

j(x)g(x) dx = ±(Zn)-112

-oo

Comparing this with (17), we have

foo -oo

f(t)[(eitE - 1 )fit] dt. (18)

208

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

Hence ](x) is equal to f(x) almost everywhere. This leads to the conclusion of the theorem. Other forms of representation of the Fourier transform in L 2 ( - oo, oo) and its inversion are given in the following theorem.

Theorem 6.2.4. If f(x)

E

L2(- oo, oo) and /( t) is its Fourier trans-

form in L2(- oo, oo ), then

1 d (2n) 1' 2 dt

foo

=

1 d (2n)li2 dx

IOQ

f(x) =

/(t)

1

e-itz -

-oof(x) A

-ix eitz -

-oof(t)

it

dx

a. e.,

(19)

dt

a.e.

(20)

1

Proof. If we differentiate both sides of (18), we obtain (20). Equation (19) is obtained from (20) by interchanging the roles of f(x) and /(t) with modifications in the signs of variables.

Now if g(t) is the Fourier transform of g(x) E L2(- oo, oo ), then, by Theorem 6.2.3, g(x) is the inverse Fourier transform of g(t) in L2( -oo, oo), and (13) and (14) can be written also as

J~oof(x)g(x) dx = J~oo/(t)g(-t) dt,

(21)

J~oof(x)g(x) d.'C = J~oo/(x)g(x) d.'t.

(22)

Finally, we shall give interpretations of the theorems obtained above in terms of the transformation from L2(- oo, oo) to L2(- oo, oo ), which transformsf(x) to its Fourier transform/(t). This transformation is called the Fourier transformation. If this transformation is written by Tf = f, then T is a bounded linear transformation, that is, with f 1 ,f2 E L 2 ( -oo, oo), a 1 , a 2 constants, T( ad1 + ad2 ) = a 1 Tf1 + a 2 Tf2 and II Tf II ; : ; C II f II with a constant C. As a matter of fact, (10) in Theorem 6.2.2 gives us

II Tfll= llfll

(23)

(here we drop the subscript 2 of the norm sign). Such a norm preserving transformation is called isometric. So the Fourier transformation is isometric. The inner product

(/,g)= J:f(x)g(x) dx,

6.2.

FOURIER TRANSFORM OF A FUNCTION IN

£2(- oo, oo)

wheref,g E L 2(-oo, oo) is now introduced [(/,f)= implies (TJ, Tg) =(/,g)

209

11/11 2]. Then (23) (24)

for any J, g E £2(- oo, oo ). Actually this is no more than the Parseval relation and is also shown from (23) in the following way. It is just what we talked about after the proofs of (13) and (14):

II/+ g 11 2 = (f + g,J +g) = 11/11 2 + llg 11 2 +(/,g)+ (g,f) = 11/11 2 +II g 11 2 +(!,g)+ (!,g) = 11/11 2 + llg 11 2 + 2 Re(f,g).

(25)

Taking -g for g,

II/- g 11 2 = II/ 11 2 + II g 11 2 - 2 Re(f, g).

(26)

From (25) and (26),

II/+ g 11 2 - II/- g 11 2 = 4 Re(f,g).

(27)

Taking Tf and Tg in place off and g in (27), we get

II T(f +g) 11 2 - II T(f- g) 11 2 = 4 Re(T/, Tg). Using (23) and comparing with (27), we get Re(f,g)

= Re(Tf, Tg).

(28)

Now we take ig in place of gin (25) and noting Re(ia) = -Im a, we have from (28), Im(/,g) = Im(TJ, Tg).

(29)

Equations (28) and (29) show (24 ). From Theorem 6.2.3, there is a one-to-one correspondence between f and TJ, or by the transformation T, the whole space L 2(- oo, oo) is mapped onto the whole space L 2(- oo, oo) with isometry (23 ). In other words, the Fourier transformation is unitary, and the inverse transformation of T, which is denoted by T- 1, is given by the inverse Fourier transformation: T-1J = l.i.m.(2n)-112 f(x)eitz dx. a-+oo

J"

-a

210

VI. £ 1 -THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

The general unitary transformation from L 2 ( - oo, oo) to L 2 ( - oo, oo) (or L 1 (0, oo) to L 2 (0, oo)) has been studied by Watson [1], Bochner and Chandrasekharan [1], and Riesz and Nagy [1].

6.3. The Class H 2 of Analytic Functions Letf(.w) be an analytic function in the unit circle U: l.w I < 1, .w = Teifl, -n ~0 < n. If

I:

1 f(Tei 6 ) ] 2

dO ~ M,

0

~ T

< 1,

(1)

for some constant M which is independent of T, then /(:) is said to belong to the class H 2 • Namely, H 2 is the class of functions/(:) analytic in I .w I < 1 for which sup O:iiir<1

r

I /(Tei 6 ) 12

dO <

(2)

00.

-n

We give here some important results on the class H 2 • Let/(:) be analytic in the unit circle U and 00

J(.w) =

L Cn.ll'". n-o

(3)

Then 00

/(TeiiJ) =

L c"TneniiJ,

0

~

T

< 1,

(4)

n-o

which is uniformly convergent for 0 ~ T < T1 < 1 for any fixed T1 • Hence it is the Fourier series of f(Te" 6 ). Therefore, (5) by the Parseval relation [Theorem 6.1.5, Eq. (24)]. We note that the Fourier coefficient C11 of f(TeU!) for a fixed 0 ~ T < 1 is zero for n < 0. Furthermore, if the integral in (5) is constant for 0 < T < T 0 for some T0 , then C11 = 0 (n = 1, 2, ... ), because if not, the right-hand side of (4) is never constant. Hencef(Teifl) = c0 • Hencef(.w) is constant in U. More generally, if (4) is constant in some interval 0 < T1 ~ T ~ T2 < 1, then/reduces to a constant. Write M 2(/:T) =

[(Zn)-1 I.,. n

I f(Tei 6 ) 12 dO

]1/2.

(6)

6.3.

THE CLASS

H2

211

OF ANALYTIC FUNCTIONS

Then we may claim that M 2 (/:r) is a strictly increasing function of O~r
An obvious property of H 2 is given by:

Theorem 6.3.1. An analytic function f(z) in the unit circle U with power series (3) belongs to H 2 if and only if (7)

This is so because by the definition, iff(z) e H 2, then L~ Icn2 l2 r 2n ~ M. Letting r---+ 1, (7) is obtained. The converse is obvious since

L

I Cn 12

r 2n ~

L

I Cn 12 •

Theorem 6.3.2. Suppose f(z) e H 2 • Then (i)

limf(rei8 ) = f(eill)

(8)

r-+1

exists for almost all -:n < 0 < :n. as

r---+1.

(9)

(iii) f(z) is represented by the Poisson integral of f(eiS):

f(z) = :n-1

J:,. f(e~)P(r, 0 -

~) dcp,

(10)

where

P(r,

~) =

(1 - r 2 )/[2(1 - 2r

cos~+

r 2 )].

(11)

(iv) f(z) is also represented by the Cauchy integral

f(z) = (2:ni)- 1

fc

[f( C)/( C- z)] dC,

(12)

where C is the positively oriented unit circle. In (12), f( C) = f(ei 8 ): C= ei8 , The integral (12) is taken in the Lebesgue sense and is determined even when f( C) is defined except on a set of 0 of measure zero. Actually, the integral in (12) is defined to be

(Z:n)-1

J:

[f(eiB)f(eiB _ z)] eill dO.

212

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

REMARK. Statement (i) of the theorem can be given in a more general form: f(rei 8) converges to f(ei 8) for almost all () as rei8 tends to ei8 nontangentially.

Proof of Theorem 6.3.2. Letf(z) e H 2 , andf(z) be given by (3). Then by the foregoing theorems,

Hence by Theorem 6.1.6, there is a function g(O) of L 2 ( -n, n) such that ( ... 0, ... 0, c0 , c1 , c2 , ••• ) is a sequence of (complex) Fourier coefficients of g(O) which is uniquely determined except on a set of measure zero. If g(O) is written by f(ei 8 ) (a matter of notation), (2n)-1

J"

f(ei8)e-in8 d() = { c,. • -n 0,

n ~ 0, n < 0.

(13)

Then 00

f(z)

=

L c,.z",

z = rei 8

n-o

"c i..J , r"eino ' n-o 00

= which is written :n;-1

J:,. f(ei~)P(r, () -

cp) dcp

from Section 4.7, Eq. (10). With Theorem 4.7.2, we obtain (i) and (iii). Next we shall show (ii). From (10), for almost all 0,

f(reiB) - f(eiB) = :n;-1

=

n-1

J:,. [J(ei~) - f(eiO)]P(r, () - cp) dcp J:,. [J(ei(8-~l) _ f(eiB)]P(r, cp) dcp.

[Note Section 4.7, Eq. (8).] Hence by Schwarz's inequality,

llf(reiB)- f(eiB)

W ~ J:,. dO :n;-1 J:,. lf(ei(8-~l)- f(eiB) 12P(r, cp) dcp

· J:,. P(r, cp) dcp = J:,. dO ! J:,. If(ei
6.3.

THE CLASS

H2

213

OF ANALYTIC FUNCTIONS

Since as ~- 0, Theorem 4.7.1 (i) gives us

II f(rei 8)- f(ei8) 1!2- 0 which shows (ii). Finally, for z = rei8, 0 (2ni)- 1

~

fc [!(C)/( C-

r -1,

as

r < 1,

z)] dC

J:,. [J(ei~)f(ei~ - rei8 )]e~ tUfJ (Zn)-1 J:,. [/(eiW-~1)/(1 - re~)] tUfJ

= (2n)- 1 =

= (2n)- 1 J" l.i.m.[/(eei< 8 -~ 1 )/(1 - re~)] tUfJ

[by (ii)]

_,. g-+1-0

Since

f"-,. e-i~eik~ d~ = { 0,2n,

n =1= k, n= k,

n > 0,

k

> 0,

the above expression is lim e-+1-0

00

00

n-o

n-o

L cnenein8rn = L Cnrnein8 =

f(z).

This shows (iv). This completes the proof of the theorem. The function J(ei 8 ) in Theorem 6.3.2 is called the boundary function of f(z) E H 2 • The Remark is obtained by considering Theorem 4.7.3. We give the following theorem which is a direct consequence of Theorem 6.3.1.

Theorem 6.3.3. If f(ei 8 ) E £2( -n, n) is a periodic function of period 2n, then it is the boundary function of a function f(z) E H 2 when and only when its Fourier coefficients en = 0 for n < 0.

214

VI. L 2-THEORY

Proof.

OF FOURIER SERIES AND FOURIER TRANSFORMS

If we. form 00

f(z) then, since

=

L c,.zn,

I z I< 1,

n-o

L:=0 I c,. 12 <

oo, we have

f(z) e H 2

and

f(rei 6 )

---+ f(ei 6 )

a.e.

as r---+ 1 - 0, by Theorem 4.7.2 (ii). Hence the theorem is immediate. The following corollary is also easy to show.

Corollary 6.3.1. If f(z) is analytic in I z the Poisson integral of some function f(ei 6 ) f(z) = (n)- 1 z = reiiJ, then f(z)

E

I:

I < 1 and is represented as E

V( -n, n),

P(r, 0 - ¢) f(ei~) dtf>,

(14)

H 2 and f(ei 6 ) is the boundary function of f(z).

In other words (14) gives a necessary and sufficient condition for an analytic function in the unit circle to belong to H 2•

Proof.

I:

I

f(rei 6 )

12

d() =

I:. I I:. P(r, () n-1

4> )f(ei~) d¢

r

d()

~ J:,. (dOfn) J:,.lf(e~) I2P(r, 0- ¢) d¢ · n- 1

I:

P(r, ()- ¢) d¢

(Schwarz inequality)

=I: (dOfn) J:,. lf(ei~) I P(r, ()- ¢) dtf> 2

= J:,.

lf(ei~) 12 d¢ n-1 J:,. P(r, ()- ¢) dtf>

= J:,.

If(ei~) 12 d¢ <

oo.

6.4. A Theorem of Szego and Smirnov We are going to show the following theorem due to Szego [1, 2] and Smirnov [1 ].

Theorem 6.4.1. Let V'(O) be a nonnegative function belonging to V( -n, n). In order that there exist a function f(z) E H 2 such that its

6.4.

215

A THEOREM OF SZEGO AND SMIRNOV

boundary function f(eiB) satisfies

If{e' 8 ) I = VJ(O) a.e.,

(1)

it is necessary and sufficient that J:, I log VJ(O)

I d() <

(2)

oo.

This can be explained, because of Theorem 6.3.3, in the following way.

Theorem 6.4.2. Let VJ(O) be a nonnegative function belonging to L 2{ -n, n). In order that there be a function f(eiB) for which the Fourier coefficient en = 0 for n < 0 and (1) hold, it is necessary and sufficient that (2) hold. We shall prove Theorem 6.4.1. Proof. Necessity. First suppose /(0) =f=. 0. Let r 1 :::::;; r 2 ~ • • • be the moduli of zeros of f(z) in I z I < 1. (Zeros are counted to be as many as their multiplicities.) Then by Jensen's formula (Theorem 1.13.1), we have, for rn ~ r ~ rn+l• log[rn 1/(0) l/(r 1 • • ·rn)] = {2n)- 1 J:,.loglf(rei8 ) I dO.

(3)

From this, it follows that (2n)- 1 J~ logl f(rei 6 ) I d() ;;;;; logl /(0)

I·

(4)

Write log+ a= { 0log a

if a> 1, if a~ 1,

log-a= { 0log a

if a~ 1, if a>l.

Then log a = log+ a

+ log- a.

Now (2n)- 1 J:,. log If(rei 6 ) I d() = (Zn)- 1 J:,. log+l f(rei 6 ) I d()

+ (2n)-

1

J~ log-1 f(rei 6 ) I d()

(5)

216

VJ. L 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

and (2n)- 1

J:, Jlogj /(reiD) II d() = (2n)- J:, log+j f(rei I dO - (2n)- J:, log-J f(rei I d() = J:, log+ If(reiO) I - (2n)- J:, log! f(rei I 8)

1

8)

1

n-1

d()

8)

1

d()

[here we used (5)]. By (4), this is equal to or less than n- 1 Since log+ a

~

J:,. log+ If(rei

8)

I d()

-

log! /(0) I·

a 2 , it is equal to or less than (6)

Now we suppose that f(O) = 0 and the origin is the zero of f(z) of mth order, m > 0. Let cf>(z) = f(z)fz"'. Then ¢>(0) -j::. 0, and (2n)- 1

J:, Jlogl /(reiD) ! I d() = (2.n)- J:, I log! cf>(rei~) I + log rm I ~ (2n )- J:, Jlogl cf>(rei II + I m log r I· 1

,-1()

0)

1

d()

Using here relation (6) with ¢> instead of J, we obtain (2n)- 1

J:, JlogJJ(reiO)JI d() ~ n- J:, I cf>(rei 1

0)

J2 d()

- logj c/>(0) I - m log r = n- 1

J:, I f(reiO)

12

dO rm - log! cf>(O)J

- m log r.

(7)

Since f(rei 0 ) __. f(ei 8 ) almost everywhere, and llf(rei0 ) 11 2 -ll/(ei0 ) 11 2 , using Fatou's lemma, (2n)- 1

f"

-n:

jlogl /(eiO)JI d() ~ lim inf (2n)- 1 r-+1-0

~ n- 1 [from (7)]. This shows (2).

J:, If(ei

8)

f"

jlogj f(rew)ll dO

-x

J2 dO- logj cf>(O) I < oo.

6.4.

217

A THEOREM OF SZEGO AND SMIRNOV

Sufficiency. Next we shall show that (2) is sufficient, that is, we are going to show that if (2) holds, then tp(O) is the modulus of the boundary function of a function belonging to H 2 • Define

J:, log tp(¢) P(r, ¢- 0) d¢,

g(O, r) = n-1

where P is the Poisson kernel [Section 6.3, Eq. (11)]. Since log tp(¢)

e LI( -n, n), from Theorem 4.7.1, we have lim g(O, r)

= log tp(O) a.e.

(8)

= P(r, 0) + iQ(r, 0),

(9)

r-+1-0

Now note that !(1

+ z)/(1

- z)

where z = rei8 , and P(r, 0) and Q(r, 0) are Poisson kernel and the conjugate Poisson kernel, respectively (Lemma 4.7.1). Define g(z)

=

n- 1

J:, log tp(¢) ![(1 + ze-i~)/(1 -

ze-i~)] d¢,

which is an analytic function in U (differentiable at z, consider

(10)

I z I < 1). Now (11)

which is also analytic in U. From (9), Reg(z) = g(r, 0), and then (12) From (8), lim

If(rei 8 ) I =

tp(O).

(13)

r-+J-0

Using the Jensen inequality (Section 1.11 ),

If(rei 0 ) I =

eg
J:, tp(t/J )P(r, ¢ -

0) d¢.

Hence, using the Schwarz inequality,

J:,.

I f(rei 8 ) 12

dO

~ J:,. •

= =

dO [ n- 1

[ n-1

J:,. tp

2(

J:,. P(r, ¢ -

¢ )P(r, ¢ - 0) d¢] 0) d¢]

J:,. dO J:, tp (¢)P(r, ¢- 0) d¢ J:,. tp (¢) d¢ J:,. P(r, ¢ - 0) dO J:,. tp (¢) d¢. 2

n-1

2

n- 1

=

2

218

VI. L 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

Here we also used

J:

x- 1

P(r, 0) dO = 1.

Hence f(z) e H 2 • Its boundary function f(ei 6 ) should satisfy I f(e' 6 ) = 1p(O) from (13). This completes the proof.

6.5. The Class

~~~

I

of Analytic Functions

We now discuss the analogs for functions analytic in the upper halfplane, Im z > 0, of what we studied in the preceding section. Let f(z) be an analytic function in Im z > 0. Suppose that f(x + iy) E L2(-oo, oo), for eachy > 0, as a function of x, namely,J(x + iy) is of L2(- oo, oo) on each line parallel to the real axis in the upper half-plane. If sup O
J""

I

f(x

-oo

+ zy) 1

2

dx

< oo,

(1)

then f(z) is said to belong to ~ 2 • That is, ~~~ is the class of functions analytic in the upper half-plane that satisfy (1 ). We shall prove the analog of Theorem 6.3.2. For this, we need the following lemma.

Lemma 6.5.1. Let f(z) e zero as I z

Proof.

~~~.

For any fixed 15 > 0, f(z) converges to ~ 15.

I ---. oo uniformly in y

Let y

~

15, and let 0 < e < 15. From Cauchy's integral formula,

f(z) = (2ni)- 1

J

IC-zl91

[/(C)/( C- z)] dC,

where the integral is taken in the positive direction along the circle I C- z I = f!· Hence

f(z)

=

Multiply both sides by We then get

(!

(Zn)- 1

J:

f(z

+ ee'"') tUf>.

and integrate with respect to

!(«5/2)~(z)=(2n)- 1

(!

over (0, 15/2).

J,,ll ede f"'_,f(z+eeilf>)dcp. 0

6.5.

THE CLASS ~ 2 OF ANALYTIC FUNCTIONS

219

By the Schwarz inequality, (1/8)~ 2 1/(z)l~(1/2)

(fat2 J"_,.l/l 2edcfode) 112 ( Ja12 J"_,.edcfode ) 112 0

0

The integrand in this integral is nonnegative and the integration is carried out over the circular domain with center z and radius ~/2, which is completely contained in the rectangle x -

( ~/2) ~ Re z ~ x

+ (~/2),

y-

(~/2) ~

Imz

~y

+ (~/2).

Therefore,

(1/8)~21/(z) I~ (~f%112) [ J"+
~ (~/4n 112 )

Writing

J:

lf(u

z--W21

[ J II+M/21 dv 11-M/21

+ iv) l

2

du

foo --oo

~C

lf(u

+ iv) 1 dv ]1/2. 2

(2)

for v > 0,

we have (3) which shows thatf(z) is uniformly bounded in Imy ~ ~For any e, choose A so large that A > 4Cfne2 • Then from (3), for Imz ~A, (4) if(x + iy) I <e. Now for each v > 0, z+
I / (~

+ iv)

12

du

-+ 0

as

I x 1- oo.

Hence

Ia12A+
2

du-+0

because of (5) and

J

z+
+ iv) 1

2

du ~C.

as

lxl-+oo

(5)

220

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

Therefore, there is an X 0 such that for x > X 0 , .4+C6/2l J 6/2

dv

Jz+!6/2l I

z-(6/2)

f(u

+ iv)

12

du < e2 nc) 2 /4.

Then from (2),

I f(z) I for

I xI >

~ (2fc)n 112 )

.-!+(6/2) (J 6/2

dv

Jz+!6/2l z-(6/2)

I f(u

+ iv) 1 du

) 1/2

2

< e

(6)

X 0 • Hence from (4) and (6),

lf(z) I < e This proves the lemma. Now we shall prove the following theorem.

Theorem 6.5.1. If f(z) e (i)

lim f(x

~ 2,

then

+ iy) = f(x)

(7)

JI-+O+

exists for almost all x, -oo < x < oo, for some f(x) (ii)

llf(x

+ iy)- f(x) 11

2

-+

E

£2( -oo, oo).

(8)

0 as y-+ 0+.

(iii) f(z) is represented by the Poisson integral of f(x): f(z)

=

n- 1

Z =X+ iy,

J:f(u)P 1(u- x, y) du,

(9)

where P 1(x, y) is the Poisson kernel

y > 0.

(10)

(iv) f(z) is represented by the Cauchy integral: f(z) = (2ni)- 1

J:

[J(C)/(C- z)] dC,

y > 0.

(11)

REMARK. (i) is replaced by the conclusion: f(~ + iy) converges to f(x) for almost all x as ~ + iy converges to x nontangentially.

Once (iii) is shown, this follows from the statement following Theorem 4.8.4. Proof of Theorem 6.5.1. Since f(x + zy) E L 2 ( - oo, oo ), for every fixed y > 0, it has the Fourier transformf(t,y) in L 2 (-oo, oo).

THE CLASS ~ 2 OF ANALYTIC FUNCTIONS

6.5.

221

Now let 0 < y 1 < y 2 • Let C0 be the contour of the rectangle, the vertices of which are given by ±a + iy 1 , ±a + iy 2 (Fig. 6.1 ). Then

f

f(z)e-itz dz = 0,

-00

Co

< t<

(12)

oo,

or

f

111 h

f(a

+ iy) exp[ -it( a + iy)] dy + J--a f(x + iy 2) exp[ -it(x + iy2)] dx a

+ i JIll111 f( -a + iy) exp[ -it( -a + iy)] dy

---+------4

-a+iy, .....

a+ iy 1

0 FIGURE

6.1.

The first and third integrals on the left-hand side converge to zero as a- oo, because of Lemma 6.5.1. Therefore,

!~~ [ J:a f(x + iy 1 ) exp[ -it(x + iy 1 )] dx - faJ(x

+ iy 2) exp[-it(x + iy 2)] dx] =

0.

(13)

But

!~ (2ni12)-1 J:a f(x

+ iyk) exp( -itx) dx =

/(t, Yk) being the Fourier transform of f(x

/(t, Yk),

+ iyk)·

k = 1, 2,

Hence from (13)

for almost all t. In other words, if y 1 is fixed, then for every y > 0,

/(t, y)e4'

=

/(t, Yt)e4'1

222

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

for almost every t. Then for almost all t, this holds for almost every y [by Theorem 1.5.2 (iii)]. Hence if we write f(t, y 1)tf111 = ](t), then

](t, y) = ](t)e-111

(14)

for almost every t. Once this holds, then it does for every y, that is, for almost all t, (14) holds for very y > 0. From the Parseval relation [Theorem 6.2.2 (10)],

J:

1f(t) l2e- 2111 dt

=

J:

1J(x

+ iy) 12 dt.

(15)

Since the right-hand side is bounded for y > 0, this implies that

f(t) = 0 For, if b > 0,

f:

1](t) 12 dt

t < 0

for

~ e-26v ~ e-2611 =

e-26v

f:

J:

J:

a.e.

(16)

1f(t) 12 e-21v dt I

](t)

1 f(x

12

e-21v dt

+ iy) 1

2

dt

[by (15)], which is ~Ce-2611 -+ 0 as y -+ oo. Hence ](t) all t < -b. Since b is arbitrary, ](t) = 0 for t < 0 a.e. Now again by the Parseval relation,

= 0 for almost

If y 1 , y 2 -+ 0+, then the integrand of the right-hand side integral converges to zero and is dominated by 41](t) 12, and accordingly the righthand side converges to zero. So does the left, which means f(x + iy) converges in L 2 ( -oo, oo) asy-+ +0. Let its limit bef(x). This shows (ii). Consider the same contour C 0 as in (12). Suppose that z = x + iy lies inside the rectangle surrounded by C0 • By the Cauchy integral theorem

f(z) = (2ni)-1

J

Co

[!(C)/( C- z)] dC,

the integral being taken in the positive direction along C0 • Then in a way similar that used to obtain (13) from (12), we may show that ~~ ) = _1_ ,,z ..,_. Joo =r.t

--oo

++ . iy zy

J(u u

1 -

1)

z

du _ _ 1_ ..,_. =r.t

Joo --oo

+ + zy

j(u . iy 2)

u

2 -

z

du,

(17)

where

If y 2 - oo, then this converges to zero. Therefore, from (17) we have, lettingy 2 - oo (other members of (17) remaining the same),

f(z) = _1_. 2nz

Joo

-oo

I f(u+ zy:- iy1) U

1 -

~ Joo

-oo

+ iy1) 11 + zy_1

Z

_

U -

J: ~ [J: +

I

•

J(u

[ Ioo

-oo

+[I

oo -oo

(18)

0+.

Z

U

IJ(u

zy1) du. z

1 -

Idu

f(u)

Z

IJ(u

!

+ zy

Now we are going to let y 1 -

I oo

f(u U

-oo

1 -

- -U - 1

-I du Z

+:y~; f(u) I du

+ 'Y1>

[(u _ x)2

lf(u

12

du

r

+ (y

2

y 12du

1 _

y)2]((u _ x)2

+ y2)

. ]1/2 Ioo + zy1)- f(u) I2du

-oo

]112

du (u- x)2

+ yz.

The first factor of the first term on the right is bounded and the second factor converges to zero as y 1 - 0+. For the last term, the first factor converges to zero as y 1 - 0+ by (ii), which we have shown already. Hence from (18) we obtain (iv), (11 ). Next we shall show (iii). Take z iy 1 for z in (18), which gives us

f(z

+ zy1) =

(Zni)- 1 J~oo

+ [J(u + iy 1)/(u- z)] du,

y 1 > 0.

(19)

On the other hand,

o = (2ni)- 1 f~oo u
+ 'Y1>t
z)] du

(20)

VI. £ 2-THEORY

224

OF FOURIER SERIES AND FOURIER TRANSFORMS

because the Fourier transform F(t) of f(u + iy 1) in £2 is zero for t < 0 (/(t) = 0 for t < 0) and the inverse Fourier transform g(t) of 1f(u- z) in L 2 is zero for t > 0 (and eiit for t < 0). This is easily seen if we take the Fourier transform of g(t) = 0, t > 0; g(t) = -ieizt. Actually,

I:a g(t) exp( -iut) dt = -i s:a exp[ -it(u = (u- x ---+

+ iy)- 1 -

(u - x

+ iy)-

X

+ iy)] dt

{exp[ -ia(u- x)] e-a71 f(u- x

+ iy)}

1

as a ---+ + oo [which is also the limit function in £2(- oo, oo)]. Hence by the Parseval relation [Section 6.2, Eq. (13)],

J~oo

J(:

~ ~)

J:

=

du

F(t)g(t) dt

=

0.

Now adding (19) and (20) side-by-side, we get f(z

+ zy 1) = ,.,~.

1

Joo (

1 -)f(u u-z

+ zy 1) du

yf(u + zy1) du • (u - x) 2 + y 2

1 Joo = 3f

-

u-z

~z-oo

-oo

Therefore,

1 Joo -

~

-oo

(

1 Joo

yj(u) u-x ) 2 y 2 du

+

= -

~

y

= lim _!_ Joo 1/t-+o+

=

.

(u-x )2 + y 2 711hm J(u -+o+

-oo

~

-oo

lim f(z

. + zy1) du

yf(u + iy1) du (u - x)2 + y2

+ zy 1) = f(z).

This shows (iii). Finally (i) is obtained from (iii). In fact, we may show that 1 Joo y lim -~ -oo (u-x )2

y-+o+

Let

+y

2

J(u) du

= J(x)

a.e.

(21)

I xI< A.

I!

(u -

JJuJ>2A

::;;; _!_ J -

~

Ju1>2A

~2 + y2 f(u) du I If(u) I du IU - X I

~ ! [IJuJ>2A

IJ(u)

2

1

du

r 2

du [fluJ>2A

(u- x) 2

]1/2 '

6.5.

THE CLASS ~ 2 OF ANALYTIC FUNCTIONS

225

which can be made as small as we want, taking A large. The part

1 --;:

I

lui<2A

y f(u) du (u- x) 2 + y 2

converges almost everywhere to f(x) if I xI < 2A (hence if I xI
E ~2•

Theorem 6.5.2. In order that f(x) E V(- oo, oo) be boundary function of a function of ~ 2 , it is necessary and sufficient that the Fourier transform /(t) in L 2 (-oo, oo) of f(x) be almost everywhere zero for t < 0. Proof. The necessity of the condition was shown in the proof of the previous theorem [see ( 16)]. Sufficiency. Suppose /(t) = 0 a.e., for t < 0. Construct f(x

+ iy) =

(2n)-11 2 I~ /(t)eil
It is easy to show that

fa(z)

= (2n)- 112

I:

y > 0.

(22)

/(t)eit• dt

is differentiable in Im z > 0 and hence analytic there. f(x + iy) is the uniform limit of fa(z) when a---+- oo, for Im z ~ !5 > 0 for any fixed positive !5 > 0, which shows the analyticity of (22). From the Parseval relation (Theorem 6.2.2), for y > 0,

I: I

f(x

+ iy) 1 dx = 2

I~ I/(t) l2r 2 'v dt ~I~ 1/(t) 12 dt

= C.

Therefore,f(z) E ~ 2 , and the Fourier transform ofj(x + iy) is given by /(t)e- 1V for t > 0 and 0 for t < 0. On the other hand, if / 1 (t) is the boundary function ofj(z), then from (14), the Fourier transform is given by / 1(t)e-1V, / 1(t) being zero for t < 0, a.e. Therefore, / 1(t) = /(t) a.e., which shows the sufficiency of the condition. Theorem 6.5.2 is a theorem corresponding to Theorem 6.3.1. 1. From (15) we see that for f(z) E ~ 2 , 11/(x + iy) 11 2 1s a nondecreasing function of y > 0 and that it converges to 11/(x) 11 2 as y---+-0+. REMARK

226

VI. £ 2-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

REMARK 2. If an analytic functionf(z), lm z > 0, is represented by the Poisson integral (9) of some function of f(x) E £2(-oo, oo), then f(z) E &2.

This is seen in the following way.

f~oo

IJ(x + iy)

12

dx =

f~oo

I:rr:- f~OOJ(u)P1(u- x,y) du rdx 1

~ J~oo (dxf:rr:) (by the Schwarz inequality and :rr:- 1 equal to

J~oo I f(u)

12

J: I

f(u) I2P 1 (u- x, y) du

J:'oo P 1( u - x, y) dx = 1),

du :rr:- 1 J~oo P1(u- x, y) dx = J~oo I J(u)

12

which is

du.

Finally, we show that if f(s) is continuous and of L 2 ( -oo, oo) on the real axis, then the boundedness of f(s) in lm s ~ 0 and the membership of f(s) in &2 are equivalent.

Theorem 6.5.3. Let f(s) be analytic in lm s > 0 and bounded and continuous in Im s ~ 0. If f(x) E L 2 ( -oo, oo), then f(s) E &2 • By Cauchy's theorem, for lm s > 0,

Proof.

A f(s) = (Z:rr:i)- 1 f -A (f(t)f(t - s)] dt

+ (2:rr:i)-

1

J

+ (2:rr:i)-

1

fA+iB A (f(z)f(z- s)] dz

-A+iB (f(z)f(z - s)] dz A+iB

+ (Z:rr:i)- r~+iB 1

[j(z)f(z - s)] dz (23)

=11+12 +13 +14 , say, where A, B > 0. Let

II2 I= as A

-+

1/(s) I ~ C in Im s

~

0.

1-1 JB f(A} iu) du I ~ _£ JB ~u 2:rr: A+tu-s - 2:rr: IA+tu-sl o

o

--+

0

oo, B being fixed arbitrarily. Similarly, as

A--+ oo.

Since f(t) E L 2 ( -oo, oo), 1/(t- s) E £2( -oo, oo) for any Im s > 0, / 1 converges as A--+ oo, therefore / 3 must converge as A--+ oo.

6.5.

THE CLASS ~ 2 OF ANALYTIC FUNCTIONS

Thus we have, letting A f(s)

= _1_. 2nt

= _1_

Zni

Ioo

oo

in (23 ),

f(t) ds- lim _1_. t- S A-+oo 2nt

-oo

IA+iB

f(z) dz

-A+iB Z -

S

J(t} ds _ _ 1_ I-.oo J(t + iB} dt t-s 2ni -+--oo t+iB-s ·

Ioo -oo

Now take any s and

-+

227

S 1,

Im s > 0, Im

S1

(24 )

> 0. Then

A [ f(t -:- iB) - f(t ~ iB) dt I I I .-.A t + zB - s t + zB - s I

-::::;;; C I s -

sl

I I.A-.A I t + z'B -

]

s ~~t t

+ z'B -

I'

s

I

where the last integral is absolutely convergent. Hence

II-.oo

-+--oo

J(t + iB} dt _ t + iB - s

::::;;; - C IS

-

Sl

I-.oo -.-oo

J(t + iB2_ dt t + iB - s

I

1

I Ioo-oo I t+z'B - s t~ t+z'B -s I' I

The last integral converges to zero as B-+ oo. That is, lim(Zni)- 1

B-+oo

I-.oo

[J(t

.,..._00

+ iB)f(t + iB- s)] dt =

C1 ,

where C1 is a constant. Then from (24), we should have f(s) = (2ni)- 1 I~oo [J(t)f(t- s)] dt

+

c1.

(25)

Since the Fourier transform in £2 of a function g(u) = eisu for u > 0, g(u) = 0 for u > 0, is given by

[(2n)112i(t _ s)]-1, we see, from the Parseval relation [Section 6.2, Eq. (13)] that f(s> = (Zn)- 1' 2 I~ /(t)eist dt + c1 = (Zn)- 112 I~ /(t>e-V 1eixtdt + c1. (26)

where s = x + iy, y > 0. Now let if>(x) be the inverse Fourier transform of a function g(t) = /(t) for t > 0, g(t) = 0 for t < 0. Then by the Parseval relation,

II (Zn)- 112 I~ /(t)e-vteixt dt- if>(x) 11:

=

I~ I /(t)e-vt dt- /(t) 12 dt,

228

VI. L 1-THEORY

OF FOURIER SERIES AND FOURIER TRANSFORMS

which converges to zero as y-+ 0+. In other words, the first integral on the right of (26) converges in L 2-norm. Hence for some sequence Yn-+ 0+, it converges almost everywhere to a function belonging to L2(- oo, oo ). On the other hand, f(x) converges as y-+ 0+ to f(x), which belongs to L 1 ( - oo, oo ). From these two facts, C1 must be zero. So we get

f(s) = (n2)-1/ll J~ J(t)e-"teizt dt.

(27)

This also shows that

llf<x + iy) II~=

(Zn)- 1' 2

which gives us f(s) e

f~ lf(t> l2e-2"t dt ~ (Zn)-1' 2 f~ lf 12 dt,

~2•

6.6. A Theorem of Paley and Wiener We are going to show a theorem due to Paley and Wiener [1, 2], which is an analog of Theorem 6.4.2.

Theorem 6.6.1. Let t/J(x) be ,a nonnegative function belonging to L2(-oo, oo). In order that there exist a function f(x) e L2(-oo, oo) such that

I f(x) I =

t/J(x)

(1)

a.e.

and the Fourier transform f(t) of f(t) e L2(-oo, oo) be almost everywhere zero for t < 0, it is necessary and sufficient that

J: [I

log t/J(x)

1/(1

+x

2 )]

dx

< oo.

(2)

Because of Theorem 6.5.2, we may say that (2) is necessary and sufficient for the existence ofj(x) e L2(-oo, oo) such that (1) hold andf(x) be the boundary function of a functionf(z) e ~ 2 • This is precisely the analog of Theorem 6.4.1. Roughly speaking, (2) means that t/J(x) cannot be very small. Suppose, for instance, that t/J(x) = O[e-B
6.6.

229

A THEOREM OF PALEY AND WIENER

Proof of the Theorem. The idea of the proof is to transpose the corresponding theorem (Theorem 6.4.1) to the half-plane. Consider the transformation from the upper half-plane Im z > 0 in the z-plane onto the interior of the unit circle I w I < 1 in the w-plane: w

= (1

+ iz)/(1 -

z = i(1 - w)/(1

or

iz)

+ w).

(3)

Also by the same transformation, the real axis in the z-plane is considered to be transformed to the unit circle: z = x -+ w = ei8 by

x

=

i(1 - ei8 )/(1

+ ei8 ) = tan(0/2).

(4)

"Also, we need the transformation of the Poisson kernels: P(r, 0) to P 1 (x, y). From Section 6.4, Eq. (9),

l Re[(1

+ w)/(1 -

w)] = P(r, 0),

and hence,

l

Re[(ei~

+ w)f(ei~- w)] =

P(r, 0- 1/J).

We may easily verify that (ei~

+ w)f(ei~- w) =

(1

+ tz)fi(t- z)

it),

t real.

if ei~

= (1

+ it)/(1 -

(5)

Hence

P(r, 0- 1/J) = l Re[(1

= P 1(t-

+ iz)fi(t- z)] = l x,y) (1

+t

{y(1

+ t 2 )/[(x- t) 2 + y 2]} (6)

2 ).

Now we shall prove the theorem. First suppose (2) holds. Then 1/J(tan lO) = tp(O), belongs to L 2 ( -:n:, :n: ), since t/J(x)

E

-:n;

< 0 < :n:,

L 2 ( - oo, oo) and

Also, (2) gives us

J:,. Jlog tp(O) J dO =

2

J~oo [Jlog 1/J(x) J/(1

+x

2 )]

dx

< oo.

VI. £ 2-THEORY

230

OF FOURIER SERIES AND FOURIER TRANSFORMS

Hence by Theorem 6.4.1, there is a function y(ei 0 ) that is the boundary function of a function y(w) E H 2 , w = reiO and is such that

I y(ei6 ) I = V'(O) a.e. and y(w)

= n- 1

(7)

J:,. P(r, 0- cf>)y(ei~) dcf>.

(8)

Define

f(z) = y((1

+ iz)/(1

f(x) = y((1

- iz)],

+ ix)/(1

- ix)].

(9)

Then (7) becomes

lf(x) I = cf>(x),

(10)

and from (8), changing the variable cf> to t by (5) and noting

dcf> = 2 dtf(1

+ t2),

we see that (8) turns out to be [because of (6)]

f(z)

=

n- 1 J~oo P 1 (t - x, y)f(t) dt,

y > 0.

(11)

Since f(t) E £2( -oo, oo) (from (10)], Remark 2 after Theorem 6.5.2 concludes that f(z) E ~ 2 , and f(t) is the boundary function of f(z). By Theorem 6.5.2, ]{t), the Fourier transform of f(t), is zero for t < 0. This completes the proof that the condition is sufficient. The necessity is shown in a similar way. Suppose that (1) holds and /(t) = 0 for t < 0. Define f(z) by

f(z)

= n- 1

Then, f(z)

J~oo P 1(t E ~ 2•

X,

y > 0,

y)J(t) dt,

-oo <X< oo.

(12)

Define

y(w) = f[i(1 - w)/(1 Then y(w) is analytic in I w

+ w)],

I w I< 1.

I < 1. If we write, by (5), y(ei~)

= f(t),

then (12) turns out to be (8). Hence by Corollary 6.3.1, y(w) and y(ei~) is its boundary function,

J:,. JlogJy(ei )JJ dO < oo, 6

E

H 2,

231

NOTES

which

JS

equivalent to

J~oo [JlogJ(f(t)JJ/(1 + t2 )] dt <

oo.

This shows the necessity of the condition. NOTES 1. For more results on Fourier series in an inner product space and for the theory of closure and completeness of a sequence of functions, see Davis [1], Chapters VIII and XI. 2. The proof of Plancherel's theorem (Theorems 6.2.1 and 6.2.2) in Section 6.2 is adopted from Titchmarsh [2], in which other proofs are also found. In order to define the Fourier transform of a function of L 2 ( - oo, oo ), first consider the Fourier transform of a function of L 1 () L 2 ( -oo, oo) by the ordinary definition. Using the fact that V () L 2 is dense in L 2 ( - oo, oo) and the extension theorem of linear transformation defined in a linear dense subspace to the whole space, the definition of the Fourier transform in L 2 ( -oo, oo) will be made. For this method of proof of Plancherel's theorem, see Bochner and Chandrasekharan [1] or Rudin [1]. 3.

As is indicated at the end of Section 6.2, a general unitary transformation from L 2 ( - oo, oo) onto L 2 ( - oo, oo) was studied by several authors. Also see Hardy and Titchmarsh [1], Plancherel [2], or Titchmarsh [1]. More detailed theory of the H• class and .p• class is given in Paley and Wiener [2], and Hoffman [1]. The theory is expanded to include the class H'~', .\)'~' (p > 0), in which

4.

J~oo if(x + iy) I'~' dx are considered in place of these with p = 2. Also see Proc. Symp. in Pure Math. XI, Entire Functions etc. Amer. Math. Soc. (1968), and Porcelli [1]. 5. More precise studies were carried out in connection with Theorm 6.6.1 by Mandelbrojt and other authors (see Mandelbrojt [1 ]).

VII Laplace and Mellin Transforms

7.1. The Laplace Transform We suppose throughout this section that F(x) is a real-valued function defined over 0 ~ x < oo and is of bounded variation in every finite interval [0, R], 0 < R < oo. We write this assumption as F(x) E V(O, oo). Form

J:

e-•z dF(x),

where s is a complex number s = u

f(s)

+ i-r.

= f ~oo e-sz dF(x) = o

lim R~

If, for a given x,

JR e-•z dF(x) o

(1)

exists as a finite number, thenf(s) is called the Laplace-Stieltjes transform of F(x). If F(x) is absolutely continuous, F'(x) = 1/J(x), a.e. (if> is locally integrable), and

f(x)

=

J;oo e-•zifJ(x) dx

exists, then f(s) is called the Laplace transform of 1/J(x). 232

(2)

7.1.

233

THE LAPLACE TRANSFORM

In particular, if F(x) is given bye,

=

F(O)

-.- -t,-(

0

z-."

An <

F(x) =an,

where A0

= 0, 0 < A1 < A2

• • ·,

X

< An+l' n = 0, 1, 2, ... '

then (1) will take the form

S = U +iT. This formal series is called the Dirichlet series. If An out to be

(3)

= n, then (3) turns

00

"' a e-ns ' £.... n

n-o

which becomes, with e-a

=

r, z

=

re-i•,

00

L a.,z",

n=O

the ordinary power series. So that the Laplace-Stieltjes transform is regarded as an extension of the power series. As a matter of fact, it will be seen that the theorems analogous to those in the theory of power series will hold in many cases. We begin with the following theorem.

Theorem 7.1.1. If for some s0 = u0

J:

e-'ox

dF(x)

+ iT

~ M,

0,

O~R
(4)

for some constant M, then

J-+oo 0

exists (converges) for every s and -+00

J0

e-sx

e-sx

dF(x)

= u +iT, with u > u0 ,

-oo

dF(x) = (s - s0 ) Joo e-<•-solx G(x) dx, 0

holds, where G(x) =

J:

e-•ou

dF(u)

<

T

<

oo,

(5)

(6)

and the integral on the right-hand side of (5) is absolutely convergent (the Lebesgue integral).

VII.

234

Proof.

I:

LAPLACE AND MELLIN TRANSFORMS

Define G(x) by (6 ). Then, for s = a

I:

e-8 z dF(x) =

e-<•-•olz

dG(x)

= e-<•-•olRG(R)

+

(s - s0 )

+ iT,

I:

a > a0 ,

e-<"-solzG(x) dx.

(7)

Since I r<s-s)oR I = e-{a-aolR, r<•-•olR ---+ 0 as R ---+ oo, and I G(R) I ~ M. The first member on the right-hand side of (7) converges to zero as R---+ oo. Also,

exists as a Lebesgue integral because

Hence, letting R

---+

oo in (7) we have the convergence of

I0-+oo e-'z dF(x) as well as the validity of (5). In particular, we have:

Theorem 7.1.2. If j(s)

exists at s = a0

+ it:

0,

= I:oo e-•z dF(x)

then it exists for every s

(8)

=

a+ iT, with a> a0 •

This implies the following statement: If (8) diverges at s = s0 , then it does so for any s = a+ it:, with a < a0 • From this and Theorem 7.1.2, we may say that there are three possible cases: (i) (ii) (iii) Res>

f(s) in (8) does not exist for any s in the complex plane. f(s) in (8) exists for every s in the complex plane. There is a real a0 such that f(s) exists for every s, such that a0 , andf(s) does not exist for any s such that Res< a0 •

If (i) is the case, we define a0 = +oo and if (ii), we define a0 = -oo. Then a0 is called the convergence abscissa of the Laplace-Stieltjes transform of F(x).

7.1.

235

THE LAPLACE TRANSFORM

We easily see that the above three cases actually occur. For example, consider

F(x) =

J:

X;;:::; 0.

eu du,

Then

If this converges at s = s0 = cr0 + ir 0 , then it should converge at s cr > cr 0 , by Theorem 7.1.2. Therefore,

J

log!n+ll

e-ax,F dx- 0

as

logn

= cr,

n- oo

must be true. Now

J

(Og(n+l)

e-axee'" dx

log n

=

JetHl

(log u)-
en

as

n - oo.

This is a contradiction. Hence case (i) really occurs. Next, take F(x) = J~ exp( -eu) du. We then see that

s= There is an R 0 such that for x for R' > R ;;:::; R 0 ,

> R 0 , -crx-

JR' e-axe-•'" dx < JR' e-x dx R

R

0

C1

+iT.

ex< -lex< -x. Then

as

R-oo.

Hence the Laplace-Stieltjes transform exists for all s, that is, case (ii) occurs. Finally let F(x) = x. Then

J:

e-•x dF(x)

=

J:

e-•x dx

which converges as R - oo when Re s

=

(e-•R- 1)/-s,

> 0 and diverges when

Res

< 0.

VII.

236

LAPLACE AND MELLIN TRANSFORMS

At G = Go, the Laplace-Stieltjes transform may or may not exist. In the last example, G0 = 0 and at s = iT,

J e->
.

0

diverges for every - oo < If

< oo.

T

F(x) = then it is easily seen that

Go

=

J:

(1

+u

2 )- 1

du,

0 and

IJRR' e-irz dF(x) I~ JR'R (1 + x

2 )- 1

dx-+ 0.

Hence the Laplace-Stieltjes transform exists at iT for every T. We now give some results on the relationship between the domain of convergence of the Laplace transform and the magnitude of a function.

Theorem 7.1.3. (i) If, for some real y, F(x)

=

(9)

o(e""')

for large x, then the Laplace-Stieltjes transform

f

-+00

0

e-•z dF(x)

(10)

is convergent for Res > y. (ii) Suppose that (10) is convergent at s = s0 then

=

G0

+ iT

0•

If G 0 > 0, (11)

for large x. If

G0

< 0, then F( oo) is finite and F(x)- F(oo) = o(e<Jo"')

(12)

for large x.

Proof

(i)

We may suppose

I F(x) I ~ Me,z for 0

~ x

< oo, M being a constant. Using integration by parts,

JR e-•z dF(x) = R'

(13)

F(R')e-•R' - F(R)e-•R

+ s JRR' e-•"'F(x) dx.

(14)

7.1.

237

THE LAPLACE TRANSFORM

From (13), I F(x)r•x I ~ Mela-vlx, a= Res. Hence the first and second terms on the right-hand side of (14) converge to zero as R, R'-.. oo if a > y. Then

IJ

R'

R

I

r•xF(x) dx ~ M

JRR' e-la-ylx dx

-+

0

as R'-+ oo, R-+ oo. Therefore, the integral (10) converges for Res > y. (ii) Suppose a 0 > 0 and (10) is convergent at s = a 0 i-r0 • Set

+

G(x)

J:

=

r"o" dF(u).

(15)

We then see that G(O) = 0 and

F(x) - F(O)

=

J:

=

G(x)e•oz- s 0

dF(u)

=

J:

e'o" dG(u)

J:

e"o"G(u) du.

Since

[F(x) - F(O)]r•ox = G(x) - sue-•ox

= sue-•ox

s:

s:

e"o"G(u) du

[G(x) - G(u)]e'o" du

+ G(x)e-•oz.

(16)

The last term converges to zero because, by assumption, G( oo) exists and a 0 > 0. For any e > 0, take X so large that I G(x)- G(u) I < e for x, u ;;:;: X. Then the first term on the right of (16) is, for x >X,

sue-•ox

s:

[G(x) - G(u)]e'o" du

+ sue-•ox

J:

[G(x) - G(u)]e'o" du,

the first term going to zero as x-+ oo (X is then fixed) since the integral is bounded, and the last being not greater than

in absolute value. Hence the right-hand side of (16) converges to zero as x-+ oo.

VII.

238

LAPLACE AND MELLIN TRANSFORMS

Therefore F(x) - F(O) = o(eao"'). Since eaox--+- oo, this is equivalent to F(x) = o(eao"'). Next let o-0 < 0. Then by Theorem 7.1.2, the integral (10) exists at s = 0. In other words, F( oo) exists. We then write

F(oo)- F(x)

=

J "'-+co dF(u) = J-+co x e'o" dG(u),

where G(x) is defined by (15). Noting that e80"'G(x)--+- 0 as x--+- oo (since o-0 > 0) and applying integration by parts, we have

F(oo)- F(x) = -e'o"'G(x)- s0 J:co e'o"G(u) du. Now as before,

-+co

[F(oo)- F(x)]e-'o"' = -G(x)- soe-'o"' J "' = soe-'o"'

e'o"G(u) du

-+co J "' [G(x) - G(u)]e'o" du

since

Since G(x)- G(u)--+- 0 as u > x--+- oo, the right-hand side converges to zero as x--+- oo and we get

F(oo)- F(x)

=

o(eao"').

This shows (ii ).

7 .2. The Convergence Abscissa A power series I; anzn has a radius of convergence

e-1 =

lim sup 1 an

(!,

given by

l 11 n,

n-+co

that is, it converges in I z I < (! and diverges in I z I > €!· If a power series is written in the form I: ane-n', then this converges for Res= q > o-0 =lim SUPn-+co log I an 1/n and diverges forO" < o-0 • This is obvious from the fact that z = e-' and I z I < e corresponds to q > log e- 1 • We are going to discuss the corresponding facts for the convergence abscissa o-0 of the Laplace-Stieltjes transform

-+co J 0 e-'"' dF(x).

(1)

7.2.

239

THE CONVERGENCE ABSCISSA

As before, we suppose throughout that F(x) e V(O, oo).

Theorem 7.2.1. (i) If lim sup[log I F(x)

1/x] =;.

(2)

z4oo

is not zero ( ±oo are allowed), then ;. is the convergence abscissa a0 of (1 ). (ii) If;. = 0 and lim..,_. F(x) does not exist, then ;. is the convergence abscissa of (1). 00

Proof. (i) First we shall show that (1) converges for Res= a>;., (If ;. = oo, there is nothing to show.) From the definition (2) of ;., F(x) = o(eA'x), where ).' is any number greater than ;., Then from Theorem 7.1.3 (i), (1) converges for a> J.'. Since).' is an arbitrary number greater than;., the integral (1) converges for a > ;., Next we show that (1) does not converge for a<;., (If ;. = -oo, there is nothing to show.) Suppose ;. > 0. If ( 1) were convergent at some a, which may be assumed to be such that 0 < a < ;., then by Theorem 7.1.3 (ii), we would have F(x) = o(ea"'), which would yield logl F(x)

I ~log ceax = log c + ax

for some constant c. This would give us lim sup[log I F(x) lfx] ~a<;., X4oa

which contradicts (2). Next suppose ;. < 0. If (1) were convergent at a < ;., then from Theorem 7.1.3 (ii) (second part), we should have F(x)- F(oo) = o(ea"'), while from assumption (2) with ;. < 0, lim log I F(x) I = -oo X4oo

should have been true, which would imply F( oo) = 0. Therefore, we must have F(x) = o(ea"'). This gives us lim sup[log I F(x) 1/x] ~ a < ;. z4oo

as before, which is a contradiction.

240

VII.

LAPLACE AND MELLIN TRANSFORMS

(ii) The first part of the proof of (i) is still good for A= 0, that is, (1) converges for e1 > 0. The additional condition that F(x) not converge as x--+ oo, means that (1) does not converge at e1 = 0. Hence the convergence abscissa must be zero. REMARK 1. In Theorem 7.2.1 (ii), the condition that limz-+oo F(x) does not exist cannot be removed. This is so because, if we consider the function F(x) = 1 - e-z, then A = 0, but the convergence abscissa of its Laplace-Stieltjes transform is easily seen to be -1. This remark shows that there is a function F(x) for which the convergence abscissa a 0 of the Laplace-Stieltjes transform is negative while A given by (2) is 0. Hence, of course, we cannot conclude that e10 is given by (2). What we can conclude is the following:

Theorem 7.2.2. If the convergence abscissa a 0 of (1) is known to be nonnegative, then a 0 =lim sup[log I F(x) 1/x].

(3)

z-+oo

Proof. Actually, this is essentially implied by Theorem 7.2.1. If a 0 = 0, and the right-hand side A of (3) is different from zero, it contradicts Theorem 7.2.1 (i). If e10 > 0 and is different from A as well as from zero, this again contradicts Theorem 7.2.1. If a 0 > 0 and A= 0, then from the first part of the proof of Theorem 7.2.1 (i), we may see that a 0 ~ 0, which is a contradiction. The following theorem (due to Ugaheri [1]) is more complete, and is of interest from a theoretical viewpoint although it may be less convenient to find the convergence abscissas of functions whose forms are explicitly given.

Theorem 7.2.3. The convergence abscissa a 0 of (1) is given by

li~~P[ log

IJ;z,

dF(x)

I/

(4)

x],

where (x) denotes the integer such that x - 1

~

(x) < x.

Proof. Write the value of (4) as A, ±oo being allowed as values of A. First we show A ~ e10 • If e10 = +oo, the conclusion is obvious. Suppose (1)

7.2.

241

THE CONVERGENCE ABSCISSA

converges at a. Write G(x)

= I: e-"" dF(x). I G(x) I ~ M. Now

Then there is a constant M such that

I z(z) dF(u) = I"'(z) e"" dG(u) = G(x)e"z - G(x)e"(zl -a I"'(z) e""G(u) du, so we may write

II z

(zl

dF(u)

I ~ M(e""' + e"<"'

+ I a I M I"' e"" du M(e""' + ea(z)) + M I e""' - ea(z) I·

=

Hence

II

z (z)

1)

(z)

I { 2M e""' 2M ea
if if

dF(u) ~

In both cases,

A.= lim sup[log z-+oo

~

0, < 0.

(J (J

Ir dF(u) 1/x] ~

(J

(z)

since a can be any number greater than a0 • This implies

(5) Now we shall show

(6) If A.= oo, there is nothing to show. In order to show (6), it is sufficient to prove that for any a> A., (1) converges at s = a, where we may suppose A. < oo. Suppose A. > - oo. From the definition that A. be equated to the value of (4 ), for every e > 0,

II:zl dF(u) I~ e 0. Choose e so small that

if

x~T>O

A.+ 2e <

a, e < 1. Write

H(x) =I"' dF(u) (z)

so that

I zw e-
=

H(x)e-
+ 2e) I"'

~

e-
(7)

Because of (7), if x

IJ z

e-
~

T,

dF(u)

(z)

I~ e-•z +I A+

2e I Jz

e-•u

+ I A+ 2e I (e-•
~

e-•z

~

e-•
+I A+ 2e I] <

C being a constant. Now let x 1 and x 2 be any numbers T m ~ 0. Then we have

~

du

(z)

e-ez)/e e-•[z-
r•
(8)

x 1 < x 2 • Write (x 2 ) = (x 1 )+m,

(when m = 0, the first sum is missing), by (8), is not greater than m-1

cL

e-e[(J:l)+nl

+ Ce-•
~

e-e(zll[JC/(1 - e-•)].

n-o

This converges to zero as x 1 --+ oo. Hence

f

-+oo

o

e-
dF(u)

converges. Hence (1) converges for s = C1 if A > - oo. The above arguments continue to hold if A= -oo because (5) still holds for any negative A no matter how large I A I is, and the arguments that follow still hold. We then have the conclusion that (1) converges for any A, which means that f10 = - oo.

7.3. Analyticity of a Laplace-Stieltjes Transform We have seen in Section 7.2 that, for any Laplace-Stieltjes transform

f(s)

=

J;"" e-•z dF(x),

(1)

there is a f10{-oo ~ f10 ~ oo) such thatf(s) is defined in the half-plane Re s > f10 (if f10 = - oo, it is the whole plane, and if f10 = oo, it is

7.3.

243

ANALYTICITY OF A LAPLACE-STIELTJES TRANSFORM

empty). We are again supposing that F(x) E V(O, oo), that is, it is a function of bounded variation over every finite interval in 0 ::;;;; x < oo. In the case of a power series, we know that the function defined by a given power series is analytic inside the circle of convergence. A similar result is also obtained for the Laplace transform. In order to show this, we first prove the following result:

Theorem 7.3.1. If (1) converges at s = s0 = 0'0 + i-r 0 , -oo < then (1) is uniformly convergent in an angular domain Res= a,

0'0

<

oo,

(2)

where K is any fixed constant greater than 1. The conclusion means that

converges uniformly for s in the domain (2) as R, R' - oo.

FIGURE

7.1.

The domain (2) is the hatched domain in Fig. 7.1, which is symmetric for the horizontal line passing s0 and in which the angle 00 satisfies (1/K) = cos 00 (0 < 00 < l:n:).

Proof. The domain (2) is denoted by LJ = LJ(K, s0 ). Since (1) converges at s0 , ( 1) is convergent at any point in LJ. Set G(x) =

J:

e- 8•" dF(u).

For any e > 0, choose R 0 such that, for x, x'

~

I G(x) - G(x') I < ef2K.

R0 , (3)

Then for R' > R

~

R 0 and for any sin £1,

IR e-az dF(x) =IR e-<s-solz dG(x) =I:' e-<s-solz d[G(x) _ G(R)] R'

R'

= e-
+ (s -

s0 )

I:'

G(R)]

e- 1'-'o'"'[G(x) - G(R)] dx,

and, by (3)

IIR' e-'"' dF(x) I; ; ; (ee-
R

R

;;;;; (ef2K) + (e Is- s0 I/2K)(a;;;;; ie + ie = e.

(4)

0'0 )- 1

(5)

This is also true for s = s0 since the integral on the left-hand side of (4) reduces to (3). (5) shows that (1) converges uniformly in L1 since the choice of R 0 does not depend on s in £1. This completes the proof. Now we show the following theorem.

Theorem 7.3.2. Let a 0 (
I:oo e-sz dF(x).

Thenf(x) is analytic in the half-plane Res>

J
0'0

I0-+oo e-""'( -x)lt dF(x)

(6)

and (7)

holds in Re s > a 0 , k = 0, 1, 2, ....

Proof. L.:!t s1 be any point in Res > 0'0 • Choose s2 in such a way (Fig. 7.2) that Re s1 > Re s2 > 0'0 and s1 lies in the angular domain L1 whose vertex is s2 and whose side makes the angle a, less than n/2, at the vertex with the line parallel to the real axis. From Theorem 7.3.1, the integral (6) converges uniformly in £1. Hence

f(s) =

L 00

n-o

I"+l e-'"' dF(x)

(8)

n

converges uniformly in L1 and, in particular, in and on the circle C with center s1 , which is contained in the interior of £1.

7.3.

245

ANALYTICITY OF A LAPLACE-STIELTJES TRANSFORM

FIGURE

7.2.

Now each of the fn(s) = J:+l rsx dF(x), n = 0, 1, 2, ... , is analytic everywhere, which is seen directly from their differentiability, and actually (9)

Then 00

f(s) = Lfn(s) n-o

is analytic in C as the sum of a uniformly convergent series with analytic functions as terms. This is no more than Weierstrass' theorem, which is shown as follows. By Cauchy's theorem, for any s inside C,

.fn(s) = (2ni)- 1

fc [Jn(z)f(s- z)] dz,

the integral being taken along C in positive direction. Since I: Fn(z) converges uniformly on C,

f(s) =

n~/n(s) =

(2ni)- 1

L {[ ~~n(z)] / (s- z)} dz

= (2ni)- 1 Jc [J(z)f(s - z)] dz, which means that f(x) is analytic inside C. Hence f(s) is analytic at s1 • Since s1 is an arbitrary point in Res> u0 ,J(x) is analytic in Res> u0 • For derivatives of f(x), we may show (7) in a similar way. Again, by Cauchy's theorem, for s in C,

f~kl(s) = [( -1 )kk!f2ni)

fc

Un(z)f(s- z)k+l] dz,

k

= 0, 1, 2, ....

VII.

246

LAPLACE AND MELLIN TRANSFORMS

Therefore, by the uniform convergence of 'Lfn(z),

~~~l(s) = [(-1)kk!f2:n:i]

n-o

J [f(z)f(s- z)k+l] dz =J
Since flfl(s) is given by (9), k =

= 0, 1, 2, ...

f-+oo e-•:z:( -x)k dF(x). 0

This gives us (7).

7.4. Inversion Formulas for Laplace Transforms We now study the inversion formulas for Laplace transforms. In fact we are goi~g to show the following two theorems.

Theorem 7.4.1. Suppose that a function F(x) e V(O, oo) is standardized as F(x) = ![(x

+ 0) + F(x -

0)]

for 0 < x < oo.

(1)

Suppose that f(s) =

has convergence abscissa lim (Z:n:i)- 1 T-+oo

.

J.

c+sT

c--&T

G0

<

s;oo e-•:z: dF(x) oo. Then for any c

[f(s)fs]e•:z: ds =

l

(2)

> 0 such that c >

F(x) - F(O),

X> 0,

![F(O+)- F(O)], 0,

X=

0,

G0 ,

(3)

X< 0.

This gives us the inversion formula for a Laplace-Stieltjes transform. The inversion formula for the Laplace transform [for the case where F(x) is absolutely continuous] is given by the following theorem.

Theorem 7.4.2. Suppose that cf>(x) is locally integrable in (0, oo) and (4)

7.4.

247

INVERSION FORMULAS FOR LAPLACE TRANSFORMS

has convergence abscissa a0 < oo, and 1/J(x) is of bounded variation in a neighborhood of x ;:;;; 0. If c is any positive number greater than a 0 , then

lim (Zni)- 1 T-+oo

c+iT + 0) + 1/J(x- 0)], Ic-tT . f(s)e•x ds = { 0H
,

X> 0, X=

0, (5)

X< 0.

We shall first prove Theorem 7.4.2, and then, using Theorem 7.4.2, prove Theorem 7.4.1.

Proof of Theorem 7.4.2.

Writings= c + i-r,

(2ni)- 1 I c+iT . f(s)e•x ds

=

(2n)- 1 IT f(c

=

(Zn)-1

c-.T

.

I

-T

T

-T

. + i-r)e
e
I-+00 e-
-oo <X< oo.

(6)

By Theorem 7.3.1, the inner integral is uniformly convergent on the segment c + iT, I T I ~ T (c > a 0) (see Fig. 7.3). Therefore the last integral in (6) can be written

(2n)- 1 I 0-+oo 1/J(u) du IT-T exp[(c

+ i-r)(x- u)] d-r,

which is equal to

(2n)...: 1 I:oo 1/J(u) exp[c(x- u)][sin T(x- u)f(x- u)] du. Choose a0 '

(7)

> 0 such that c > a0' > a0 • Then from Theorem 7.1.3 (ii), F(x)

=I:

1/J(u) du = O[exp(a0 'x)]

FIGURE

7.3.

248

VI I.

LAPLACE AND MELLIN TRANSFORMS

for large x. Now write, for every fixed x,

r/>(u, x)

=

r/>(u)

I:oo [
=

Actually this exists and

r/>(u)

=

O[exp(a0 ' - e)u]

(8)

for large u. This is seen in the following way

I

R
X-V

u

X-V

=

+IR F(v) [- exp e(x-v)+eexpe(x-v)]dv u (x- v) 2 x- v O[exp(a0 ' - e)R]

tfX -=~-=--='----'---=-

R-X

+ exp ex

O[exp(a0 '

-

e)u]

---''----''-'-=---------''----"-

u-x

+ exp ex I: O[exp(a

0' -

e)v] dv.

From this, letting R--+ oo, (8) is immediate. Now

T(x - u) du I RR'
=

-r/>(u) sin T(x- u) lu=R

= O[exp -(e- O'o')R]

R'

+ T I R r/>(u) cos T(x- u) du

+ r[ r/>(R) I: cos T(x- u) du + r/>(R')

I: cos T(x- u) du]

[by the second mean value theorem, r/>(u) being of bounded variation on every finite interval]

T(x - u) du I RR'
-(e- a 0 ')R]

= O[exp

-(e- a 0 ')R]

+ r/>(R) 0(1) + r/>(R') 0(1)

(Note here that 0 is independent of T.)

7.4.

249

INVERSION FORMULAS FOR LAPLACE TRANSFORMS

Hence for any e > 0,

I -!

fRo cp(u) exp c(x- u) sin;~-;: u)

I

du < e

for some R0 (;;:;; I x I + 1) which is independent of T. Now from Fourier's single integral theorem [Theorem 4.3.1 (ii)],

fRo cp(u) exp c(x- u)

. 1 hm ,.,_

T~oo ~

(where cp 1 (u)

X-U du fRo cp (u) exp c(x- ·u) sin T(x- u)

0

. 1 hm -2 T-+oo :n;

=

=

sin T(x- u)

1

-Ro

X -

cp(u), u ;;:;; 0 and cp 1 (u)

lim ,.,~

fR

du

u < 0)

cp(u) exp c(x- u) sin T(x- u) du

X-U

o

T~oo ~

=

= 0 for

U

Hc/JI(x

+ 0) + cp (x 1

0)],

which is just the right-hand side of (5). This completes the proof of Theorem 7.4.2. Proof of Theorem 7.4.1.

f:

Since F(R) gives us

=

e-'z dF(x)

For any Res=

=

CT

e-'RF(R) - F(O)

> 0,

+s

CT

f:

>

CT0 ,

e-szF(x) dx.

o(eaR), R-oo from Theorem 7.1.3 (ii), the above equality f ~oo e-sz dF(x) = -F(O) 0

+ s f-+oo e- zF(x) dx 0

8

~oo

f0

= s

e-'z[F(x) - F(O)] dx,

that is, if we write F 1 (x)

=

F(x) - F(O),

(9)

then ~oo

f(s)fs

=

f0

e-'zF1 (x) dx.

(10)

Actually this integral is absolutely convergent for Re s > CT > 0 for any CT > CT0 since F 1(x) = o(eaz). (Since CT is arbitrary, we of course have that the integral (10) is absolutely convergent for Res= CT > 0, CT > CT0 .)

If we apply Theorem 7.4.2 to (10), we have

lim (2ni)- 1 I T-+oo

c+iT . [f(s)fs]esz ds c-oT

=

l[F1(x

+ 0) + F (x- 0)] 1

for x > 0. But

l[F1 (x

+ 0) + F (x- 0)] = 1

l[F(x

+ 0) + F(x- 0)]- F(O)

= F(x) - F(O).

For x = 0, x < 0, it is easy to see that (3) is true. In Theorems 7.4.1 and 7.4.2, we have taken c > 0 in (3) and (5). If cr0 < 0 and we choose c such that cr0 < c < 0, we may still have some sort of inversion formulas.

Theorem 7.4.3. Let F(x)

V(O, oo) and be standardized. Suppose that (2) has convergence abscissa cr0 < 0. Then f6r any c such that cr0 < c < 0, X> 0, . { F(x)- F(oo), , C+IT hm (2ni)- 1 I . [f(s)fs]e'z ds = ![F(O) + F(O+)]- F(oo), X= 0, T-+oo c-IT F(O) - F( 00 ), X< 0. (11) E

This theorem will be obtained from the following theorem in which F(x) is absolutely continuous. We need the conditions in order to appeal to the inversion formula for the Fourier transform.

Theorem 7.4.4. Suppose that if>(x)e-cz E £1(0, oo) and if>(x) is of bounded variation in a neighborhood of a point x ( ~ 0). Then lim (2ni)-1 Ic+iT f(s)e'z ds T-+oo

c-iT

= { Hif>(x + 0) + if>(x- 0)], tt/>(0+),

X> 0, (12) X=

0.

For x < 0, under only the condition that if>(x)e-cz E £1(0, oo), lim (2ni)- 1 IT f(s)e'z ds = 0. T-+oo -T

(13)

In this theorem, c may be negative or zero. If c > 0, Theorem 7.4.2 is the better result.

7.4.

251

INVERSION FORMULAS FOR LAPLACE TRANSFORMS

Suppose cf>(x)e-cz e £1(0, oo). Then

Proof of Theorem 7.4.4.

s = c +it, is the Fourier transform (as a function of t) of a function

X> 0.

(2n)II2r zcf>(x) g(x) = { 0 '

X

'

<0,

Therefore, from Theorem 4.3.4 (the inversion formula for the Fourier transform), if cf>(x)e-cz is of bounded variation around x,

(2n)-11 2 lim T-+oo

JT

f(c

-T

+ it)eizt dt =

![g(x

+ 0) + g(x- 0)].

(14)

If x > 0, then the right-hand side is

(2n)1 12 t[cf>(x

+ 0) + cf>(x -

O)]e-cz

and the left-hand side is

(2n)-112i-1 lim T-+oo

I

r+iT

. f(s)e•z ds e-cz.

c-IT

Hence (14) gives the first relation of (12). If x = 0, then the right-hand side is i cf>(O+), from which we have the second relation of (12). If x < 0, then g(x) = 0 and this gives us (13). In this case g(x), being 0 for x < 0, is always of bounded variation and the condition on c/J(x) is not needed except for the integrability of cf>(x)e-cz over (0, oo). Now we turn to the proof of Theorem 7.4.3. In order to do this we need the following simple relations. If c < 0, then ,

hm (2ni)-1 T-+oo

0'

J . (e'zfs)dx= { -i, -1 C+iT

c-•T

'

X> 0, X=

0,

(15)

X< 0.

This is shown in the following way. Consider the function g(x) = 0 (x > 0), g(x) = e-cz (x < 0), where c < 0. Then the Fourier transform l(t) of g(x) is

(2n)-112

J:oo e-cze-itz d.'C =

_ [(2n)I12(c + it)]-I.

VII.

252

LAPLACE AND MELLIN TRANSFORMS

Hence by the inversion formula (Theorem 4.3.4 ),

IT

lim (2n)- 1

T~oo

0,

[ei 1"'f(c +it)] dt

= { -!,

-e ~x ,

-T

X> 0, X=

0,

X< 0,

from which (15) is obtained.

Proof of Theorem 7.4.3. Because of Theorem 7.1.3 (ii), F(oo) exists and F(x)- F(oo) = o(e""') for any G such that Go< G. Therefore, taking G so that G0 < G < c, we see that [F(x)- F(oo)) e-cx E £1(0, oo).

(16)

Nowfors = c +it,

I: e-•x dF(x) =I: e-'"' d[F(x)- F(oo)) = e-""'[F(x)- F(oo)] 1: + s I: [F(x)- F(oo)]e-•z dx. Therefore,

f(s)

=

I0~oo e-•x dF(x)

=

F(O)- F(oo)

+ s Ioo [F(x)- F(oo)]e-•z dx, 0

where the last integral is absolutely convergent because of (16) and can be written in the form {f(s)- [F(O)- F(oo)]}fs =I~ [F(x)- F(oo)]e-""' dx. Since F(x)- F(oo) belongs to V(O, oo), Theorem 7.4.2 applies to get lim

T~oo

(2ni)- 1 Ic+~T ({f(s)- [F(O)- F(oo)]}fs)e""'ds c-&T

F(x)- F(oo),

= { ![F(O+)- F(oo)), 0,

X> 0, X=

0,

X< 0.

Taking (15) into account, this relation is no more than (11 ).

7.5. The Laplace Transform of a Convolution We have already defined (Section 3.9) the convolution of two functions belonging to M or M 0 • We are now interested in the convolutions of functions defined over [0, oo) that are of bounded variation over every finite interval.

7.5.

253

THE LAPLACE TRANSFORM OF A CONVOLUTION

Let F 1(x) and F 2 (x) be two such functions. If we define F 1 and F 2 for x < 0 to be zero, then the convolution becomes, at least formally, !:

Jx F (x -

l

~ ;-r,,: ,

/'

7: :tl (u,..

'

fo1Jt-r£.

1

/

(/f..lr~ /]~~;,...., :

u) dF2 (u),

X> 0.

(1)

0

+

Actually this definition is valid even if F 1 and F 2 are not bounded at oo. So we set up the definition of convolution: Let F 1 (x) and F 2 (x) be functions defined over [0, oo) and of bounded variation over every finite interval in [0, oo); that is, F 1 (x) and Fix) belong to V(O, oo ). The function ( 1), with the value 0 at x = 0, is called the convolution of F 1(x) and F 2 (x) and is denoted either by F 1 • F 2 or by (F 1 • F 2 ) (x),which is defined over [0, oo). Moreover, we assume throughout this section that any function F belonging to V(O, oo) satisfies F(O)

=

(2)

0,

_F(x) = l[F(x

+ 0) + F(x- 0)]

(3)

for x > 0. We denote the class of functions satisfying (2) by V 0 (0, oo). Then as before, we have

(4) (5)

where F 1 , F 2 , F 3 E V0 (0, oo). We also recall that, if VF(x) is the total variation of F(x) over [0, x], then generally we can write (see Section 4.9)

VF(x) = and

J:

a(x) dVF(x) =

J:

J: I

dF(u)

a(x) I dF(x)

E

1"0 [0, oo)

I

1.

0

~a<

b,

where a(x) is any bounded measurable (possibly complex-valued) function over [a, b]. First we shall show the following theorem.

Theorem 7.5.1. If F 1 (x), F 2 (x) E V0 (0, oo), then F 1 • F 2 also belongs to V 0 (0, oo) and

VII.

254

LAPLACE AND MELLIN TRANSFORMS

Here V0{0, oo) is the class of functions vanishing at zero which have a finite total variation over the whole half-line [0, oo ).

Proof.

Write G(x) = (F1

•

F 2 ){x). Let

be any finite number of points. Suppose that F 1{x) is extended for x < 0 so that for x < 0. Then, obviously for any u > 0 and any positive integer n

Now n-1

L I G(xH1) -

i-o

=

=

n-1

~0

~1

&-0

G(xi)

I

JZt+l F1(xi+1- u) dF (u) - JZt F 1(x 2

0

IJoo F1(xi+l -

~ Joo ~1 I F 1(xi+ 1 0

u) dF2(u) -

0

0

1 -

u) dF2 (u)

Joo F 1(xi- u) dF2(u) I 0

u)- F 1(xi- u)

t-O

II dF2(u) I·

By (7), this is not greater than

which gives us the required proof. We are going to discuss the Laplace transform of the convolution of two functions of V0 (0, oo). In doing so we need the following definition. If

J~ e-az I dF(x) I <

oo,

a being real, then the Laplace-Stieltjes transform

f(s) =

f

-+oo

0

e-•z dF(x),,

(8)

7.5.

255

THE LAPLACE TRANSFORM OF A CONVOLUTION

which is convergent at s = G + i-r, is called absolutely convergent at s = G + iT. In this case we write f(s) = J~ e-"'" dF(x).

Theorem 7.5.2. Let F 1 (x), F 2 (x) e V0 [0,

=>·

If

(9) and (10) are absolutely convergent at s0 , then (11)

where F = F 1 • F 2 , and the right-hand side integral, the Laplace-Stieltjes integral of F, is absolutely convergent, and we have

where Re s0 = Jlroof.

G0 •

~rite

(13)

and define

A(x,

G0 )

is the convolution of A 1(x, A(x,

G0 )

G0)

= [A 1

•

and A 2(x, A 2 ](x,

G 0 ),

G 0 ).

or (14)

VII.

256

LAPLACE AND MELLIN TRANSFORMS

Using integration by parts,

I: [

A(x, s0 ) =

e-•olz-ulF1 (x - u) dF 2 (u)

+ s I~" e-•ovF (v) dv] e-'o" dF (u)

=

I:

0

1

2

e-'o"'F1(x- u) dF2(u)

+ So f"' e-'o" dFz(U) I"' e-•olw-u) Fl(w • 0

e-'o"'F(x)

=

I: I:

+s

0

= e-'o"'F(x) + s0

0

e-•ow dw

I:

u) dw

F 1 (w - u) dF2(u)

e-•owF(w) dw.

=I: e-•ow dF(w).

(15)

In particular,

A(x, a 0 ) =

I:

(16)

e-aow dF(w).

Therefore, by Theorem 7.5.1 and in view of (16) and (14),

I~ e-aow I dF(w) I ~I~ e-aou I dF1(u) I I~ e-aou I dF2(u)

1,

which is (12). Next consider which we write as

Since A 2(x, s0 ) X-+ oo.

A(x, s0 )

-

-+ / 2 (s0 )

/ 1

as x

-+

oo, the last term converges to zero as

(s 0 )A 2 (x, s0 )

=I: [ I~" e-•ov dF (v)] e-'o" dF (u)- I: J (s )e-'o" dF (u) = =

I:

1

[A 1(x- u, s0 )

I + I"' X/2

o

z/2

= /1

-

2

/ 1

(s0 )]e-'o" dF2 (u)

+ /2,

X> 0,

1

0

2

7.5.

257

THE LAPLACE TRANSFORM OF A CONVOLUTION

(17) For any E > 0, take X 0 so large that for x > X 0 , I A 1(x, s0 ) Now let x > 2X0 • Then, since x-u> xf2 in (17),

I A1(x -

u, So) - /1(s0) I <

-

/ 1

(s0 ) I <

E.

E.

Consequently,

Hence 1 1 --+- 0 as x--+- oo. We see, for / 2 , that

I l2l ~ J;~;z/2 I A1(x -

u, So) -jl(so)

I e-aou I dF2(u) I

~ [max I Al(v, O"o) I+ 1/l(so) 11 o;;>v
r

e-aoU

z/2

I dF2(u) I·

The multiplying factor in this expression is finite and the integral converges to zero as x --+- oo since / 2 (s0 ) is absolutely convergent. Hence, if x--+- oo, / 2 -0. Together with what we obtained above,

as x

--+-

oo. Therefore we have (18)

Since, from (5), A(x, s0 )--+- f(s 0 ), (18) gives us (11). This completes the proof. We may check this proof to find that the following theorem holds.

Theorem 7.5.3. If the Laplace-Stieltjes transforms

are convergent and one of them is absolutely convergent, then

J

-+00

0

e-'oz dF(x)

VII.

258

LAPLACE AND MELLIN TRANSFORMS

exists and (19)

If F 1 (x) and F 2 (x) are absolutely continuous and

then Theorems 7.5.2 and 7.5.3 take the following forms.

Theorem 7.5.4. Suppose that l/J 1 (x) and l/J 2(x) belong to V over every finite interval in [0, oo) and write (20) If e-•ozifJ 1 (x) and e-•ozifJ 2 (x) belong to V(O, oo), then e-•ozifJ(x) E V(O, oo) and

Theorem 7.5.5. Suppose that l/J 1 (x) and l/J 2 (x) belong to V over every finite interval in (0, oo) and

are convergent. If one of the integrals of / 1 (s0 ) and / 2(s0 ) is absolutely convergent, then

f

-+oo

0

e-•ozifJ(x) dx

is convergent and (22) where l/J(x) is given by (20).

7.6.

259

OPERATIONS ON LAPLACE TRANSFORMS

7.6. Operations on Laplace Transforms and Some Examples We begin with the following simple theorem.

Theorem 7.6.1. Let F(x) belong to V 0 (0, oo). (i) If the Laplace-Stieltjes transform f(s) of F(x) converges m Res > c, then the Laplace-Stieltjes transform f 1 (s) of Fl(x) =

{~(ax- b),

x ~ bfa x < bfa

(a> 0, b

~

0),

(1)

converges for Re s > ca and (2)

(ii) If the Laplace-Stieltjes transform f(s) of F(x) converges in Res > c, then f(as +b), a > 0, is the Laplace-Stieltjes transform of (3)

for Res > (c- b)fa. Proof of (i ).

JoR e-•x dF1(x)

=

JR e-•x dF(ax- b) b!a

=

J:R-b e-•lx+bl!a dF(x)

=

e-sbta

IaR-b e-sxta dF(x), 0

from which (i) is immediate. Proof of (ii ).

J: e-•x dF1 (x) = J: e-•x d[

J:

e-buta dF(ufa)]

= J: e-'ze-bzta dF(xfa) =

from which (ii) is obvious.

faR e-!as+b>x dF(x), o

VII.

260

LAPLACE AND MELLIN TRANSFORMS

For convenience we also give similar results for the Laplace transform

f(s)

=

f

-+oo

0

e-•xcp(x) dx.

(4)

Theorem 7.6.2. Let cp(x), x > 0, be locally integrable. (i) If the Laplace transformf(s) of (4) converges for Res> c, then the Laplace transform f 1 (s) of

tPI(x) = { cp(ax- b), 0,

x ~ bfa, x < bfa,

(a> 0,

b

~

0),

(5)

converges for Res > cfa and is given by

(ii) If the Laplace transformf(s) in (4) converges for Res > c, then the Laplace transform of (6)

isf(as +b), a> 0, for Res> (c- b)fa. The proofs will be left to the reader. Now we consider the Laplace transforms of the integrals or derivatives of a function. Let cp(x) be locally integrable in (0, oo) and let its n-tuple integral 1 fx 1 f 0x dxn-I fx,._ (7) dxn_ 2 • • • 0 cp(u) du 0 be denoted by

u: r

du cp(u).

Theorem 7.6.3. If the Laplace transform f(s) [in Eq. (4)] of cp(x) is convergent for Res> c ~ 0, then the Laplace transformfn(s) of

u:

r

du cp(u)

is convergent in Re s > c and j,.(s) = s-"f(s),

n;;;;; 0.

(8)

7.6.

261

OPERATIONS ON LAPLACE TRANSFORMS

Proof.

I:

It is sufficient to prove the case n = 1.

e-sz dx

I:

t/J(u) du =

I:

r!J(u) du

= s- 1

e-•z dx

e-'"t/J(u) du - s- 1e-sR

=

11- 1z,

~

0, from Theorem 7.1.3 (ii),

say. Let Rs >c. Since c

I:

I:

I:

r!J(u) du = o(e"R),

I:

t/J(u) du

a= Rs,

as R - oo. Hence 12 - 0. 11 converges for Rs > c by assumption. This shows the theorem.

Theorem 7.6.4. Suppose that t/J(x) has the (n - 1)th derivative t/Jin- 1l(x), which is absolutely continuous in [0, oo). If the Laplace transform fn(s) of t/J 1nl(x) which exists almost everywhere, converges for Res > c ~ 0, then the Laplace transforms of r!Jin- 1l(x), ... , t/J'(x), t/J(x) converge also for Re s > c and fn(s) = snf(s) - t/J(O)sn-1 - t/J'(O)sn-2 - ... - r!Jin-ll(O),

(9)

wheref(s) is the Laplace transform of t/J(x) given by (4).

Proof. It is sufficient to prove the theorem with n is absolutely continuous. For Re s > c ( ~ 0),

=

1. Suppose t/J(x)

From Theorem 7.1.3 (ii), t/J(R)e-•R = 0(1) as R-oo. Therefore the above equation becomes, as R - oo,

which is (9) with n = 1. Now we give some simple examples of Laplace transforms. Let t/J(x) be locally integrable on (0, oo) and let f(s) be the Laplace transform:

f(s) = for some right half-plane.

I:oo e-•zr!J(x) dx

VII.

262

LAPLACE AND MELLIN TRANSFORMS

By direct computation, the following results (i), (ii), and (iii) are immediate. (i)

For cf>(x) = 1, 0 < x < oo, Res> 0.

f(s) = 1fs, (ii)

For cf>(x) = 0, 0 < x < a, cf>(x) = 1, x

f(s) = e--fUfs, (iii)

(10) ~

a,

Res> 0.

(11)

For cf>(x) = eaz (a real),

f(s) = 1/(s- a),

Res> a.

(12)

We also see (iv)

For a(x) =

X"- 1,

a > 0,

f(s) = F(a)fs«,

Res> 0.

(13)

(For s", we take a branch such that s" = 1 at s = 1.) This is shown as follows:

JoRe-•zxiZ-1 dx = f'Ro e-"(uiZ-lfsa) du,

Res> 0,

where the path of the integration of the right-hand integral is the segment joining 0 and the point sR in the complex domain (see Fig. 7.4). Because sR

FIGURE

7.4.

of the Cauchy integral theorem, the right-hand integral is (see Fig. 7.4) (14)

where the second integral comes from the integral of e-"(u 1fs along the segment joining A and sR. Note that I () I < 7&/2 since Re s > 0. 11 -

11 )

7.6.

263

OPERATIONS ON LAPLACE TRANSFORMS

We may easily show that the second integral in (14) converges to 0 as A-+ oo. Hence

for a> 0 by the definition of the gamma function. This shows (iv). Write (13) in the following form, letting s = a+ it, where a > 0:

J~ e-
(15)

Note that the left integral is an absolutely convergent integral which, being multiplied by (2n)- 112, is thought of as the Fourier transform of the function

e-azX"-1 g(x) = { 0, ,

X> 0, X

(16)

~0.

Hence, from the inversion formula for the Fourier transform (Theorem 4.3.4) [note that g(x) is of bounded variation around any x =1=- 0], we have the following result : For a> 0, a> 0, lim 7'-+oo

I

T

-7'

[eitzJ(a +it)"] dt =

{

[2nfF(a)]e-azx"- 1

X> 0,

0,

X< 0.

'

(17)

If a > 1, then the left-hand integral may be replaced by

J:

[eitzj(a +it)"] dt.

If we take the real and imaginary parts of both sides of (15), we have

J~ e-az cos tx x"- 1 dx = [F(a)f(a2 + t 2 )"' 2] cos[a arctan(t/a)],

(18)

J~ e-az sin tx x"- 1 dx = [F(a)f(a2 + t 2 )"' 2] sin[a arctan(t/a)],

(19)

where arctan is the principal value, -n/2 < arctan u < n/2 and a > 0, (] > 0, - 0 0 < t < oo. We are now going to let a-+ 0 in (18) and (19). Suppose now that 0
VII.

264

LAPLACE AND MELLIN TRANSFORMS

second mean value theorem, we have, for any A > u-P creasing for x > 0):

f

A a-P

e-azxrr.- 1 cos xt dx

=

exp(- u 1-P) u-P
=

J"

a-P

(e-azx~- 1

is de-

cos xt dx

A)

exp( -u1-P) u-P
which for a fixed t =F 0 converges to zero as u That is,

fa-Poo e-azxrr.- cos xt dx

=

1

--+-

La-P'

0 uniformly in A.

o(1).

(20)

Now consider I =

J

,-P

0

e-azx"- 1 cos xt dx -

fa-P 0

x"- 1 cos xt dx,

and

as since p

<

+ 1)-

(a

1•

u--+- 0+

(21)

Putting (20) and (21) together we see that

f 0oo e-azx"- 1 cos xt dx - fa-P x"- 1 cos xt dx --+- 0 0

(22)

as u--+- 0+. Now, since

J x"B

A

1

cos xt dx

=

A"-- 1

J cos xt dx--+- 0 e'

as A-+-oo

A

for t =F 0 (the second mean value theorem), A

f-+00 x"0

1

<

~'

< B, the integral

cos xt dx

exists. Therefore from (22),

f -+oo x"- 1 cos xt dx = 0

lim n-+0+

foo e-azx-rr.t cos xt dx 0

7.7.

265

THE BILATERAL LAPLACE-STIELTJES TRANSFORM

for t ::;C 0; that is, by (18), f -->-oo x~- 1 cos tx dx

= lim [F(a)f(u 2 + t 2 )~' 2 ] cos[a arctan(t/u)] a-->-0+

0

= [F(a)/J t J~] cos(na/2),

0
(23)

t > 0, 0 < a < 1.

(2+)

Similarly, we have J:oo x"- 1 sin xt dx

= [F(a)ft"] sin(na/2),

Adding these, or using (15) directly, we also see that -->-00 f o x"-1eixt dx = [F(a)ft"]e-i'""l2>,

In particular, if a f

+OO 0

= i, we have, since

(cos xfx 112 ) dx

f-->-00

=

0

0
t > 0, F( l)

(25)

= n 112 ,

(sin xfx 112 ) = (n/2)- 1' 2 .

(26)

or -->-oo f 0 cos x 2 dx

=

f+oo 0

sin x 2 dx

= l(n/2)112.

(27)

These are called Fresnel integrals.

7.7. The Bilateral Laplace-Stieltjes Transform Suppose throughout that F(x) E V( -oo, oo), that is, F(x) is a function of bounded variation over every finite interval in (- oo, oo ). For convenience, F(x) is supposed to be standardized as

F(x) = l[F(x

+ 0) + F(x -

0)],

-oo

<X<

00.

The integral

f(s) = J-->-oo e-•x dF(x), -+-oo

s=

0'

+iT:,

(1)

is called the bilateral Fourier-Stieltjes transform of F(x), where a and are real numbers and where (1) means

J e-•z dF(x), R-->--oo R lim

R'-+oo

R'

T

(2)

266

VII.

LAPLACE AND MELLIN TRANSFORMS

which exists for some x, or more precisely, both integrals

-+oo I 0 e-•z dF(x)

and

converge for some common s.

Theorem 7.7.1. If (1) converges at s1 = <1 1 + i1: 1 and s2 = <1 1 < <1 2 , then it converges in the strip <1 1
Since

<1 2

+ i1:

2,

-+oo I 0 e-•z dF(x)

converges at s = s1 , s1 = 7.1.2.

<1 1

+ iT

I o-+-oo e-•z dF(x)

1,

it does so for Res >

= -

<1 1 ,

by Theorem

I-+oo e-<-slz dF( -x) 0

converges at -s = -s2 = -<1 2 - i1: 2 , and hence, again by Theorem 7.1.2, it converges in Re(-s) > -<1 2 or in Res< <1 2 • Hence in the common domain <1 1 < Re s < <1 2 , (1) converges. This shows the theorem.

If <1 1 = -oo or <1 2 = +oo, the strip of convergence becomes the whole plane or some half-plane. It could happen that ( 1) converges on a single vertical line Rs = <1 1 • For instance take

F(x) = I~ (1

+u

2 )- 1

du.

Then

f(s) = J:oo [e-'z/(1

+x

2 )]

dx,

which converges on Rs = 0, but never converges for s with Rs #- 0. If

F(x) =

{I:J:

duf(u

+ 1)",

duf(i u I + 1)",

x ~o.

X< 0,

and 0
I:oo [e-'z/(1 + x)"] dx + J:oo [e'z/(1 + x) ] dx.

7.7.

THE BILATERAL LAPLACE-STIELTJES TRANSFORM

267

Each of these integrals diverges at s = 0, but each converges at Rs = 0, s =1= 0 because of Section 7.6, Eq. (24). It is obvious that (1) does not converge for any s with Rs =I= 0.

Theorem 7.7.2. For the Laplace-Stieltjes transform ( 1), there are the following three possibilities: (i) There are 0' 1 < 0' 2 , such that /( s) in ( 1) con verges in O't < Rs < 0'2. (ii) f(s) does not converge at any points. (iii) f(s) does not converge at any points except at some or all points on a vertical line. This is immediate from the above considerations. If in (i), a 1 = -oo, a 2 = oo, then the convergence domain/(1) is the whole plane. If one of 0' 1 and 0' 2 is infinite, then the convergence domain is a half-plane. Note that if F(x) = x, J:oo r•:e dx converges for Res > 0, and J~oo r•:e dx converges for Re s < 0. And for Re s = 0, both diverge. Hence J::'oo r•:e dx diverges everywhere. From Theorem 7.3.2, the following theorem is immediate.

Theorem 7.7.3. If (1) converges in a 1
Theorem 7.7.4. If lim sup logJ F(x) Jfx

:e-+oo

= A1 =I= 0,

lim sup logJ F(x) Jfx = A2 =I= 0,

(3)

z-+-oo

and A1 < A2 , then ( 1) converges for A1 < Re s < A2 and diverges for Re s < A1 and Re s > A2 • We also get the inversion formulas for the bilateral Laplace transform from theorems given in Section 7.4.

Theorem 7.7.5. Suppose that (1) converges in the strip a 1 < Res< a 2 • (i) If 0' 1 < c < -oo < x < oo,

0' 2

lim(Zni)- 1

7'-+oo

and c < 0, then F(-oo) is finite and for all

Jc+iT c-iT

[J(s)fs]e•zds = F(x)- F(-oo).

(4)

VII.

268

(ii)

If a 1

LAPLACE AND MELLIN TRANSFORMS

< c < a 2 , and c > 0, then F( oo) is finite and for all

-oo <X< oo,

c+iT

lim (2ni)- 1 J . [J(s)fs]e•x ds

T~

c--tT

= F(x)- F(oo).

(5)

Recall that F(x) E V(-oo, oo) and is standardized. For absolutely continuous F(x), we have, as we had for Theorem 7.4.4, the following theorem. ~(x)rcx ELl( -oo, oo) and ~(x) is of bounded variation in a neighborhood of a point x (- oo < x < oo ). Then c+iT lim (2ni)- 1 (6) . f(s)e•x ds = t[~(x + 0) + ~(x - 0)].

Theorem 7.7.6. Suppose that

f

T~

c-oT

Here we give the proof of Theorem 7.7.5. Theorem 7.7.6 can be shown in quite the same way as in the proof of Theorem 7.4.4. Proof of Theorem 7.7.5.

f(s) =

Write

J:: f

r•x dF(x)

f1(s)

+f

-+oo

=

0

=

r•x dF(x)

2

+ Jo

+-oo

r•x dF(x)

(s),

say. Then both f 1(s) and f 2 (s) converge in a 1 < Res < a 2 , and hence f 1(s) converges in Res> a 1 and f 2(s) converges in Res< a 2 • Let c > 0, a 1 < c < a 2 • Then from Theorem 7.4.1,

"T lim (2ni)- 1 r~

T~

c--tT

[/

1(s)fs]e•x ds =

{ F(x) - F(O), t[F(O+)- F(O)], O•

X> 0, 0,

X= X

<0.

[In fi(s), F(x) will be defined to be zero for x < 0.] Now

fc+iT [/ (s)fs]e•x ds = 2

c--iT

(writing -s for s) =

J-c-iT [/z( -s)fs]r•x ds -c+iT

f-c+iT [ -f -c--iT

2(

-s)fs]e•<-x> ds,

(7)

7.8.

269

MELLIN-STIELTJES TRANSFORMS

where - /2 (

fo

-s) = -

~-~

e+s.r: dF(x) =

f-+oo e-•.r: dF( -x). 0

So - /2 ( -s) is the Laplace-Stieltjes (one-sided) transform ofF( -x), which converges for Res> -a2 • Since -c > -a 2 and -c < 0, we get, from Theorem 7.4.3, lim(2ni)- 1 7'-+oo

=

f

c+iT

[ /2 . c->T

lim(2ni)- 1 T-+oo

f

(s)fs]e•.r: ds

-c+iT •

[ /2 (

-s)fs]e•<-.r:l ds

-c;T

F(+x)- F(-oo)

for for for

= { l[F(O) + F(O-)]- F(-oo) F(O)- F( -oo)

-x > 0, x = 0, -x < 0.

(8)

Adding (7) and (8), we find that the left-hand side of (4) is l[F(x + 0) + F(x -- 0)] - F( -oo) for any x. This gives us (4). (5) is proved in just the same way.

7.8. Mellin-Stieltjes Transforms Suppose that F(x) is a function defined over (0, oo) and belongs to V(O, oo), that is, F(x) is of bounded variation over every finite interval [a, P] (0
m(s) =

f

-+oo

-+0+

xs- 1 dF(x).

(1)

If this converges for some s, m(s) is called the Mellin-Stieltjes transform of F(x). Here s = a+ iT, a and T being real. Write x =ell. If x e (0, oo), then y e (-oo, oo). We write F(x) = F(e11) = G(y). Then G{y) is defined over (-oo, oo) and G{y) is easily seen to be of bounded variation over every finite interval in (- oo, oo); that is, G(y) e V oo, oo). Suppose as before that F( x) is standardized:

<-

F(x) = l[F(x

+ 0) + F(x- 0)],

0 <X<

Then G(y) is also standardized. Now, if we consider H(y)

=I: e-ll dG(y),

-00


00.

VII.

270

LAPLACE AND MELLIN TRANSFORMS

then H(y) also belongs to V <- oo, oo) and is standardized. Then (1) becomes

m(s) =

f

-+oo

-+-oo

(2)

e'V dH(y).

Hence the Mellin-Stieltjes transform is reduced to the Laplace-Stieltjes transform, with a slight difference of notation (s instead of -s). Therefore we have, from the theorem for the bilateral Laplace-Stieltjes transform, the following theorem.

Theorem 7.8.1. For the Mellin-Stieltjes transform m(s) in (1), there are three possibilities: (i) There are 0' 1 < 0'2 such that m(s) converges in the strip 0' 1 < Res < 0' 2 and m(s) is analytic there. (ii) m(s) does not converge at any point s. (iii) m(s) does not converge at any point s except at some or all points on a vertical line. For the inversion we have from Theorem 7.7.5:

Theorem 7.8.2. Suppose that (1) converges in (i) If 0'1 < c < finite number and lim (Zni)- 1 T-+oo

Jc+iT.

0' 2

[m(s)f( -s)]x-'+1 ds 0' 2 ,

< Res <

=

F(x) - ax,

c

0 < x < oo.

(3)

< 0, then limz-+o+ F(x)fx = Pexists as a finite

1

lim (Zni)- 1 Jc~T [m(s)f(-s)]x-'+ ds = F(x)- px.

T-+oo

0' 2 •

and c > 0, then limz-+oo F(x)fx = a exists as a

c--tT

(ii) If 0'1 < c < number and

0' 1

(4)

c--tT

From now on we suppose that F(x) is a bounded nondecreasing function such that F(O+) = F(O) = 0. In this case we shall write F(x) E M 0 (0, oo). The function F(x), which is made up of F 1 (x), F 2(x) e M 0 (0, oo) in the following way, corresponds to the convolution in association with the Laplace-Stieltjes transform: (5)

The integral exists for all 0

~

x and it is easily seen that F(x) e M 0 (0, oo ).

7.8.

271

MELLIN-STIELTJES TRANSFORMS

Theorem 7.8.3. Let m1(s) in D 1 : a 1' ~ Res ~ a 2' and m2(s) in D 2 : a~' ~Res ~ a~', be the Mellin-Stieltjes transforms of F 1 (x), F 2(x) e M 0(0, oo), respectively. Then in D 1 n D 2 , m1(s)m 2(s) is the MellinStieltjes transform of F(x) given by (5). D 1 n D 2 is supposed to be nonempty. It could be a single vertical line in the case where m1 (s) and m2 (s) are supposed to converge on the whole line. We note here that the Mellin-Stieltjes transform of a function belonging to M 0 (0, oo) is absolutely convergent in the strip of convergence.

Proof.

Lets e D 1 n D 2 • For 0

< e < R,

which is, by Lemma 3.10.1,

J~ X"- 1 dF(x) = J~ dF2(y) J~ X"- 1 dF1(xfy) =

Joo 0

y•-1

dF.(y)

JR!v

u•-1

dF1(u).

(6)

8/1/

Note that m2(s) converges on the vertical line in D 1 n D 2 , which implies the absolute convergence of Res= a. Hence using the bounded convergence theorem, we may let R e-+ 0+ in (6), which gives us the conclusion of the theorem:

-+

oo,

(7) Now consider the Laplace-Stieltjes transform

f(s) = J~ e-•z dF(x)

(8)

of F(x) E M 0 (0, oo). Since F(x) e M 0(0, oo), it is absolutely convergent on Res= 0. Hence the right half-plane of convergence contains the right half-plane Re s ~ 0. (If the half-plane of convergence is exactly Re s > 0, then on the border where Re s = 0, the Laplace transform is absolutely convergent.)

VII.

272

LAPLACE AND MELLIN TRANSFORMS

We are now going to give a simple but direct relationship between the unilateral (ordinary) Laplace-Stieltjes transform J(s) and the MellinStieltjes transform m(s).

Theorem 7.8.4. Let F(x) e M 0 (0, oo). Suppose that J(s) is the Laplace-Stieltjes transform of F(x) and m(s) is the Mellin-Stieltjes transform in a 1 < Re s < a 2 , a 2 > 0. Let k be any nonnegative integer less than [a 2]. Then, in C11-

m(k

+ s) =

k < Res <

ct 2 -

[a2],

(9)

[(-1)k/F(1- s)] I~ x-•j(x) dx.

(10)

For this theorem, see Zolotarev [1].

Proof. f(s) is defined over 0 ;::;; Res < oo and, in 0 < Res < oo, is differentiable infinitely often (actually analytic). Now we shall prove the case where k = 1 : f'(s)

= - I~ e-•xx dF(x),

Res> 0.

Consider, in the domain (9),

r

y-•j'(y) dy = -

r

y-• dy

I: y-•e-vx dy -I~ x• dF(x) I: u-•e-u du.

= - I~ x =

I~ xe-VX dF(x)

dF(x)

Since Res > a 1 = 1, J;' x• dF(x) is convergent and hence, as we noted before, this is absolutely convergent and, since F(x) is continuous at x = 0, we may apply the dominated convergence theorem, letting e---+ 0+, R---+ oo. (Note that J;' u-•e-u du is convergent since Res < 1.) Then

I~ y-"f'(y) dy

=

-I~ x• dF(x) I~ u-•e-u du

=

which is just (10) with k = 1. The same arguments apply to the general k.

-F(1 - s)m(s

+ 1),

7.9.

273

THE MELLIN TRANSFORM

7.9. The Mellin Transform We now consider the Mellin-Stieltjes transform of a function that is absolutely continuous. Let ifJ(x) be integrable over any finite interval (a, {J), 0 < a < fJ < oo. When m(s)

=

I

-+00

-+O+

l/J(x)x•- 1 dx

(1)

converges for somes= a+ i-r, Eq. (1) is called the Mellin transform of ifJ(x). Letting F(x) =

{J: l/J(u) du,

-I:

ifJ(u) du,

x

~a,

0

< x
a being any fixed positive number, the general theory of Mellin-Stieltjes transforms applies to the Mellin transform of ifJ(x). Therefore, either (1) has some convergence strip (which may be reduced to a single line) or (1) never converges. We do not intend to repeat the results obtained from the theory of Mellin-Stieltjes transforms given in the preceding section, but we will be interested in (1 ), which is absolutely convergent on some vertical line. Suppose that t/J.(x)x- 1 E Ll(O, oo), s = a+ i-r. Then

m(s)

= I~ l/J(x)xs- 1 dx

(2)

(which is absolutely convergent). Write x = eY as in Section 7.8. Then

I~ I l/J(x) I xa- 1 dx = I~oo If(ell) I eav dy. Hence the Fourier transform (with -t for t in the ordinary definition of Fourier transform, that is, the inverse Fourier transform) of ifJ(e")eaY is given by

In other words, the Mellin transform m(s) of ifJ(x) is no more than the Fourier transform of (2n)112l/J(eY)eav. Hence by the inversion formula for

VII.

274

LAPLACE AND MELLIN TRANSFORMS

Fourier transforms (Theorem 4.3.4) we obtain the following theorem as we did for the bilateral Laplace transform in Theorem 7.7.4 or 7.7.6. Theorem 7.9.1. Suppose that ifJ(x)xa- 1 e Ll(O, oo) and ifJ(x) is of bounded variation in a neighborhood of a point 0 < x < oo. If m(s), s = t1 iT, is the Mellin transform of ifJ(x), then

+

lim(lm)-1 Ja~T m(s)x-• ds = l[ifJ(x

7'-+oo

a--17'

+ 0) + ifJ(x -

0)].

(3)

The proof is much the saine as that of Theorem 7.4.4. The (C, 1)-summability theorem for Fourier transforms (Theorem 4.8.1) gives us Theorem 7.9.2. Let ifJ(x)xa- 1 e Ll(O, oo) and m(s) be the Mellin transform of ifJ(x). Then for almost all 0 < x < oo,

lim(lm)-1 Ja~T [1- (It 1/T)]m(s)x-• ds = ifJ(x).

7'-+oo

(4)

a--.7'

Moreover (4) holds for every continuity point x (0 < x < oo) of ifJ(x). The L2 theory of Fourier transforms also applies to the Mellin transform. Actually, we have the following theorem. Suppose that xa-112ifJ(x) e L2(0, oo). Define

(i)

Theorem 7.9.3.

ma(s) = Ja ifJ(x)x'-1 dx,

(5)

1/IJ

s = t1 +iT, a> 0. Then there exists a function m(t1 +iT) which belongs to L 2(oo, oo) as a function of -oo < T < oo, such that m 11 (t1 +iT) converges to m( t1 + iT) in L2(- oo, oo); that is,

J:

I

m11 (t1 +iT)- (t1 +iT) 12 dT- 0

(ii) If we write ifJa(x)

=

(lm)-1

J:

as a- oo.

m(s)x-• tis,

(6)

(7)

then lim a-+oo

Jooo I ifJ(x) -

ifJa(x) 12 xlla-1 dx = 0.

(8)

7.9.

275

THE MELLIN TRANSFORM

(iii)

We have

I~ I tfo(x) 12 x2a-l dx = (2n)- 1 I~oo I m(a +iT) 12 d-r:. Proof. Then

Write x

=

e", log(lfa)

(9)

= -a, log a= a, a-+ +oo, as a-+ oo.

and t/J(e11)ff111

E

L 2 (-oo, oo)

smce

Hence

where g(y) = t/J(e11)ff111 e £2(- oo, oo ). Hence the theory of Fourier transforms of £2 functions (Section 6.2) applies to get the above theorem. The following Parseval relations are also obtained in a similar way from Section 6.2, Eq. (21 ).

Theorem 7.9.4. If xa- 112tfo(x) and

xn-a>-ll~(x)

a being real, then

I~ tfo(x)tp(x) dx =

(Zni)- 1

I:::

belong to L 2 (0, oo),

m(s)n(1 - s) ds,

(10)

where m(s) and n(s) are the Mellin transforms of tfo(x) and tp(x), respectively. More about Mellin transforms will be found in the work of Titchmarsh [2]. Finally, we shall give some simple examples. If 0 ~ x
then m(s) If

=

a•fs for Res t/J(x)

= { ~:

x

(11)

~a,

> 0.

= { log(afx), 0,

0 x

~

x
~a,

(12)

VII.

276

LAPLACE AND MELLIN TRANSFORMS

then for

m(s) = a'fs2

Res> 0.

If ~(x) = e-z, then m(s) = F(s) for Res > 0. This suggests the importance of the Mellin transform in the theory of gamma functions. The Mellin transform has some particular applications in number theory which we do not intend to discuss here. Consider the so-called Riemann zeta function defined by 00

C(s) =

L (1/n"),

Res> 1.

(13)

Res> 1.

(14)

n-1

Now take

~(x) =

1/(ez - 1 ). Then

m(s) = F(s)C(s), This is shown in the following way:

(Interchange of the integral and summation signs is permitted, since e-zx•- 1 e LI(O, oo) for Re s > 1 and the series L e-kz ~ Ax for small x and is bounded for large x.)

foo [(x'- )/(ez- 1)] dx = f 1

o

k-o

(1

+ k)-' Jooo e-uus-

1

du

= C(x)F(s).

NOTES 1. Except for Theorem 7.2.3, most of the material in Sections 7.1-7.7 is due to Widder [1].

2. We have shown, for the Laplace-Stieltjes transforms (and hence for the Laplace transform), the existence of the convergence abscissa u•. We may show that there is a u,. and u,. such that if Re s > u,., then the Laplace-Stieltjes transform converges absolutely, but if Res < u,., it does not; if Res > u,., then the Laplace-Stieltjes transform converges uniformly for Res > u,. + e for any e > 0. It does not converge uniformly in u ~ u,. - e. Here for Uc, u,., and u,., we have the relation u. ~ u,. ~ u,.. (See Widder [1] and Doetsch [1].) We note that there are no corresponding facts in power series; that is, for a power series in the circle of convergence, the series converges absolutely, and in any closed circle interior to the circle of convergence, a power series converges uniformly.

277

NOTES

3.

The inversion formula given in Theorem 7.4.1 (and then Theorem 7.4.2) is an analog of the representation of the coefficients of a power series:

/(z)

= a,.z";

where C is any circle in the circle of convergence. Another representation a,. is given by /'" 1(0)/n!. Studies of the analog of this formula for the Laplace transform were exhaustively made by Widder and others. (See Widder [1], p. 288, 290 and Hirschman and Widder [1].) One of them is the following. Write Lk,u[f]

=

(-1)kjlkl(kfu)(kfu)k+l,

k being a nonnegative integer. Equation (2) in Section 7.4 is supposed to be convergent at some s. Then F(x) - F(O+) =

lim k-+oo

J"' Lk,urfl du. 0

In the case of Section 7.4, Eq. (4), t/>(x)

= lim

L.~:,u[/]

k-+oo

for almost all x

>

0.

4. Another inversion formula is the following. If F(x) is a bounded nondecreasing function with F(O)

=

lim

~

t-+oo

o~nsz

0, then

(-1)"(t"/n!)f<"'(t)

=

l[F(x

+ 0) + (x-

X> 0.

0)],

(See Widder [1], p. 295.)

5.

The inversion formula for Laplace transform can also be handled from a symbolic viewpoint, which has a particular connection with the convolution operator. (See Hirschman and Widder· [1] and Widder [2].)

6.

For the Laplace transform of the vector-valued function, see Hille [1], Chapter X.

7. The Mellin transform is very useful in the theory of the gamma function as was stated in Section 7.9 after Eq. (12). Write F(a b c -x)

' ''

=

ab 1 - -- x

c·t!

+

a(a

+ 1)b(b + 1) c(c+1)2!

x1

-

(1)

which is defined in 0 < x < 1. (When x > 1, its function, obtained by analytic continuation from formula (1 ), is also denoted by F(a, b, c, -x).) In 0 < Re s < min(Re a, Re b), the Mellin formula holds: m(s) =

F(s)F(a - s)F(b - s)

F(c)

r(c- s)

F(a)F(b)

---=,---..,...---

where Re a > 0, Re b > 0, c =f= 0, -1, ... , and m(s) is the Mellin transform of F(a, b, c, -x). (See Titchmarsh [2], p. 191. Also see Whittaker and Watson [1], p. 286.)

VIII More Theorems on Fourier and Laplace Transforms

8.1. A Theorem of Hardy In this chapter, we shall present more theorems, more or less deeper results in the theories of the Fourier and Laplace transforms that will be used later in relation to probability theory. We shall begin with an important theorem given by Hardy which is associated with Wiener's remark given in Section 6.6, that a pair of functions f(x) and its Fourier transform /(t) cannot both be very small at infinity. In other words, if f(x) and/(t) both converge to zero very fast as I x I - oo and I t I - oo, respectively, then f and J must be zero identically. For example, take an extreme case in which f(x) = /(x) = 0 for I x I > X, for some X > 0. Then

/(z) = (2n)-1' 2

J~x e-iz"J(u) du 278

8.1.

279

A THEOREM OF HARDY

is analytic in the whole z-plane, i.e., it is an entire function, which is obvious, since f(z) is differentiable in the whole plane. From the wellknown elementary fact that an analytic function which is zero on a set that has a limiting point in the domain of analyticity, /(z) = 0 for all z since /(z) = 0 for z = x > X. The following Hardy theorem and its corollaries give the most precise interpretation of the above remark (Hardy (1 ]).

Theorem 8.1.1. Suppose that f(x) is defined over ( -oo, oo) and for some a> 0, (1) and that the Fourier transform

/(t) = (Zn)-I/2

J:

e-iz'f(x) dx

satisfies (2)

for some nonnegative integer m and for some -00

Psuch that ap

6;

!. Then

< t < oo,

(3)

where P(t) is a polynomial at most of degree m. The typical case is that where a=

p=

l, which gives us the following:

Corollary 8.1.1. If f(x) and its Fourier transform/(x) are both 0[1 x 1m exp(-lx2 )] for large lxl, then each is of the form exp(-lx2 )P(x), where P(x) is a polynomial at most of degree m. The following particular case of this corollary is easily verified.

Corollary 8.1.2. Ifj(x) and/(x) are both O[exp(-lx2 )] for large I xI, then f(x) =/(x) = Cexp(-lx2 ), where C is a constant. Corollary 8.1.1 is derived from Theorem 8.1.1 as follows. By Theorem 8.1.1, /(t) = exp(- it2 ) P(t). Let Hn(x) be the Hermite polynomial defined in Section 4.11. Now any polynomial of degree m can be repre-

280

VIII.

MORE THEOREMS ON TRANSFORMS

sented by a linear combination of Hermite polynomials Hk(x), k = 0, 1, 2, ... , m, and hence

where ~k( t) = exp( -lt2 ) Hk(t). By the italicized statement in the end of SeCtion 4.11, the Fourier transform of ~k(t) is given by (-1)k~k(t). Therefore f(x), the inverse Fourier transform of ~k(t), is given by

where Q(x) is a polynomial at most of degree m. We shall show the theorem, after first giving a lemma.

Lemma 8.1.1. Let f(z) be an entire function. If f(z) = 0(1 z lnezlzl),

a> 0,

(4)

for large I z I and (5) for large x, n, and m being nonnegative integers, then

= ezzP(z),

f(z)

(6)

where P(z) is a polynomial of at most degree m. Proof.

Consider the Phragmen-LindelOf function

h(6) =lim sup(1/r) loglf(rei 0 )

I·

r-+oo

(See Section 1.15.) From (4), log I f(rei 0 )

I ~ log(Krnezr)

for some constant K > 0. That is, log I f(rei 0 ) I

~

log K

+ n log r + ar.

Therefore,

h(6)

~a.

From (5), we have similarly h(O) = lim sup r-• I f(r) r-+oo

I~

-a.

(7)

8.1.

281

A THEOREM OF HARDY

Let 15 be any number, 0 < 15 < n and write

H(O)

=

-a cos() +a tan(l5/2) sin 9.

Then we see that

h(O)

~

-a = H(O),

h(n - 15)

~

a

=

H(n - 15).

Hence by Theorem 1.15.3

h(O) holds for 0

~

()

~

~

H(O)

n - 15. In particular,

h(n/2)

~

H(n/2) =a tan(l5/2).

Since 15 is arbitrary, this gives us H(n/2)

~

(8)

0.

Now define F(z) = euf(z) and G(z) = F(z)f(z + i)m. Then G(z) is analytic in lm z ;;;; 0 and I G(z) I ~ e• 1•1 for some c > 0. Moreover, on the real half-axis, G(x) is bounded because of (5) and on the imaginary axis, from (8), lim sup(1/y) log G(iy) =lim sup r- 1 I logf(ir)

I ~ 0.

r-+oo

Hence, by the theorem of Polya-Szego (Theorem 1.15.4), G(z) = 0(1), namely, m Im z;;;; 0. A similar argument leads to: m

lm z

~

0.

Therefore, F(z) = 0(1 z lm) in the entire plane. Liouville's theorem then shows that F(z) is a polynomial at most of degree m.

Proof of Theorem 8.1.1. Write

Now we turn to the proof of the theorem.

F(z) = f(z) = (2n)-11 2 J~oo e-izuf(u) du.

(9)

VIII.

282

MORE THEOREMS ON TRANSFORMS

From (1) the integral in (9) exists for all complex z and is seen to be an entire function. From (1 ), for real u, for

I u I> 1,

where C is a constant. Therefore,

I F(z) I ~ ~

1 (2.n)-112 f -1

elz1(2.n)-1!2

I/(u) I eizui du + C(2.n)-112 foo lui me-"'" eizui du 1

-oo

f -1 1/(u) 1du + C(2/n)112 foo0 ume-"'"eiziu du. 1

1

The last integral is d )mfoo exp( -au 2 2 )1/2 ( "'djZT C ( --;0

2 )112 ( dldz I ~ C ( --;2 )1/2 ( = C ( --;-

=

o( I z I

m

+ I z I u) du

)m exp ~ 1 z 12 Joo -oo exp [-a(u -

I z 12 Joo dldz I )m exp ~ -oo

exp

1z 1 )

2(l

2] du

exp -av 2 dv

I~ 2 ) •

Hence we get (10) Now consider F 1 (z) = F(z)

+ F( -z).

(11)

Then F 1 (z) is an even entire function and from (10) we have (12)

From (2), for real x, (13)

Suppose that m is even.

8.1.

283

A THEOREM OF HARDY

Write z 2 = w, F 1 (z) = G(w), m = 2n. Since F 1 (z) is even, G(w) is again an entire function (which is easily verified) and from (12), noting

ap

~

!,

Also from (13), for real E G(E) = 0(1 E l"e-llll 1).

Hence by Lemma 8.1.1, G(w) = e-flwQ(w), where Q(w) is a polynomial of w at most of degree n. Thus we get (14) where P 1 (z) is a polynomial at most of degree m. If m is odd, then we take m + 1 in place of m in the above and again we have (4) with P(z) a polynomial at most of degree m + 1. Next we consider F 2 (z) = F(z) - F( -z).

Then F 2 (z) is odd. Consider zF2 (z) = t/J(z), which is even, and for this (z) we again have the same estimations as (12) and (13), with m + 1 for m. Then we obtain

where R(z) is a polynomial of w = z 2 and the constant term of R(z) must be zero since t/J(O) = 0 [from t/J(z) = zF2 (z)]. Hence R(z)fz is a polynomial P 2 (z) of degree m + 1 at most. Therefore, (15) From (14) and (15),

where P(z) is a polynomial at most of degree m + 1. But F(t) = /(t) for real t and assumption (2) shows that the degree of P(z) must be at most m. This completes the proof.

284

VIII.

MORE THEOREMS ON TRANSFORMS

8.2. A Theorem of Paley and Wiener on Exponential Entire Functions We call an entire function f(z) an exponential entire function if

f(z) = O(e·41zl)

(1)

for some A > 0. We shall show the following theorem due to Paley and Wiener [2], p. 12.

Theorem 8.2.1. Let f(z) be an entire function such that f(x) E £2( -oo, oo) on the real axis. In order that (1) hold, it is necessary and sufficient that the Fourier transform](t) in £2(-oo, oo) of f(x) vanish almost everywhere for I t I > A.

Proof.

The sufficiency of the condition is obvious since, in this case,

f(z)

=

(2n)- 112

t.d eiztj(t) dt,

(2)

from the inversion of the Fourier transform in £2(- oo, oo) and]( t) = 0, I t I > A, and (2) implies

I f(z) I ~

t.d I](t) I dt.

(2n)-112e..41zl

We now prove that the condition is necessary. Consider the entire function

(eiAzfe)

J:+e f(w) dw =

e > 0.

G(z)

We then see, for a fixed e > 0, G(z)

=

O(eAizl) O(e"Hizl+•>)

=

O(eA<21zl+•>);

that is, G(z) is exponential entire. If z = x is real,

I G(x) I =

I(eiAxfe) Jx

= 0

x+a

(I

oo

-oo

I

f(w) du = e-112

lf(w) l2 du

)1/2

= 0(1).

(Ix+•x If(w) 12 du )1/2

(3)

8.2.

A THEOREM OF PALEY AND WIENER

If z

=

285

iy is imaginary and y > 0, then

I G(ty) I

=I (e-Avfe) I::+- f(w) dw I= O(e-Av) O(eAv+A•) =0(1).

Hence G(z) is bounded on the whole real axis and on the positive imaginary axis. Therefore, together with (3), from the Phragmen-LindelOf theorem (Theorem 1.15.2), G(z) is bounded on the upper halfplane. Therefore, by Theorem 6.5.3, G(z) E ~ 2 and then by Theorem 6.5.2, the Fourier transform G(t) of G(x) is zero almost everywhere on t < 0. Hence we may write G(x) = (2n)-11 2 l.i.m. IT G(t)eilz dt. T-+oo

[Note that G(t)

E

0

L2(-oo, oo).] Thus

e- 1 Iz+- f(w) dw

=

e-iAzG(x)

=

(2n)-11 2 l.i.m. IT G(t)ei
z

T-+oo

=

(2n)- 1' 2 l.i.m. I T'-+oo

T'

G(t

-A

0

+ A)eitz dt.

(4)

Now consider/( -w) instead of f(w) in (3) and write

I:+- f(w) dw = G1(z).

e-iAze-1

(5)

Then as before, G 1 (z) is shown to belong to ~ 2 in the lower half-plane (this definition will be clear), and then G 1 ( -z) is in ~ 2 (in the upper half-plane). Therefore, G 1 ( -x) = (2n)-11 2 l.i.m. IT' G 1 ( -t)eitz dt, T 1 -+oo

0

where G 1 ( -t) is the Fourier transform of G 1 ( -x). Accordingly,

G1(x) = (2n)-1' 2 l.i.m.

T1 -+oo

r

which is written from (5) as e- 1 Iz.ts j(w) dw

= (2n)- 1' 2 l.i.m. T 1'-+oo

:z;

-Tl

G 1(t)eitz dt,

r, -Tt

G1(t- A)eitz dt.

Therefore, from (4) and (5), the Fourier transform/.(t) of e- 1

I~ f(w) dw

(6)

VIII.

286

= e- 1 J:+o f(w) dw belongs to £2( -oo, oo)

vanishes for ltl >A. Now J.(x) since

I:

lf.(x)

12dx ~ e- 1

MORE THEOREMS ON TRANSFORMS

I:

dx

I~

lf(w)

12dw =I~'"' lf(x) 12dw.

The Parseval relation then shows that

11/.(x)- f(x) ll2 = ll.f.(t)- /(t) ll2, where

11/.(x)- f(x) II~~ e-1 = e- 1

I:

dx

I~

lf(w)- f(x)

I: I: I dw

f(u

+ x) -

12 dw

f(x)

12 dx- 0

J

as e - 0. Therefore, (t) converges to /(t) in L 2-norm as e - 0. Since /.(t) = 0 for I t I > A, /(t) = 0 for I t I > A. This completes the proof. We now show a similar theorem for the Fourier-Stieltjes transform. For simplicity we consider the characteristic function of a distribution function.

Theorem 8.2.2. Let f(z) be an exponential entire function, where f(z) =

O(eAizl ),

A >0,

(7)

and suppose thatf(z) is the characteristic function of a distribution function F(x) on the real axis z = t. Then

F(x) = {

~

for for

x >A, x <-A.

(8)

Conversely, for the characteristic function f(t) of a distribution F(x) satisfying (8), there is the entire functionf(z) satisfying (7), whose values on the real axis z = t are identical with f(t). For the proof we give, first the following lemma.

Lemma 8.2.1. Let f(t) be the characteristic function of a distribution function F(x). Then the Fourier transform of f(t) sin atfat,

a> 0,

8.2.

287

A THEOREM OF PALEY AND WIENER

in £2(- oo, oo) is given by

(2n)ll2 (1/2a)[F(x + a) - F(x- a)].

Proof of the Lemma.

Since f(t) is bounded,

f(t) (sin atfat) e £ 2 (-oo, oo). If

lim (Zn)-1' 2 T-+oo

IT

f(t)(sin atfat)e-izt dt

(9)

-T

exists, then by the remark after Theorem 6.2.2, it is the Fourier transform of f(t) (sin at fat) in £2(- oo, oo) almost everywhere. Hence it suffices to show that (9) is equal to (2n} 112 (1f2a)[F(x + a) - F(x- a)] almost everywhere. Now (2n)-11 2 =

I:T f(t)(sin atfat) e-izt dt (Zn)-112a IT f(t}[(e-i
which is seen, by the inversion formula for the Fourier-Stieltjes transform [Theorem 4.4.1, or Section 4.4, Eq. (2}], to converge to (2n} 112 [F(x +a) - F(x- a)] if x +a, x- a are continuities of F(x). Since the set of discontinuity points is countable, this proves the lemma.

Proof of Theorem 8.2.2.

Let e be any positive number and consider

if> (t) = f(t) (sin etfet) e £ 2 (-oo, oo). Then, because of (7}, ¢>.(z)

= f(z)

(sin ezfez)

=

O(e
Therefore, from Theorem 8.2.1, the Fourier transform ~.(x) of ¢>.(t) vanishes for I xI >A+ e. By Lemma 8.2.1, ~.(x) is given by (2n} 112 X (1/2e)[F(x + e)- F(x- e)] almost everywhere. Thus

F(x

+ e) -

F(x - e) = 0

for

I xI> A+ e

almost everywhere. Hence F(x) is constant for I x 1 >A+ e, which shows F(x) = 1, for x > A + e, and F(x) = 0 for x < -A - e. Since e is arbitrary, this gives us the required proof of necessity.

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MORE THEOREMS ON TRANSFORMS

The converse part is obvious, because if (8) holds, f(t) becomes

J

A+O

-A-il

.

e•xt dF(x)

and

JA+O

-A-il

.

A

e'"x dF(x) = O(e 1•1).

Another proof of Theorem 8.2.2 which is direct and does not appeal to Theorem 8.2.1 will be given as the proof of Theorem 11.5.1.

8.3. Theorems of Ingham and Levinson In Section 8.1, we have shown Hardy's theorem in relation to Wiener's remark that a function and its Fourier transform cannot both be very small. We are going to give another result due to Ingham [1]. Suppose that f(x) = 0 outside a certain interval. Then obviously its Fourier transform /(t) cannot satisfy the condition without being identically zero : /(t) = O(e--'ltl) (1) for large

I t 1. Because, if ( 1) were satisfied, then f(z) = (2.n)-I12 J~oo ei•tf(t) dt

would exist as a Lebesgue integral in the strip I y I < <5 in the complex domain and there be analytic since it is differentiable. Hence f(t) is zero in an interval, and therefore it is zero everywhere - oo < x < oo. Now consider the condition that

/(t) = O(e-liCtl)

(2)

and ask how O(t) could be large for large I t I, for the function f(x), and be zero outside some interval without being identically zero. The following theorem was given by Ingham [1], also see Levinson [1].

Theorem 8.3.1. (i) Suppose that O(t)ft is a nonincreasing function of t > 1 such that O(t)ft --+- 0 as t --+- oo and

I~ [O(t)ft 2 ] dt <

00,

(3)

Then there exists a functionf(x) whic_h vanishes outside a certain interval (-I, I), and for which its Fourier transform/(t) satisfies (2) for large I t I·

8.3.

289

THEOREMS OF INGHAM AND LEVINSON

(ii) Let O(t) be a nonnegative function fort > 1. If there is a function f(x), which is not identically zero and which vanishes outside a certain interval (-I, 1), and for which its Fourier transform f(t) satisfies (2), then (3) holds.

Proof. (i) We may suppose that O(t)- oo as t-oo. Otherwise we may replace O(t) by O(t) + (1 + t) 112 • Let I be any fixed positive number, and let :E f.!n be a convergent series of positive terms such that :E f.!n ~ I. Now consider the function

n sin entfent, 00

/(t)

=

/(0)

= 1.

(4)

n=1

We are going to show that/(t), defined in this way, is the Fourier transform of some function f(x) which vanishes outside the interval (-I, 1). Since (5) the infinite product converges uniformly in every finite interval of t. Moreover,

1/(t) I ~ where

11

TI (lfent)

t > 0,

for

(6)

n-1

is any positive integer, so that

/(t) = 0(1 t !-•)

(7)

as t ---+ oo. It is obvious that /(t) is not identically zero. Now, from Section 4.11, Eq. (6),

f n (sin (!.,tfent)eixt dt = 0 oo

-oo

Writing for n

n

for

v-1

~

2,

I xI

>I.

(8)

n

fn(t) =

n sin entfent,

•-1

It I >

(9) 1,

which belong to V( -oo, oo), fn(t)---+ /(t). Hence by the Lebesgue dominated convergence theorem, we have, from (8) that for

x >I.

(10)

Therefore, f(x), defined by

f(x) = (Zn)-112

J:

./(t)eizt dt,

(11)

is zero on I x I > I [therefore it belongs to £1(- oo, oo) and is continuous], and /(t) is the Fourier transform of f(x). We next show that/(t) satisfies (2) by placing further conditions on the choice of en . Since O(t)ft2 is also a decreasing function and (3) holds, I: O(n)fn 2 < oo because

EO(n)fn ~ Er

[O(t)ft 2 ] dt =

2

n-2

n-2

n-1

Now choose en with I: en

~

I"" [O(t)ft

2]

1

dt <

I, in such a way that en+l

en

~

~

00.

en and (12)

aO(n)/n 2

for n ~ n0 , n0 some integer and a some number, a >e. This is possible if, for instance, en is taken to be aO(n 0 )/n 0 2 for n ~ n0 and aO(n)fn 2 for n > n0 , where n0 is chosen so large that 00

Len= a[O(no)fno] +anL O(n)/n

2

~ l.

0 +1

Then from (6),

for t > 0, 11 ~ n0 • Take 11 = [O(t)]. For sufficiently large t, n0 < (since 11/t = [O(t)]/t ---+ 0) and

11


11 2 /aO( 11 )t = [O(t)]11/aO( 11 )t ~ ((O(t) + 1)fat)( 11/0( 11)) :::;;;; a-1 [1 + (11/t0(11))] ~ e- 1 for sufficiently large t (since

8(11)---+

oo, 11/t---+ 0). Thus

1/(t) I ~e-. <

e--lllll+l.

This completes the proof of (i) of the theorem. (ii) Let f(x) be a function which is not identically zero and which vanishes outside (-I, 1). Suppose (2) is satisfied. Write

](z)

=

(Zn)-112

(z f(x)e-wz dx,

(13)

8.3.

291

THEOREMS OF INGHAM AND LEVINSON

where w

=

t

+ iT. Without loss of generality we suppose I= 1. Obviously f(z) = (elzl).

(14)

We apply Carleman's theorem (Theorem 1.13.2) in the upper halfplane. We then see that logl /(t)f( -t) I (t- 2 - R-2) dt

(Zn)- 1 r

+ (nR)-

1

I:

log Ij(Rei~) I sin () d()

+ 0(1) > 0

for large R. Accordingly, (Zn)- 1 I~ log+ I f(t)]( -t) I

~! + R- 1 I: log+ I/(Rei 8 ) I sin() d() + 0(1)

> -(Zn)- 1 rlog-lf
(15)

where log- ·a= log a- log+ a (as in Section 6.3). Since/(t) = 0(1) as It 1- oo, log+ I/(t)/( -t) 1 t- 2 dt =

(Zn)- 1 r

o(J~ t-2 dt) =

0(1 ),

and (nR)- 1

I:R log+I/(Rei

I sin () d() =

8)

o( R- I: R d()) = 0(1) 1

using (14). Therefore the right-hand side of (15) is 0(1). Hence 0(1)

=

-(Zn)- 1

~ -(2n)- 1

~-

43

I~ log-1/(t)./(-t) 1 (t- 2 - R-2) dt

I

R/2 1

(Zn)- 1

log-1/(t)/(-t) 1 (t- 2 - R-2) dt

IR/2 0

log-1/(t)/(-t) 1 t- 2 dt

Now from (2),

/(t) ~ Ce-Bom, C being a constant, and then

log/(t)f(-t) ~ 2log C- 20(1 t 1),

(16)

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MORE THEOREMS ON TRANSFORMS

from which we have -log- /(t}f(-t) ~ -2 log C

+ 20(1 t 1).

Then from ( 16 ), 0(1)

~

C 1 + (3/4n)

I

R/2

[O(t)ft 2 ] dt,

0

which gives us (3 ). This completes the proof. REMARK 1. In the first part of the theorem, we may choose /(t) to be nonnegative for all t.

en in

In fact, we may do this, choosing the to be equal in pairs.

the proof of the theorem

2. A slightly more general statement than (ii) of Theorem 8.3.1 holds. Let O(t) be a nonnegative function fort > 1. If there is a functionf(t) which is not identically zero and which vanishes on (- oo, -I) and for which its Fourier transform /(t) satisfies (2), then (3) holds. REMARK

In place of (13) we consider (supposing I= 1) that

/( -z)

=

(Zn)-112

I:

f(x)e+izz dx.

Then/( -z) is analytic in lm z > 0 and satisfies

/( -z) = O[exp(lm z)] for Im z > 0. This enables us to get (nR)- 1

I:

log+ I/(- Rei8 ) I sin 0 dO

=

o( R- I: R dO) = 0(1 ). 1

In the above statement the interval (-oo, -1) on which f(t) vanishes may be replaced by (I, oo). Now we have a theorem of Levinson, which has some connection with Ingham's theorem (Theorem 8.3.1 ).

Theorem 8.3.2. Letf(x) belong to L 2 (-oo, oo) such that f(x) = O(e-B)

(17)

as x--+- +oo, where O(x) is a nondecreasing continuous function for x > 1,

8.3.

293

THEOREMS OF INGHAM AND LEVINSON

with the property

J~ [O(x)fx 2 ] dx =

(18)

oo.

If the Fourier transform /(t) of f(x) is almost everywhere zero in some interval (a, b), then/(t) and hencef(x) must be zero almost everywhere in -oo < t < oo and -oo < x < oo, respectively. In fact, this is a special case of the more complete theorem of Levinson [2-4]. We shall prove this theorem, reducing it to Theorem 8.3.1 (ii). The idea of the method is Levinson's. First we need the following simple lemma.

Lemma 8.3.1. Suppose that O(x) is a function in Theorem 8.3.2. Then

J~ {min[x, O(x)]Jx2 } dx Proof.

=

(19)

oo.

(20)

Suppose

J~ {min[x, O(x)]fx2 } dx < Let E

oo.

= {x:

O(x) > x, x > 1}. Then oo

>

JE x-

1

dx

+ JEc [O(x)fx

2]

dx.

(21)

Hence, write n-1

where (an, bn) are distinct intervals; this is possible since E is an open set. From (21), (22) Therefore,

J

E

[O(x)fx 2 ] dx = oo,

from which we have

(since O(n) is nondecreasing and bn

E

Ec), and

(23) But from (22), bnfan--.. 1 and log(bnfan)/[(bnfan)- 1]--.. 1 as n--.. oo. Hence (22) implies that L [(bnfan)- 1] < oo, which contradicts (23).

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MORE THEOREMS ON TRANSFORMS

We need another lemma.

Lemma 8.3.2. Let

Then

s~1

for all real u, y,

S

~

4e-cu-yl/

S S

~

S

(25)

for

y > 0,

u > 0,

(26)

4-e'"-11112

for

y < 0,

u < 0,

(27)

~

2rul2

for

y < 0,

u > 0,

(28)

~

2eul2

for

y > 0,

u < 0.

(29)

2

Proof. Condition (25) is obvious. Since the integrand in S is analytic except at x = 0, x = 1, we see from Cauchy's integral theorem, that l J 0 [(ez-

.

.

1)/(e- ez)]-•ue-•zv dx =

JiCHil 0

.

.

[(&- 1)/(e- et)]-•ue-•zv dz

+ Jli
[(&- 1)/(e- eZ)]-iurizv dz.

(30) In the first integral on the right, z runs along the segment joining 0 and !(1 + i) and similarly in the second integral. (More exactly, we apply Cauchy's theorem, to a circumference which excludes the points 0 and 1 by small circular arcs and then let the radii of these arcs shrink to zero.) The first integral on the right-hand side of (30) is 11

= (1 + i)

J:'

2

{exp[x(1 + i)] - 1 }-iu

· {e- exp[x(1 + i)]}iu exp[ -i(1 + i)xy] dx. Now we see that (9

= _am

ez(l + i) - 1 e- ez' 1 +il

= _ am

ez COS X - 1 + iez sin X p e- ez cos x - iez sin x _ sin x (cos x- e-z)- 1 +sin x (e 1-z- cos x)- 1 - - arctan sm . x (cos x - e-z ) - 1 sm . x (e1 -z - cos x ) - 1 - 1

p

sin x sin x = arctan ----.,.... + arctan - : :----cos X - e-z e1-z - COS X smx cos x - e-z'

;?; arctan - - - - -

8.3.

295

THEOREMS OF INGHAM AND LEVINSON

and if 0

~

x

~

l, then x sin x

f?J ~ arctan

~cos

~

-X-

x- e-z and

arctan 2

. Sin

1

2

~

1 1 arctan tan 2 = 2,

where "amp" means "the amplitude of." Hence

which is ~ e-u 12 for y < 0 and is ~ 2e-(u-W2 for y > 0. The second integral on the right-hand side of (30) is

12 = (1 - i)

J

1

1/2

{exp[x

· {e- exp[x = (1 - i)

J

1/2

+ i(1 -

+ i(1 -

x)] }iu exp{ -i[x

[exp(1 - x

0

x)] - 1 }-iu x)]y} dx

+ ix)- 1]-iu

+ ix)]iu exp[ -i(1 -

· [e - exp(1 - x

+ i(1 x

+ ix)y] dx.

Here we see that

e

1

=-amp

+

e1 -z+iz - 1 e1 -z cos x - 1 ie 1 -z sin x . amp--.,....--------:--:------:-e- e1 -z+tz = e- e1 -z cos x- ie 1 -z sin x

+

e-z cos x - e- 1 ie-z sin x = -amp 1 - e-z cos x- ie-z sin x

e-z sin x (e-z cos x - e- 1 )- 1 + e-z sin x (1 - e-z cos x)- 1 1 - e-z sin x (e-z cos x- e- 1 )- 1 • e-zsin x (1- e-zcosx) 1 e-z sin x e-z sin x = arctan + arctan ~1 ---::---1 e-z cos x- ee-z cos x sm x smx sm x = arctan -------::---:-1 + arctan ~ arctan - - - - cos x - ezez - cos x ez - cos X = arctan

~

--;---::---;o.__----;-----::---!,__~-=---::--:---'~---=-----7-

arctan sin xezx

~

. 1) arctan ( 2e- 112 sm 2

~

arctan ( tan 21 ) = 21 .

As before, we have the same estimates. We then have (26) and (28). In order to get (27) and (29), start with

J:

[(ez _ 1)/(e _ ez)]-iu e-izv dx =

J

(l+i)/2 0

+J 1

[(ez- 1)/(e- ez)]-iu e-izv dz .

(1-,)12

[(ez- 1)/(e- ez)]-iu e-izv dz

(31)

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MORE THEOREMS ON TRANSFORMS

in place of (30) and proceed in just the same way, in which -i takes the place of i and, for instance, e has a different sign but its absolute value is not less than !. In this way we may easily obtain (27) and (29). Proof of Theorem 8.3.2. Suppose that /(t) is zero in some interval. It is sufficient to show that if (17) and (18) hold, then/(t) = 0 almost everywhere. Consider l f•0 dv fl+v t-v J(u) du =

A

f.(t),

where e is a sufficiently small fixed positive constant. If /(t) is zero in (a, {3), then/.(t) is also zero in (a+ e, {3- e), and if /.(t) is zero almost everywhere, then /(t) is also zero almost everywhere. Hence it suffices to show that /.(t) is zero almost everywhere. Suppose that /.(t) is zero in (a, b) but not zero almost everywhere. We shall show that this assumption leads to a contradiction. We may suppose that a= 0, and

/.(t) = 0,

0


0

+ 'YJ,

1 > 'YJ > 0,

and suppose /.( t) =f:: 0 on a set of positive measure in ('YJ, 1). In other words, /.(t) is not identically zero almost everywhere in (0, 1). It is obvious that /.( t) E L 2 ( - oo, oo ). Also we may show without difficulty that the inverse Fourier transform of /.(t) is given by

[(1 -cos x)fx 2 ]f(x). Consider the transformation

t = log[(eH 1

+ 1)/(1 + e')],

which transforms the t-interval (0, 1) to the s-interval (- oo, oo ). We have e8 = (el- 1)/(e- et).

Then /.{log[(e'+l + 1)/(1 + e')]} = 0 in (-oo, -1), where I= log [(e-'~- 1)/(e- e-'~)]. Now define G(s)

= /.{log[(es+ 1 + 1)/(1

+ e')]} dt/ds

where dtfds is seen not to be zero for all -oo < s < oo. Hence G(s) is not identically zero almost everywhere and is zero on (- oo, I).

8.3.

297

THEOREMS OF INGHAM AND LEVINSON

The inverse Fourier transform G(u) of G(s) is G(u)

= (2n)- 1' 2 J : G(s)e-ius ds = (2n)- 1: 2 J: /,(t)[(e 1-1)f(e-e1 )]-i"dt =

(2n)-li 2

J [(e 1

1 -

1)/(e- e- 1)]-i"dt l.i.m. JA j,(y)e-ivt dy .4-+oo

0

-.4

= (2n)-112 J~""J,(y) dy { [(et _ 1)/(e _ e-t)]-iue-ivt dy. Let u > 0. We split G(u) into three parts:

+ J"'0 2 +

G(u) = (2n)- 1 (Jo

-oo

] 1

=

Jo

-oo

[(1 -cos ey)fy 2]f(y) dy

J""u/2 ) =

Jl [(e 0

1 -

(2n)- 1 (

J1

+ ] 2 +fa)·

.

1)/(e- e- 1 )]-iue-tvt dt.

From (26),

I ] 1 I ~ 2e-"12 J:oo I ]21 ~ 4 J:'2 [(1

[(1 - cos £Y)/y 2]j(y) dy = O(e-"12 ), - cos ey)fy2] I j(y) I e-lu-vl/2 dy

[by (25)], and Using (24),

I lal ~ 2 f""u/2 r

2 lf(y) I dy,

which is, from (17), 1lal·~

o( J:2 [e-o
= O(e-Oiu/21) for large u. Hence, altogether we have

I G(u) I ~

Cte-"'4

+ C2e-o
as u- (X) for some constants cl and c2. In the same way, we may show, using (27) and (29) that I

for u - fore,

=·

G(u)

1

~

C 1e-lul/4

+ C 2e-O
In other words, this relation holds for

I u I - =· There-

I G(u) I= O(exp{-min[! I u 1. O(ll u 1)]}).

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Since G(u) is not identically zero and vanishes on ( -oo, l), we conclude from Remark 2 made after the proof of Theorem 8.3.1, that

I~ {min[u/4, O(uf2)]/u 2 } du <

oo

or

I~ {min[x, 0(2x)]fx2 } dx <

oo.

But 0(2x) satisfies (18) and, from Lemma 8.3.1, the above integral must diverge. Hence we get the contradiction. This completes the proof.

8.4. Singularities of Laplace Transforms Let u0 be the convergence abscissa of the Laplace-Stieltjes transform

f(s) =

I:oo rsz dF(x),

(1)

where F(x) is of V(O, oo), that is, a function of bounded variation over every finite interval of [0, oo). For convenience F(x) is supposed to be standardized:

F(x)

=

i[F(x

+ 0) + F(x -

0)]

for

0 < x < oo,

q + i-r, u, T real. Suppose 0'0 to be finite. We have shown (Theorem 7.3.2) thatf(s) is analytic in the right half-

s=

FIGURE

8.1.

plane Res > u0 • We study here the analyticity on the vertical line of convergence s = 0'0 • Let s0 be a point on the convergence line, or Re s0 = u0 (see Fig. 8.1 ). If there exists a function cp(s) that is analytic inside some circle C(s0 ) with center s0 , andf(s) = cf>(s) in the common part (half-circle) of this

8.4.

299

SINGULARITIES OF LAPLACE TRANSFORMS

circle and the right half-plane Re s > a0 , then s0 is called a regular point of f(s). The points on the line Res = a 0 which are not regular points are called singular points of f(s). 4>(s) is called an analytic continuation of f(s) to the domain which is the common part of e(s0 ) and the half-plane to the left of s = a 0 • Usually it is not easy to find the locations of singular points or regular points. But there are some special cases in which we can identify a particular point as a singular point. For power series, similar terminologies are applicable on the circle of convergence. One such theorem is the following: If an ~ 0 for n > N 0 for some N 0 , and the power series 1::'=o anzn has a radius of convergence R, 0 < R < oo, then z = R is a singular point (Vivanti's theorem). (See Titchmarsh [3], p. 214.) The analog of this theorem for the Laplace-Stieltjes transform is true, in which the proof is also carried over. (See Widder [1].)

Theorem 8.4.1. If F(x) is a monotonic function, then s singular point of the Laplace-Stieltjes transform (1 ).

= a 0 1s a

Proof. Without loss of generality, we may suppose that a0 = 0 because if not, we may consider f(s

+ C1o) = f0

-+oo

where G(x)

=

J:

e-sx

e-aou

dG(x),

dF(u),

which is again a monotonic function. We may suppose that F(x) is nondecreasing. Otherwise we may take -F(x). We are going to show that the origin is a singular point when a 0 = 0. Suppose tentatively that the origin is a regular point. Then there is a function 4>(s) which is analytic in I s I < 15 and 4>(s) = f(s) in Res > 0, Is I < 15. Choose a point (! > 0 inside I s I < 15 such that a circle 11 with center (! is interior to I s I < 15 and contains the origin inside it; take a point -n(rJ > 0) which is inside the circle ee (see Fig. 8.2). The Taylor series 4>(z) about (!,

e

()()

L [4> 1•,(e)fv!](z- e)•, •-0

(2)

VIII.

300

FIGURE

converges in C11 , where, since

(!

MORE THEOREMS ON TRANSFORMS

8.2.

> 0 and cfo(e) = f(e ),

by Theorem 7.3.2. Thus (2) becomes

Since (1) is convergent at z at z = -1], that is

=

-1],

the above series also converges

converges. The integral is actually absolutely convergent since F(x) is nondecreasing and the integrand is nonnegative. Furthermore, (z (! )•fv! is also positive and hence by the monotone convergence theorem we may interchange L and J to get the fact that the series

+

roo e-ex f 0

•-O

[(

'YJ

+ e)•fv!] x• dF(x)

(3)

(the arrow sign is dropped) is convergent. Now (3) is

which must be convergent. This shows thatf(s) converges at -1], which contradicts the assumption that the origin is the convergence abscissa. This completes the proof.

8.5.

ABELIAN THEOREMS FOR LAPLACE TRANSFORMS

301

8.5. Abelian Theorems for Laplace Transforms We know that if sn---+ s, then (s 0 + · · · + sn)f(n + 1)---+ s. Theorems which assert, from the convergence of a sequence, the convergence of some sort of average of the sequence, are called Abelian theorems. The same terminology is applied also for a function. In the use of this terminology, sometimes the conver:gence is replaced by some asymptotic relation; for instance, such as the fact that sn ,_ n implies (s0 + + sn)f(n + 1) ,_ ln, as n---+ oo. Abel's continuity theorem (Theorem 4.5.3), which states that the convergence of (1)

to s implies that 00

lim x-+t-o

L anxn =

n-o

s,

(2)

is also an Abelian theorem. Here we study the analogs of the above theorem for the LaplaceStieltjes or Laplace transforms. As before, we suppose throughout that F(x) E V(O, oo) and we consider the Laplace-Stieltjes transform with the real parameter: (3)

which is real valued. A direct extension of Abel's continuity theorem is:

Theorem 8.5.1. Suppose that F(O) = 0. If F(x) converges to a as x---+ oo, then (3) exists for a > 0 and /(a) converges to a as a---+ 0+. Proof.

For any R

I:

> 0,

e-ax dF(x} = e-aRF(R}

+a

I:

F(x}e-ax dx.

(4)

If we let R---+ oo, then, since F(x)---+ a as x---+ oo, e-aRF(R)---+ 0 and, for a > 0, the last integral converges absolutely. Hence

I-+oo e-ax dF(x} = a Ioo F(x}e-ax dx = Ioo F(ufa)e-u du. 0

0

0

For each u, F(ufa)---+ a as a---+ 0+ and since F is bounded, the last

VIII.

302

MORE THEOREMS ON TRANSFORMS

integral converges to

a J~ r"du =a by the dominated convergence theorem. To show a more general theorem, we consider a nondecreasing function

cfo(x) defined for large x which is written as

a;::;; 0,

cfo(x) = x«L(x),

(5)

where L(x) satisfies

L(cx)JL(x)--+ 1

as

x--+ oo

(6)

for every c > 0. Such a function L(x) is called a function of slow growth. This concept was introduced by Karamata [1]. One example of cfo(x) is

cfo(x) = x"(log x'f

for

x > 1, a ;::;; 0, fJ ;::;; 0.

The simple property of L(x) stated in (6) actually leads to a rather definite representation of L(x) (Note 5). For the detailed study of the behavior of L(x) at oo, see Karamata [1] and also Feller [1], II; VIII, 8.9. However, we do not need the particular properties of such a function except for the following simple consequence of the definition. For any A.> 0, (7) L(xy)JL(y) :5:: C:x!l for x > 1,y;::;; A, for some A= A(7J), C being a constant. Choose n such that 2n-1 < X ~ 2n. For any 1 > 7], let A be such that L(2x)JL(x) < 1 + 7J, for x > A. Then

L(xy) L(xy/2) L(xy) = L(xy/2) . L(xy/2 2)

~ (1 + 7J)nL( £,: )· Since X"'L(x) is nondecreasing,

Hence

L(xyJ2n-1) ( xy) L(xyJ2n) . L

zn

(8)

8.5.

303

ABELIAN THEOREMS FOR LAPLACE TRANSFORMS

We then see from (8) that L(xy)fL(y)

~ ~

2(1 + 7J)n ~ 22(1 + 7J)n-1 ~ 22(1 C(1 + 7J)Iogz ~ C.xlogll+ril ~ Cx'l.

+ 7J)Iogz!log2

This shows (7).

Theorem 8.5.2. Suppose that F(O) = 0. Let lf>(x) be a nondecreasing function for large x which has the form (5). If (9)

F(x) '""""'lf>(x) as x---+- oo, then f( a) exists for a > 0 and

/(a)'""""' F(a

+ 1) ¢>(1/a)

(10)

as a---+- 0+. Proof.

From (4),

I:

e-"z dF(x)

= e-"RF(R) + a

I:

e-"zF(x) dx.

(11)

Since F(x) '""""'x«L(x), F(x) = O(x"'+>'~) for any 7J > 0 [which we can see from (7)], taking y to be a large fixed constant. Hence the first term on the right-hand side of (11) converges to zero for a > 0 as R ---+- oo, and the integral on the right converges absolutely for a > 0 as R ---+- oo. Therefore,/( a) exists for a > 0 and

/(a)= a I~ e-z"F(x) dx, where the integral is a Lebesgue integral. Now [f(a)fF(a

+ 1)¢>(1/a)] =

[afF(a

= [F(a

+ 1)] I~ e-"z[x"'L(x)fa-"'L(1fa)] dx

+ 1)]-

1

I~ e-uu"'[L(ufa)fL(1fa)] du.

For each u and small a, L(ufa)fL(1fa) ~ Cu'~ for any A. > 0. For large u (and since Cu'~+"'e-u is integrable), we may apply the dominated convergence theorem to get, using (6),

+ 1)]- Ioo e-uu"'[L(ufa)/L(1fa)] du = [F(a + 1)]- I~ e-uua. du = 1.

lim [F(a a-+o+

1

0

1

This shows the theorem.

VIII.

304

In this theorem, if L(x) following corollary.

Corollary 8.5.1.

=

MORE THEOREMS ON TRANSFORMS

C (constant), x > 0, then we have the

Suppose F(O) = 0. If a

F(x) ,.._, Cxrz

as

~

0 and if

x - oo,

(12)

then

/(a),.._, CF(a

+ 1)a-rz

a- 0+.

as

(13)

If F(x) is absolutely continuous, the theorem and the corollary will take the following forms, respectively.

Corollary 8.5.2. If (x) is the function in Theorem 8.5.2 and 'f'(x) is locally integrable in (0, oo ), then

I:

as

'f'(u) du ,.._, (x)

x-oo

(14)

+ 1)¢(1/a).

(15)

implies that

/(a)=

I

-+oo

0

Corollary 8.5.3. If a

I:

~

e-az'f'(X) dx ,.._, F(a

0, 'f'(x) is locally integrable, and as

'f'(u) du ,.._, Cxrz

x - oo,

(16)

C being a constant, then

/(a)=

f

-+00

0

e-a~(x)

dx ,.._, CF(a

+ 1)a-rz

as

a- 0+.

(17)

8.6. Tauberian Theorems The direct converse of Abel's continuity theorem that the convergence of (1)

to s, implies 00

lim x-+1-0

L

n-o

anxn = s

(2)

8.6.

305

TAUBERIAN THEOREMS

is not true. [The series in (2) is convergent for 0 ~ x < 1, and (2) does not hold.] The theorems, with some additional conditions on an, that deal with the converse of the Abelian theorems are called Tauberian theorems, and were already introduced in Section 4.5. The conditions needed are called Tauberian conditions. Actually Tauber [1] was the first to have shown the theorems that are the converses of Abel's continuity theorems. In fact Tauber showed that if an= o(1fn)

(3)

and (2) hold, then (1) converges to s. He also showed another theorem: If (2) holds and a1

+ 2a + · ·· +nan= o(n), 2

(4)

then (1) converges to s. Littlewood [1] replaced (3) by an= 0(1/n),

(5)

which was a rather surprising extension of Tauber's theorem. Since then, a tremendous number of results on Tauberian conditions have been made. (See Hardy [2].) Karamata [2] gave a quite ingenious and simple proof of this result. Another striking result is the Hardy-Littlewood generalization of condition (5). They showed that (5) can be replaced by nan> -k,

n = 1, 2, ... ,

(6)

where k is a constant (Hardy [2], Hardy and Littlewood [3], and Landau [1 ]). Schmidt [1] (also Hardy [2]) showed the following Tauberian theorem, writing Sn = Lk-o ak. If (7) lim inf (sn - sm) ~ 0 m--Joooo

when n > m, (n- m)fm---.. 0, then (2) implies the convergence of (1) to s. Condition (7) is more general than (6), as is easily seen. Wiener [4, 6, 7] unified the Tauberian theorems proving the so-called general Tauberian theorems in which he formulated the Tauberian the-

VIII.

306

MORE THEOREMS ON TRANSFORMS

orems in the most general framework, adopting the convolution forms and using Fourier transform theory. However, this subject is beyond the scope of this book. Besides, so far, mostly Tauberian theorems for Laplace transforms have been used in probability theory. We begin with the Laplace-Stieltjes transform analog of Tauber's original theorems. We deal with the Laplace-Stieltjes transform

f(u) =

s;<» e-az dF(x)

with a real parameter u, where F(x)

V(O, oo).

E

Theorem 8.6.1. Suppose that F(O)

(8)

=

0, (8) exists for u > 0, and

lim f(u) = s. a->-o+ If

JZ+l I dF(u) I = z

(9)

as

o(1Jx)

X--+00,

(10)

then

F(oo) = lim F(x) = s.

(11)

z->-<»

Proof.

Write

JZ+l dF(u) = g(x), z

X ~0.

(12)

From Theorem 7.1.3 (ii), g(x) = o(e,z) for any y > 0 as x--+ oo. Now take any u > 0. Then the integrals in the following relation exist and

J~ e-azg(x) dx = -u-1e-azg(x) ~~ + u-1 J~ e-az d[F(x + 1)- F(x)] =

u- 1g(O)

+ (eaju) J~ e-az dF(x)-

u- 1 J~ e-az dF(x)

(taking into account that in choosing y < u, e-azg(x)--+ 0 as x--+ oo),

J~ e-azg(x) dx = u-lg(O)- (eaju)

J:

e-az dF(x)

+ [(ea- 1)/u] J~ e-az dF(x) =

J:

{[1- ea
+ [(ea-

1)Ju]J(u).

8.6.

307

TAUBERIAN THEOREMS

Using (9), the right-hand side converges to

J:

s-

(1 - x) dF(x)

as

a- 0+,

that is,

lim a-+O+

J"" e-azg(x) dx =

s-

0

JI (1 -

x) dF(x) = s-

0

JI F(x) dx.

(13)

0

[Integration by parts is used with F(O).] Now consider

f ila g(x) dx- f"" e-azg(x) dx 0

fila (1 - eaz)g(x) dx-

=

0

J""

lla

0

e-azg(x) dx. (14)

Since g(x) = o(1fx) by (10), the second integral on the right is

e-azo(1fx) dx ~ f oo Ila

(J

o(1) f"" e-az dx = o(1), lla

and for the first integral, it is, in absolute value, not greater than a

f 0ila xI g(x) I dx ~ a fA.0 xI g(x) I dx

~

a

J:

xI g(x) I dx

+ ea fila A. dx + e,

where e is any number greater than 0 and A is taken so that I g(x) e(1/x). Hence letting e- 0, a- 0,

f

ila 0

I<

xlg(x)ldx=o(1).

Therefore, (13) converges to zero as a- 0+. From (14), we then see that lim

JR g(x) dx =

R-+oo

s-

0

JI F(x) dx.

(15)

0

Now

J:

g(x) dx

J:

=

+ 1)- F(x)] dx JR+I F(x) dx- JR F(x) dx

=

JRR+I F(x) dx-

=

[F(x

I

= F(R)

0

+ JR

R+I

II F(x) dx 0

[F(x) - F(R)] dx-

II F(x) dx, 0

VIII.

308

MORE THEOREMS ON TRANSFORMS

where the last member but one is, in absolute value, not greater than

II:+l dx I: dF(u) I~ I:+l I dF(u) 1. which converges to zero by (10). Comparing the above with (15), we obtain lim F(R)

=

s.

R4oo

The following is a special case of the foregoing theorem, where F(x) is absolutely continuous.

Theorem 8.6.2. If cf>(x) belongs to Lin every finite interval of (0, oo) and the integral

I

_,.oo

0

e-azcf>(x) dx = f(a)

exists for a > 0, then the conditions lim /(a)= s

(16)

a4o+

and

cf>(x) = o(1fx) imply the convergence of

I:

as x---+ oo,

(17)

cf>(x) dx

as R---+ oo and

I

_,.oo

0

cf>(x) dx = s.

(18)

Also it is obvious that Tauber's first theorem is a particular case of Theorem 8.5.1. The analog of Tauber's second theorem is the following:

Theorem 8.6.3. If F(O) = 0, Eq. (8) converges for a > 0, and (9) holds, then (11) holds if and only if G(x)

=

I:

u dF(u)

=

o(x)

as x---+ oo.

(19)

8.6.

309

TAUBERIAN THEOREMS

Proof.

"Only if" part.

Using integration by parts,

J: F(u) du.

G(x)

=

xF(x) -

G(x)fx

=

F(x) - x- 1

Hence

Since F(x)

---+-

J: F(u) du.

s as x---+- oo, G(x)fx---+- s - s = 0.

"If" part.

r

e-ax dF(x)

= f~

(e-""fx) dG(x)

= [e-"RG(R)/R]- e-"G(l)

+a

r

[G(x)fx]e-"" dx

+

r

[G(x)fx 2]e-"x dx.

From (18), the integrals on the right-hand side converge absolutely as R ---+- oo and the first term converges to zero for a > 0. Hence

s;oo e-"x dF(x) =

-e-"G(l) +a f~ [G(x)fx]e-"x dx

+ J~ [G(x)fx2]e-ax dx. Since

I(] f~ [G(x)fx]e-ax dx I ~ (] [for a large R such that

r

[G(x)fx]e-ax dx

I G(x)fx I <

(20)

+ ea J; e-"x dx

e for x > R], we have

Ia J~ [G(x)fx]e-"x dx I ~a J~ [G(x)fx]e-"x dx + e. Letting a---+- 0+ (R being fixed), we have a J~ [G(x)fx]e-ax dx---+- 0

as

a---+- 0+

and the left-hand side of (20) converges to s-

J:

dF(x)

=

s- F(l).

We have, from (20), lim Ioo [G(x)fx 2 ]e-ax dx = s- F(1) a-+o+

+ G(1).

1

Now we apply Theorem 8.6.2, noting that G(x)fx2 = o(1fx), and obtain

I~ [G(x)fx 2 ] dx = s - F(1) - G(1 ). Since

I~

[G(x)fx 2 ] dx

~~+I~ [dG(.~)fx]

=

-[G(x)fx]

=

-[G(R)fR]

=

+ G(1) + I~ dF(x) -[G(R){R] + G(1)- F(1) + F(R),

and G(R) = o(R), we have, letting R

---+

oo,

lim F(R) = s, R-+oo

which was to be proved. When F(x) is absolutely continuous we have the following theorem:

Theorem 8.6.4. If if>(x) is locally integrable in (0, oo) and (16) holds, then

I:

ucp(u) du

= o(x).

(21)

Now we are going to show the converse of Theorem 8.4.2 with an additional condition. It is a much deeper Tauberian theorem than the above Theorems 8.6.1 or 8.6.3. Theorem 8.6.3 gives a necessary and sufficient condition, but the condition

I:

dF(u) = o(x)

does not seem very convenient to conclude that

I:

dF(x)

converges. It would be better if a condition were imposed on F(x) itself.

8.6.

311

TAUBERIAN THEOREMS

Theorem 8.6.5. Suppose that F(O) = 0. Let cp(x) be a nondecreasing function for large x such that cp(x) = x"L(x), a > 0, for large x where L(x) is a function of slow growth. If F(x) is nondecreasing and J(a) = J~ e-ax dF(x) exists for a > 0 and

/(a)"' F(a

+ l)cp(1/a)

as

a ---+ 0+,

(22)

then as

F(x) "'cp(x)

x---. oo.

So, in this theorem, the condition that F(x) be nondecreasing, is the Tauberian condition. Since F(x) is nondecreasing, the Laplace-Stieltjes transform/( a) is absolutely convergent.

g(x)

FIGURE

8.3.

The following is substantially the proof of Karamata [2]. (Also see Hardy [2], Section 7.11.) Proof.

Suppose a

> 0. Let (Fig. 8.3)

g(x)

=

{

x-1

e- 1 0

0, ,

~X~

~X~

1, e- 1 •

Let g. (X )

g_,(x)

= { =

x- 1

0,

x- 1 { 0,

+ e,

e- 1 0

~X~

e ,

e- 1 0

~X~

-

~X~

~X~

1, e- 1 ; 1, e- 1 •

VIII.

312

MORE THEOREMS ON TRANSFORMS

Further, construct a continuous function h,(x), which is defined by 0

~

x

~ e- 1 -

e

~

x

~

e- 1

~

x ~

0,

h,(x)

= { linear,

e- 1

-

g,(x),

e,

e- 1 , 1,

where e is an arbitrarily small positive number, e- 1 - e > 0 and h,(e- 1 - e)= 0, h,(e- 1 ) = g,(e- 1 ). Because of the Weierstrass approximation theorem, there is a polynomial P,(x) such that I h,(x)- P,(x) I < e. Then obviously g(x)

+ e ~ h,(x) ~ P,(x) + e;

that is, g(x)

< P,(x).

In the same way define h--a(x) as h_,(x)

=

0, { linear, g_,(x),

e-

1 -

0

~X~

e

~

x

~

e-1 ~X~

e- 1 e-1, 1.

E,

Then there is a polynomial p.(x) such that p,(x)

< g(x),

so that altogether, p,(x)

< g(x) < P.(x).

(23)

Besides, taking e small enough, we may have

I P,(x)

- p,(x)

I ~ !5,

(24)

where !5 is a preassigned small positive quantity.

Noting that F(x) is nondecreasing, from (23),

I~ e-azp,(e-az) dF(x) ~I~ e-azg(e-az) dF(x) ~ I~ e-azP,(e-az) dF(x). Since p,, P, are polynomials and the integral f( a) exists for all a > 0, the integrals on both extreme sides exist. Note that

Ioo e-azg(e-az) dF(x) = f1/a dF(x) = F(1/a). 0

0

8.6.

313

TAUBERIAN THEOREMS

Then the above inequalities turn out to be

Now by the assumption (22),

J~ e-aze-naz dF(x)

=

J~ e-
as a--+ 0+, for every nonnegative integer n. The right-hand member is F(a

+

1)(n

+

1)-"a-"L[1/(n

+

l)a].

Since L is of slow growth, for each n, L[1f(n

+ 1)a] ,...., L(1fa)

as

a--+

0+.

Therefore,

J~ e-aze-naz dF(x),...., T(a

+

= a!fo(1/a)

1)(n

+

1)-"!fo(1/a)

J~ e-ze-nzx"-1 dx,

0'-

0+.

Since p,(e-az) is a polynomial and hence a linear combination of rnaz, we have, from the above relation,

Similarly,

Hence from (26) we have, for any positive 'YJ,

a!fo(1/a) J~ e-zx"- 1p,(rz) dx (1 - 'YJ) ~

F(1fa)

~ a!fo(1/a) J~ e-zx"- 1P,(rz) dx (1

+

'YJ)

for small a > 0; that is,

(1 - 'YJ)a J~ e-zx•- 1p,(rz) dx ~ [F(1/a)/!fo(1/a)]

~ (1

+ rJ)a Je-zx"- 1P,(e-z) dx.

(27)

On the other hand, if we take F(x) =

x~

in (26),

a I~ e-azp.(e-az)x~- 1 dx ~ (1/a~) ~a I~ e-azP,(e-az)x~- 1 dx,

that is, a I~ e-zp,(e-z)x"- 1 dx ~ 1 ~a I~ e-zP,(e-z)x~- 1 dx.

From (25 ), the difference between two extreme sides is less than the arbitrary quantity T(a + 1)<5. So, taking a look at (27),

I [F(1/a)/
+ 2rJ)T(a + 1)<5,

which gives us the required proof.

Theorem 8.6.6. Theorem 8.6.5 still holds if a = 0. The proof of Theorem 8.6.5 fails when a = 0, but the theorem keeps holding even if a= 0. We do not go into details in this case. We give just the idea. First, as in the proof of Theorem 8.6.5, we may show the following theorem, the proof of which is left to the reader.

Theorem 8.6.7. If F(x) is a nondecreasing function with F(O)

=

0,

/(a)= I~ e-az dF(x) "'L(1/a), and g(x) is continuous on [0, 1], then

I~ e-azg(e-az) dF(x) "'g(1)L(1/a). Then in this theorem, taking g(x)

=

x-1 (1 - log x-1 ),

e-1

~

x

~

1;

g(x)

=

0,

0

~

x

~

e- 1,

we may show that Theorem 8.6.6 holds, after some manipulation. (For details, see Hardy [2], pp. 169-170.)

8.7. Multiple Fourier Series and Transforms Let us denote the m-dimensional Euclidean space by Rm. The inner product of two points x = (x 1 , • •• , xm) andy = (y 1 , ••• , Ym) is defined by (1)

8.7.

315

MULTIPLE FOURIER SERIES AND TRANSFORMS

and the norm of x, by

II x II =

m

L X1c /c-1

(

2

)1/2

(2)

•

Let the interval in Rm, -n < x1c < :n, k = 1, 2, ... , m be denoted Em. Suppose that f(x) is periodic with period 2:n for each x1c, and that f(x) = f(x 1 , ••• , xm) E V(Em), that is, f(x) is integrable over Em:

f

Em

J" ··· J"

lf(x) I dx =

-,.

-,.

IJ(x1, .. . , Xm) I dx1 · · · dxm < oo.

Construct

c,.

=

Cn 1 , ... ,n,.

= (Z:n)-m X

=

J

(2:n)-m

Em

f(x) exp -i(n, x) dx

J:,. ··· S:,.J(x1•· · ., Xm)

exp -i(n 1x1

+ · ·· + nmxm) dx

1 • • •

dxm

(3)

for -oo < n < oo or n1c = 0, ±1, ±2, ... , k = 1, 2, ... , m. en is called the Fourier coefficient of f(x). The Fourier series of f(x) is defined by

J(x) ,._

L Cneiln,.zl,

(4)

n

This is a natural extension of the Fourier serJes of a function of V( -:n, :n) to the multidimensional case. It is easy to see that the sequence {ei(n,.z> }, n = (n 1 ,. •• , nm), (n1c = 0, ±1, ... , k = 1, 2, ... , m) constitutes an orthogonal sequence on Em. The partial sum of (4) is N1

SN= SN 1•···· Nm =

where [N = (N1 ,. variable, by

"f..J

n~--Nl

•• ,

Nm

...

" "-.J

nm=-~Vm

cn 1, ... ,nm expi(nx 1 1 +···+n mx) m'

Nm)], which is given just as in the case of one

t = (t 1 ,. •• , tm)· However, the convergence of SN as N- oo is more involved than in the one-variable case. N - oo here means that each n1c- oo, k = 1, ... , m. As a matter of fact, the convergence of (5) at a point X 0 is not determined by the property ofj(x) in a neighborhood of x0

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MORE THEOREMS ON TRANSFORMS

only. While in the case of R 1 , the convergence at x 0 depends only on the property of a function in a neighborhood of x 0 (Theorem 4.1.1 ). However, we may say that the convergence at x 0 depends only on the property of a given function in a set called a cross neighborhood of x 0 • A cross neighborhood is the set of x = (x 1 , • •• , x111 ) such that at least one of I xk - Xo,k I < <5, k = 1, ... , m holds, where x 0 = (x 01 , • •• , x 0 ,n)· (See Tonelli [1], Chapter 9 and Zygmund [5], Vol. II, p. 305.) It is more convenient to consider the spherical partial sum

SR =

L

c, exp i(n, x)

llnii~R

L

n~+···+n!,;:;;R•

Cn 1, ... ,nm

exp i(n 1 X1

+ ··· + n

111

X111 ).

(6)

(See Bochner [2, 3].) If the sequence (n 1 , • •• , n 111 ) is arranged in the order of II n 11 2 = n 1 2 + · · · + n1112 (n's with the same II n II may be arranged in arbitrary orders) and according to this order of n, we consider the sequence of orthogonal functions {ei
JRm

IJ(x) I dx = J~oo

The function/(t), t

E

•· ·

J:

IJ(x 1 , ••

• , X 111 )

I dx

1 • • •

dxm <

oo.

(7)

R 111 , defined by

[1/(2.n) 111' 2] J Rmf(t) exp -i(t, x) dt

= [1/(2.n)m!2]

J: ...

X exp -i(t 1x 1

J:J(x 1 , . . . ,x111 )

+ · ·· + t

111

X111 ) dx 1

• • •

dx,.1

(8)

is called the Fourier transform of f(x). It is quite easy to see that /(t) is bounded and continuous. Moreover lim /(t) lltll4ooo

= 0.

(9)

8. 7.

317

MULTIPLE FOURIER SERIES AND TRANSFORMS

+ ··· +

tm 2 ---+ oo, tk---+ oo for some k. Suppose for definiteSince t 12 ness that t 1 ---+ oo. Then the right-hand side of (8) is written [1/(2n)m' 2 ] X

J:

exp it 1x 1 dx 1

exp -i(t 2 x 2

[

J~oo

···

+ · ·· + tmXm) dx

J~ooj(xt•· . . , Xm)

2 • • •

dxm ]·

The inner (m - 1)-tuple integral is a function E V(R 1 ) (as a function of x 1 ). Hence by the ordinary Riemann-Lebesgue lemma, ](t) ---+ 0. Also we may easily see that

ST(t) = [1/(Zn)m12 ] J

lr

/(t)eiCt,x) dt,

(10)

where T = (T1 , ••• , T m) and IT is the interval in Rm: - Tk < k = 1, 2, ... , m, is equal to

t~c

<

T~c,

We shall show the theorem which corresponds to Fourier's single integral theorem and the inversion formula. But from a consideration of the convergence of multiple Fourier series stated above, we expect that some strong conditions will be needed for this. We shall give a theorem due to Bochner [5], p. 254, Theorem 66. A result that is the analog of a consequence of the second mean value theorem that was needed in the one-dimensional case will be shown first.

Lemma 8.7.1. Let f(x) = f(x 1 , ••• , xm) be a nonnegative and nonincreasing function of each variable X~c in ak ~ X~c ~ h~c , k = 1, 2, ... , m, and let C~c(x~c) be a function of a single variable such that (12)

for a" ~ u ~ h~c, k = 1, 2, ... , m, where C~c is a constant. Then we have

0

~

t' •••tm j(x a,

am

~f(a., ..

. , am)

1 , • •• ,

Xm)

0 g(x~c) dx

k=l

1 • • •

dxm

m

n ck.

(13)

k-1

Proof. If k = 1, this is a direct consequence of the second mean value theorem. We shall prove (13) by mathematical induction. Suppose that

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MORE THEOREMS ON TRANSFORMS

the theorem is true for m = v - 1, and we show that it is also true for m = v. Write

Then

Now f(x 1 , • •• , x._ 1 , x) is, for every fixed x, a function of v - 1 variables, which is nonnegative, nonincreasing for each xlt , 1 ~ k ~ v - 1. Since (13) holds form= v - 1,

Obviously J(x) is nonincreasing as x increases. Hence by the second mean value theorem,

0

~ J ~](a)

r

"•

g.(x) dx

for some

~.

a ~~~b.

Now we shall show the theorem.

Theorem 8.7.1. If the function f(x 1 , • •• , Xm) is nonnegative, bounded, and nonincreasing in each variable in alt ~ xlt, k = 1, 2, ... , m, then

where T = (T1 , • •• , T m) - oo means that all Tit - oo and means Proof.

It is easily seen that the integral on the left-hand side of (14) is

8.7.

319

MULTIPLE FOURIER SERIES AND TRANSFORMS

convergent from Lemma 8.7.1, in which Ck is replaced by

[where t/J(x 1 , ••• , xm) stands for f(a 1 than

+x

1 , ••• ,

am+ xm)] is not greater (15)

we have Ak ~ ~k numbers. Since

~

Bk, by Lemma 8.7.1, Ak, Bk being any large positive

where TAk ~ 'YJ ~ T tBt. expression (15) converges to zero as Bu ... , Bm -+ oo and A 1 , A 2 , ••• , Am -+ oo. This shows that the left integral of (14) converges. If some Ak = 0 in (15), then the corresponding one is

J:t (sin Ttxkfxk) dxk

= 0(1 ).

Consider (16)

which is obviously equal to the integral on the left-hand side of (14), where u = (u 1 , ••• , um), and form

Jj =

( -2 n

)mJ . .. J-+oo . .. J 1/J (---, u ••• ,--Um) n m-sin- d uk u 1

a

Tm

Tm

k-l

Uk

in which at .least one of the integrals (jth) in the multiple integral has the integration range (a, oo), a > 0. If a is large enough, then

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MORE THEOREMS ON TRANSFORMS

Hence, as in (15), if>(A 1 , ••• , Am) and sin T,.xkfxk being replaced, respectively, by if>( ... , afT, ... ) and sin ukfuk, we have

I]; I~

E

for an arbitrarily given e > 0. Then I = (; = Ii

r J: t~>(

;:. ... , ;: )

fl

si:kuk du

+ L Jj

+ Lli•

say, where A = (a, ... , a). We have

L I]; I <

2m-Ie.

Fixing A, we let Tk- oo, k = 1, ... , m. Then by the bounded convergence theorem,

k-t, ... ,m

= (~)m tf>(O+, •• •'0+) n

fi JAo _s_in_u.::....k duk. uk

k-I

Letting a - oo, we have lim

T~~o-+oo

I= ¢>(0+, ... , 0+),

k-1,2, ... ,m

which is what we wanted to show. Suppose now that f(x 1 , ••• , Xm) is defined m Rl,! 1: 0 k = 1, 2, ... , m, and is continuous there, and

f(x) =/(xi,· · ·, Xm) as

-

~ xk ~

0

oo,

(17)

II x II- oo. Furthermore, let f have the partial derivative g(x) = /z1,,,z,.(xl, ... , Xm),

which is supposed to belong to LI(Rl,! 1) and be continuous. We shall show that, in R<J.>, such anf(x) can be represented as

f(x) = /Ul(x) - j<2>(x),

(18)

8.7.

321

MULTIPLE FOURIER SERIES AND TRANSFORMS

where J
< 0,

Then

g(x)

= g
g<2l(x).

Construct the function

i = 1, 2. This integral exists, because of the assumption that g(x) E U(R).!l). Then h
h(x)

= =

f'

Xm

s:

since lim fu 2 , ••• ,u,. (u 1 , u 2 , ••• , um) = 0, u 1 -+oo

by (17). By repeated integration, we finally get (19)

Hence, if m is even, define

/m(x)

=

=

h
j<2l(x)

=

h(2l(x),

j<2l(x) = hOl(x).

h<2l(x),

and if m is odd, define

j
Then we always have, from (19),

f(x)

fUl(x) - f'2l(x).

=

Hence the statement containing (18) is proved. Once we obtain this, it is easy to show from Theorem 8. 7.1 that the following theorems hold.

Theorem 8.7.2. Suppose that f(x) is continuous in the whole space lim f(x)

= 0,

(20)

llxfl-+oo

and that f(x) has the partial derivative fx 1 , ••• ,x,.(x) in Rm, which is continuous and is in Ll(Rm)· Then lim

Tk-+oo

.n-m

J

R,;

f(x 1

+u

1 , ••• ,

x,,

+ um)

TI (sin Tku Juk) du 1

k-1

k=-l, ... ,m

(21)

Theorem 8.7.3. Letf(x) satisfy the conditions in the preceding theorem. Moreover, suppose thatf(x) E Ll(Rm)· Then for the Fourier transform in Rm,

J

](t) = (Z.n)-m/2 Rm f(x)e-i
(t, x)

L

=

tkxk,

k-1

we have the inversion formula

f(x) =

lim (2.n)-m12 T~c-+oo

JTm JTm-1 -Tm -Tm-1

···

J

T1

-T1

A

•

f(t)e•
(22)

k-l, ... ,m

Proof of Theorem 8.7.2. This is immediate from Theorem 7.8.1 because, if we divide Rm into 2m octants Rl:;1, v = 1, ... , 2m, then for each octant,

Jn;.;<•l f(x + u) TI (sin Tkukfuk) du = J u f(x! ± Xm ± vm) fi k-1

Rm

1

V 1 , ••• ,

k-1

(sin Tkvkfvk) dv,

(23)

where v = (v 1 , ••• , vm), R!,! 1 is the first octant, 0 ~ vi < oo, i = 1, 2, ... , m, and in (23) only one set of signs is adopted.

8.8.

NONDECREASING FUNCTIONS IN

R 111

323

Then from Theorem 7.8.1, each integral (23) converges to (n/2)"'f(xi, ... , x111 ) as TI, ... , T 111 --+- oo since it is continuous. This gives Theorem 8.7.2. The integral on the right-hand side of (22) is

Proof of Theorem 8.7.3.

(2n)-m

I~;m

= n-m I

I:~l exp(il:ttXk)dt I~oo

... Rm

= n-m I Rm

...

I~oof(u)exp(-il:tkuk)du

TI [sin Tk(uk- xk)f(uk- xk)] du f(x + u) TI (sin Tkukfuk) du. f(u)

k-I

k-I

Hence the theorem follows from Theorem 8.7.2.

8.8. Nondecreasing Functions and Distribution Functions in Rm Let F(xi, . .. , x 111 ) be a function defined over them-dimensional Euclidean space R 111 • We write LI~~1 F(xi•· .. , x111) = F(xi•· . . , xk-I• xk

+ hk, xk+I•· . . , X

111 )

- F(xi, x 2 , • • • , xk•· . . , X 111 ) so that, for instance, Ll~~~ Ll~~~ F(xi, . .. ,

X111 )

+ hi, X2, ... , X [F(xi +hi, x 2 + h2, x

= LI~~1 [F(xi =

111 )

-

3 , ••• ,

F(xi, X2, ... , Xm)] X111 ) - F(xi, x 2 + h, x 3 , ••• , X111 )]

-[F(xi +hi, x 2, ... , x111 ) - F(xi, x 2, ... , xk)]. Generally, A(ml , , ,

LJhm

AUlF(x I••

L1h 1

• ·•

X ) m

+ hi, x2+ h2, ... , X + h F(xi, x 2 + h 2, ... , X + h - · · · - F(xi + hi, X2 + h2, · · ·, Xm-I + hm-I, XIII) + F(xi, x x + h X + h + ··· + F(xi + hi•· · ., Xm-2 + h111-2• Xm-I• Xm)- ''' + (-l)"'F{xi, x2, ... , X

= F(xi

111 ) -

111

2 ,

3

3 , ••• ,

111 ).

111

111 )

111

111 )

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32+

MORE THEOREMS ON TRANSFORMS

A function F(x 1 , ••• , xm) is called nondecreasing if for any

h 1 ;?; 0, ... , hm ;?; 0.

In particular, a nondecreasing function is nondecreasing for each variable = 1, ... , m. But note that the converse is not true. For instance, consider if x 1 + x 2 ;?; 0, (1) if x 1 + x 2 < 0. xk, k

It is easy to see that F(x 1 , x 2 ) is non decreasing for both x 1 and x 2 • However,

Ll 32 Ll 3 1 F(-1, -1) = F(2, 2)- F(-1, 3)- F(3, -1)

+ F(-1, -1) =

-1.

Let M 1 be the class of all open intervals in Rm. Consider the realvalued function F(I) defined over M 1 • Let / 0 be a given interval, ak < xk < bk, (k = 1, ... , m). Let/~ and / 06 be the intervals, ak- t:5 < xk < bk + 1:5, (k = 1, ... , m), and ak + t:5 < xk < bk - t:5 (k = 1, ... , m), respectively, where t:5 < min 1 :;;k;:;;;m(bk - ak)· If F(I~) ---+

as

F(/0 ),

F(lii")

---+

F(/0 )

(2)

0+, then / 0 is called a continuity interval of F(I). Let F(x) be a function defined in Rm. Define, for an interval t:5 -

k = 1, 2, ... , m, the interval function F(I) by (3) This interval function is called the interval function generated by a function F(x). If F(x) is a nondecreasing function, then by the definition of nondecreasingness, F(I) ;?; 0. Suppose that a probability space (.Q, B, P) be given and consider them random variables X 1 (w), ... , Xm(w). The ordered set X(w) = [X1 (w), ... , X ... ( w)] is called an m-dimensional random variable or a random vector. Let X= (x 1 , ••• , Xm) E Rm:

8.8.

NONDECREASING FUNCTIONS IN

Rm

325

defines a function on Rm. This function has the following properties: (i) (ii)

F(x)

=

lim

F(x 1 , ••• , xm) is nondecreasing. F(x 1 , .•• , Xm)

= 1.

Xk-+oo

k=l, ... ,m

(iii)

lim.,t-+-oo F(x 1 , ••• , xm)

(iv)

F(x 1 ,.

• •,

= 0, for each k = 1, 2, ...

m.

Xm) is left-continuous in each variable xk.

Let all xi go to infinity except x1 . Then lim F(x 1 , ••• , Xm) = F(+oo, .. . , oo, x1 , oo, ... , oo) = F1(x1)

(4)

Zk-+oo fr*j

exists. This function F1(x1) is called the probability marginal distribution function or simply the marginal distrib.ution function. The function satisfying (i )-(iv) is usually called the probability distribution function or simply the distribution function (of X). However, here we call any function satisfying (i)-(iii) a distribution function as we did in Section 3.1. For the distribution function F(x), an interval

k= 1, ... ,m, is a continuity interval of F(I) generated by F(x) if and only if ak and bk are continuity points of Fk(x1), k = 1, ... , m. Then the number of continuity intervals is countably infinite. Suppose a distribution function F(x) over Rm. As before we may construct the interval function F(I) generated by F(x). Let [- 6 be defined as in (2). Set tP(/) = li~-+o+ F(I- 6 ), where tP(/) is an interval function. for any finite union E of disjoint intervals, tP(E) is defined by additivity. Then we get an additive function defined over a field containing intervals. Then by the extension theorem of measure, we may define a probability measure defined over all Borel sets in Rm , where a set belonging to a small u-field which contains all open intervals is called a Borel set in Rm. It will be seen immediately that F(I) and tP(/) are identical for all continuity intervals I. tP(E) is called the measure generated by F(x). The above discussion easily extends to a bounded nondecreasing function F(x), in which the measure tP(E) generated by F(x) is a finite measure. Let E be any Borel set and let g(x) be any Borel measurable function on Rm. Let tP be a measure generated by F(x). According to the general

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MORE THEOREMS ON TRANSFORMS

definition, we may consider the integral JEg(x) d(/J, which is supposed to exist. The Stieltjes integral

IE g(x) dF(x) =IE g(x 1 , ••• , x,,) dF(x 1 , ••• , xm) is defined to be JEg(x) d(/J. If G(x) is a bounded Borel measurable function the integral always exists.

8.9. The Multiple Fourier-Stieltjes Transform Let F(x 1 , x 2 , ••• , xm) be a bounded nondecreasing function. The class of such functions is denoted by M = M(Rm)· The class of distribution functions is denoted by D = D(R,n).

f(t) = I

R,.

exp i(t, x) dF(x)

is called the Fourier-Stieltjes transform of F(x) E M, where t = (t 1 , ••• , tm), x = (x 1 , ••• , xm)· Since eW.zl is bounded by 1 in absolute value, there is no problem at all in defining (1 ). If F(x) is a distribution function, f(t) is called the characteristic function of F(x) or of the random variable in Rm whose distribution function is given by F(x). It is obvious that for F(x) EM,

/(0) =/(0, 0, ... , 0) = F(oo, oo, ... , oo)- F(-oo, -oo, ... , -oo) (2) /(-tl, -t2, ... , -tm) =f(t 1 , t 2, ... , tm)

(3)

and that f(t) is a continuous function. The last statement is obtained by use of the bounded convergence theorem. In this section we are content with showing the inversion formula for the Fourier-Stieltjes transform. There does not appear to be any difficulty in showing it in comparison with the corresponding theorem in the onedimensional case (Theorem 4.4.1 ).

Theorem 8.9.1. Let f(t) be the Fourier-Stieltjes transform of a function F(x) EM in Rm. Then if/: ak < xk < bk> k = 1, ... , m, is a con-

8.9.

327

THE MULTIPLE FOURIER-STIELTJES TRANSFORM

tinuity interval of F(l) generated by F(x), we have

F(I) = lim _1_ IT

T-+oo (2n )m -T

... IT -T

TI k=l

exp( -itkbk) - exp( -itkak) itk

• J(tl•· · ., tm)dt~ • • • dtm. Proof.

(4)

The integral on the right-hand side is

= I~oo

... I~oo dF(x, ... ' x)

in which 1

-2 n

IT -T

_ - -

exp( -itkbk) - exp( -itka,.) . d . exp ztkxk tk ztk 1 2n

IT -T

sin tk(xk- bk) d tk tk

+

1 2n

IT -T

sin tk(xk- ak) d tk tk·( 6 )

This converges as T- oo to - U(xk - bk) + U(xk - ak), where U(x) is a (one-dimensional) standardized unit distribution function : 0,

U(x)

= { 1,

i.

X< 0, X> 0, X=

0.

[see Section 3.1, Eq. (4), Lemma 2.6.3]. Thus the limit of (5) is for for for

ak < xk < bk , xk < ak or xk > bk , xk = ak> bk.

Therefore the limit of the right-hand side of (4 ), when T - oo, is

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MORE THEOREMS ON TRANSFORMS

which is the measure of the interval I generated by F(x) when I is a continuity interval. This completes the proof. The formula (4) is called the inversion formula. As is obvious from the above proof, the right-hand side can be replaced by lim Tk-+oo k-l, ... ,m

1 (2n)m

JTm

JT1

-Tm

-T1

fi

k-I

exp( -itkbk) - exp( -itl!lk) itk (7)

An implication of Theorem 8.9.1 is that the Fourier-Stieltjes transform determines uniquely the measure generated by the given function of M. NOTES 1. The proof of Theorem 8.1.1 is the original proof given by Hardy [1]. Another proof is also found in the same paper. 2. The proof of Theorem 8.2.1 is adopted from Paley and Wiener [2]. A proof apparently independent of Phragmen-Lindelof's theorem was given by Rudin [1]. 3. The proof of the first part of Theorem 8.3.1 is due to Ingham [1] and the proof of the second part is that of Levinson [1]. The first part is also shown by the use of the Bernoulli convolutions (see Wintner [3]). See also Kawata [5], p. 140. 4. Some theorems which are thought of as for the intervening cases of Hardy's theorem (Theorem 8.1.1) and Ingham's theorem (Theorem 8.3.1) were given by Morgan [1]. Let p > 2, A > 0. Let p' be the conjugate of p and A' = a' sin a, where a'= [p'(Ap)P'- 1]- 1 , a= n/[2(p- 1)]. Let I and/. be even entire functions (/.is the cosine transform of f). If j'('x) = O(e-h:P) and /.(t)

=

O(exp -(A'

+ e)xP')

for some e > 0, then I = J. = 0 identically. Another result is: For any p > 2, A > 0, m real, there exist functions l(x) and /.(t) satisfying l(x) = O(xm exp( -AxP)), /.(t) = O(tm' exp( -A'xP')), where m' = (2m - p 5.

+ 2)/(2p -

2).

A function L(x) is of slow growth if and only if L(x)

where e(x) -+ 0, a(x) -+ c, c result is given there.

=

a(x) exp(J: [e(y)/y]

dy),

< oo as x-+ oo. See Feller [1], II., p. 274, a more general

NOTES

329

6. For the general Tauberian theorems, also see Wiener [4, 6, 7], Widder [1], Bochner and Chandrasekharan [1], and Pitt [1]. 7. For the Tauberian theorems (Theorems 8.6.5 and 8.6.6), there is another very interesting proof due to Hardy and Littlewood. (See Hardy [2], 7.1.2. Also see Landau [1], 5.9.) 8.

The proofs of the theorems in Section 8.7 were adapted from Bochner [5].

IX Convergence of Distribution Functions and Characteristic Functions

9.1. Hally Theorems and Convergence of Nondecreasing Functions Now we shall study, for the most part, some important topics in probability theory. The methods used are taken from the Fourier analysis already given in previous chapters. It would be convenient to begin with fundamental theorems on the sequence of nondecreasing functions. As before, we denote the class of distribution functions and of bounded nondecreasing functions by D and M, respectively. Although the probability distribution function of a random variable X; P(X < x) is leftcontinuous, we here call a nondecreasing function F(x) with F( -oo) = 0, F( oo) = 1 a distribution function.

Lemma 9.1.1. Let {Fn(x), n = 1, 2, ... } be a sequence of functions of M defined on ( -oo, oo). Let E be any set which is dense in ( -oo, oo). 330

9.1.

331

HELLY THEOREMS

If Fn(x) converges to some function G 1 (x) defined on Eat points of E, then there is a nondecreasing function G(x) defined at every point on (-oo, oo) such that G 1 (x) = G(x) on E and Fn(x) converges to G(x) at every continuity point of G(x). Proof. Since Fn(x) is nondecreasing, G 1(x) is also nondecreasing on the dense set E. Let x be any point of Ec and choose xn e E to be increasingly convergent to x. This is possible since E is dense. Define G(x) by limn-+oo G 1(xn)· If x e E, we write G(x) = G 1 (x). It is obvious that G(x) is nondecreasing in (-oo, oo). We shall now show that Fn(x) converges to G(x) at any continuity point of G(x). Let x be a continuity point of G(x) and let x', x" be points in E such that x' < x < x". Then

Letting n

--+

oo,

G(x')

~

lim inf Fn(x)

~

lim sup Fn(x)

Since x is a continuity point of G(x), letting x' G(x)

~

lim inf Fn(x) n-+oo

~

G(x").

n-+oo

n-+oo

~

lim sup Fn(x)

t x, x" ~ x through ~

E,

G(x),

n-+oo

which shows the conclusion of the lemma.

Theorem 9.1.1. Every sequence {Fn(x), n = 1, 2, ... } of uniformly bounded functions of M contains a subsequence {Fnk(x), k = 1, 2, ... }. which converges to some function F(x) eM at every continuity point of F(x). This theorem is called Helly's selection theorem or Helly's first theorem. Proof. Let R be the set of all rational numbers in (- oo, oo ). R is dense. Arrange R in a sequence r 1 , r 2 , . . .. Consider {Fn(r1 ), n = 1, 2, ... }, which is just a sequence of real numbers. It has at least one limiting point since Fn(r 1 ) is bounded. Hence we may choose a subsequence F 1,n(r 1 ) which converges to some number (J)(r1). Now form the sequence F 1,n(x) of uniformly bounded functions of M and consider {F1n(r 2 ) }. We then select from it a subsequence {F2n(r2 )} which converges to a certain number (J)(r 2 ). Then {F2,n(x)} is subse-

a

332

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

quence of F 1,n(x) and Fn(x) as well, and then F 2,n(x) converges to IP(r 1 ) and to IP(r 2 ) at x = ru and x = r 2 , respectively. Continue this procedure of selection and get {F1,n(x), n = 1, 2, ... }, .. . ,{Fm,n(x), n = 1, 2, ... }, ... ,

in which each sequence is a subsequence of the preceding sequence, and {Fm,n(x), n = 1, 2, ... } converges to IP(r) at r = r 1 , r 2 , ••• , rm. Now define the "diagonal sequence"

k

=

1, 2, ....

Then Fk,k(x) is a subsequence of all {Fm,n(x), n = 1, 2, ... }, m = 1, 2, .... Hence lim Fnk(r) = IP(r) k-+oo

for all rational numbers r. The number IP(r) is defined over R. Since Fn(x) is uniformly bounded, IP(r) is bounded on R. Then by Lemma 9.1.1, there is a function F(x) on (-oo, oo) such that F(r) = IP(r) on R and Fnt(x) converges at every continuity point of F(x). This proves the theorem. If a sequence {Fn(x), n = 1, 2, ... } of functions of M converges to a function F(x) of M at all continuity points of F(x), we say that Fn(x) converges weakly to F(x). Reviewing Lemma 9.1.1, a sequence {Fn(x)} of nondecreasing functions, converging on a dense set, converges weakly to a function of M. The limit function is not uniquely determined, but the two nondecreasing functions, which are identical on a dense set, are identical everywhere if both functions are known to be left-continuous everywhere (or right-continuous), or both are standardized. Therefore, as in the case of the probability distribution function, if a nondecreasing function is always supposed to be left-continuous in advance, then the limit function in Lemma 9.1.1 is uniquely determined. We also remark that two functions of M, which are identical except at their discontinuities, generate the same measure for Borel sets since, for an open interval (a, b), the same measure is associated with them. Actually, F(x) and G(x) are two such functions. Then F(a 0) = G(a 0), F(b - 0) = G(b - 0). This gives one interpretation of weak convergence but there is another explanation (see Theorems 9.1.3, 9.1.4).

+

+

9 .1.

333

HELLY THEOREMS

We now prove a theorem called Helly's second theorem or the HellyBray theorem (the first part of Theorem 9.1.2).

Theorem 9.1.2. (i) Let Fn(x) be a sequence of nondecreasing functions in [a, b], -oo
r

f(x) dFn(x)

lim n~

a

=

r

(2)

f(x) dF(x).

a

(ii) Conversely, if (2) holds for any continuous function f(x), and Fn(x) E M for some function F(x) E M, then Fn(x) - Fn(a) converges weakly to F(x) - F(a). Here, needless to say, Fn(a), Fn(b), F(a), and F(b) are supposed to be finite. ( 1) implies the uniform boundedness of Fn(x ). Proof. (i) Let a= Xm,o < Xm,l < · · · < Xm,km = b be a set of division points of [a, b], m = 1, 2, ... , and let these points be continuity points of all functions F(x), Fn(x), n = 1, 2, .... Let IA be the indicator of a set A. Write a~ x ~

at

b,

x =b.

This is a step-function which converges uniformly to f(x) as max,.(xm,•+l- Xm .• ) - 0, as m-oo. Suppose that if m is made larger, then the partitions of the interval [a, b] become smaller and, for any e > 0, suppose 1/m(x)- f(x) I < e when m ~ m0 • Let I Fn(x) I ~ C for a ~ x ~ b, n = 1, 2, .... Then

IJ:Jm(x) dFn(x)- J:J(x) dFn(x) I ~ for m

~ m0

,

J: I

fm(x) - f(x) I dFn(x)

~

eF,.(b) < Ce,

(3)

uniformly for n. Also J:Jm(x) dF(x)- J:J(x) dF(x) < eC

(4)

334 IX. form

CONVERGENCE OF DISTRIBUTION AND CHARA<;TERISTIC FUNCTIONS

~ m0 •

For each m,

where we have used (1) as well as the choice of Xm,• to be continuities of F(x). Then b

k,.-1

II

•-0

J fm(x) dFn(x) = L f(xm,.)[Fn(Xm,•+l) -

Fn(Xm,.)]

converges to

that is, (5) as n -. oo for each m. Now

I= f:J(x) dF,(x)- f:J(x) dF(x) =

s:

[f(x)- fm(x)] dFn(x)

+

J:

Choose m

~

+ u:fm(x) dFn(x)- s:fm(x) dF(x)]

[fm(x) - f(x)] dF(x). m0 and fix m. Then from (3) and (4 ),

I II~ 2eC

+I s:fm(x) dFn(x)- s:fm(x) dF(x) I·

Let n -. oo. We then have from (5) lim sup I I

I ~ 2eC.

n-+oo

This proves (i) of the theorem. (ii) Suppose that (2) holds for any continuous functionf(x). Let Ebe any continuity point of F(x), a < E < b. Choose e > 0 such that E+ e < b and form a continuous function

f(x)

=

1, { e-1(-x + E +e),

0,

a~

x

E~ x E+ e

~

E,

+

~ E e, ~ x ~b.

(6)

9 .1.

335

HELLY THEOREMS

Then

Letting n

-+

oo, from (2),

sup[Fn(~) - Fn(a)] ~

lim

:;

n-+oo

r

f(x) dF(x)

a

s:~ dF(x) = F(~ +e)- F(a).

In a similar way, considering, for a+ e <

~.

(7)

e > 0,

x < ~- e, ~- e ~X~~'

a~

~ ~x,

we have lim

inff[Fn(~)-

Fn(a)]

~ F(~-

e)- F(a).

(8)

n-+oo

Since

~

is a continuity point of F(x), (7) and (8) give us lim

Fn(~)

- Fn(a) = Fa) - F(a)

n-+oo

for a<~< b. For ~ = b, Fn(b) - Fn(a) -+ F(b) - F(a) is easily seen, taking f(x) = 1 in (2). This proves (ii). Now consider a relation similar to (2) in the case where (a, b) is replaced by (- oo, oo ). Two kinds of theorems are known.

Theorem 9.1.3. (i) Let f(x) be continuous over (- oo, oo) and lim.z-+±oof(x) = 0. If Fn(x) E M converges to F(x) and Fn(x) is uniformly bounded, then (9) (ii) If for Fn(x) e M, F(x) e M, (9) holds for any continuous function f(x) with lim-+±oof(x) = 0, then Fn(~)- Fn(7J) converges to F(~)- F(7J) at continuity points~. 7J of F(x).

Proof. (i) For a given e, choose A such that If(x) I < e for I x I ~ A and F(x) is continuous at ±A. Let I Fn(x) I ~ C, C being a constant

336

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

independent of x and n. Then I F(x) I

II

lzl>.-l

f(x) dF11 (x)

I~ e I

-

IZI51;.4

~C.

Then

dFn(x) ~ e[F,(oo)- F,(-oo)] ~ eC, (10)

and also

I

Jlzl;;;.(

f(x) dF(x)

I~ eC.

(11)

Now

II~oof(x) dF,(x)- I:J(x) dF(x) I ~ If~A f(x) dF,(x) f(x) dF(x) I

r.

+ I{.,1~.4 f(x) dF,(x) I + IJlzi;;;A f(x) dF(x) I

~

ItA f(x) dF,(x) -

rA

f(x) dF(x)

I+ 2eC,

from (10) and (11) and the first member of the last expression, converges to zero by Theorem 9.1.2 (i). This proves (i). (ii) Choose f(x) to be, for 1J < ~. e > 0,

r 1(x- 1J f(x) =

{

+ e),

'1'}-e~X~'f}

!~~(-x +~+e),

1J~X~~ ~ ~ x < ~ elsewhere.

0,

+ e,

Proceed as in the proof of Theorem 9.1.2 (ii). We then see that lim

sup[F11 (~)-

F,('YJ)]

n->oo

~ F(~

+ e) -

F(1J- e).

Similarly, lim inf[F11 (~)- F,.('YJ)] n-+oo

~ F(~-

e)- F('f}

+ e).

From these we have F,(~)- F,('YJ)---+ F(~)- F(rJ). Another analog of Theorem 9.1.2 is the following Theorem 9.1.4. Let M 0 be a subclass of M such that any function f(x) E M 0 satisfies F(-oo) = 0, as was already introduced in Section 3.1. We consider functions of M 0 • For the general case we may take F( x) - F(- oo) in place of F(x).

9.1.

337

HELLY THEOREMS

Theorem 9.1.4. A necessary and sufficient condition that a sequence Fn(x) of functions Fn(x) E M 0 converge weakly to a function F(x) e M 0 and F n ( =) - F( = ), is that, for any bounded continuous function f( x ),

(12) In particular, we have the following theorem.

Theorem 9.1.5. A sequence Fn(x) of distribution functions converges to a distribution function F(x) if and only if (12) holds for any bounded continuous function f(x). Proof of Theorem 9.1.4. Necessity. Let e > 0 be any positive number. Choose A such that A and -A are continuity points of F(x) and

F(=)- F(A) < e,

F(-A) <e.

(13)

There is an N such that

I Fn(=)- F(=) I <

for n

~

E,

(14)

I Fn(A)

- F(A)

I < e,

(15)

I Fn(-A)-

F(-A)

I< e,

(16)

N. Then

I Fn(=)- Fn(A) I ~I Fn(=)- F(=) I+ IF(=)- F(A) I

+ I F(A) -

Fn(A)

I<

3e,

by (13)-(15), and Fn( -A). ~ I Fn( -A) - F( -A) I + F( -A) < 2e.

Hence

u:

+

r:) Jf(x) I

dFn(x)

~ c(J: +

r:)

dFn(x)

= C{[Fn(=)-Fn(A)]+ [Fn(-A)]} < where C =max

5Ce,

lf(x) 1. Thus

IJ~oof(x) dFn(x)- tA f(x) dFn(x) I< 5Ce.

(17)

338

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTiONS

Since, by Theorem 9.1.2, J~Af(x) dFn(x) converges to J~Af(x) dF(x), (17) shows (12). Sufficiency. The proof is similar to those of Theorems 9.1.2 and 9.1.3. Proceed just as in those proofs, choosing f(x) = 1 for -ex> < x < E; f(x) = e- 1 ( -x + E+e) forE~ x ~ E+ e;f(x) = 0 forE+ e < x <ex> and / 1 (x) = 1, for -ex>< x ~ E- e; / 1 (x) = e- 1 (-x +E) for E- e ~ x ~ E; / 1(x) = 0 forE< x <ex>. Finally we give a simple theorem:

Theorem 9.1.6. If a sequence {Fn(x)} of functions of M 0 converges weakly to a continuous function F(x) of M 0 and Fn(=) converges to F(=), then Fn(x) converges uniformly to F(x) over (-ex>, ex>). Proof. For any e > 0, there exist numbers -A, B, and the division points of [-A, B], -A = x 0 < x 1 < · · · < xk = B such that F(-A) < e, F(x;) - F(x;_ 1 ) < e,

F(=)- F(B) <

E,

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

(18) (19)

Since Fn(x) converges to F(x) at every point, there is an N 0 such that for n ~N0 , j = 0, 1, ... , k (20) F(x;) - Fn(x;) < e, (21)

F(cx:>)- Fn(=) <e.

If x ~ -A, then 0 ~ F(x) ~ F( -A) < e and 0 ~ Fn(x) ~ Fn( -A) ~ F(-A) + e < 2e, from (18) and (19). Therefore,

I F(x) - Fn(x) I < 2e If

X;_ 1 ~

(22)

x <X; for some j, then

that is, F(x;_ 1 )

~

F(x) < F(x;_ 1 )

+ e.

Fn(x)

~

Fn(x;) < F(x;)

+e<

Fn(x)

~

Fn(X;- 1 ) > F(x;_ 1 )

(23)

From (19),

for n

~

-

F(x;_ 1 )

+ 2e,

e,

N 0 • Hence

(24)

9.2.

339

DISTRIBUTION FUNCTIONS WITH BOUNDED SPECTRA

Thus (23) and (24) give us

I F(x) - Fn(x) I < 2e

for

n

~

N0•

(25)

Finally, if x ~ B, we have also the same relation as (25), because in the above argument we may take B for Xj-I, and oo for Xj. Together with (25) and (22), we get for n ~ N 0 ,

I F(x) - Fn(x) I < 2e

for

- oo

<

x

< oo,

and N 0 is independent of x, which is what we wanted to show. Finally we give the following remark which is obvious from the proof above. If {F,.(x)} is a sequence of nondecreasing functions defined over [A, B], if F,.(x) converges weakly to a continuous function F(x) as n- oo, and if F,.(A)- F(A), F,.(B)- F(B), then F,.(x) converges uniformly to F(x) over [A, B]. REMARK.

Here A or B can be -oo or +oo, respectively.

9.2. Convergence of Distribution Functions with Bounded Spectra We are going to discuss the convergence of distribution functions to a distribution function in terms of characteristic functions. To begin, we consider the sequence of distribution functions with bounded spectra. Suppose the spectrum is contained in ( -.n, .n), without loss of generality, that is, the distributions are supposed to be circular distribution functions. Let F(x) be such a distribution function: F(x) = 0 for x < -.n, F(x) = 1, x > .n. The characteristic function is

J:,

eitx

dF(x)

=

f(t).

However F(x) is uniquely determined by J(n)

=

J:,

einx

dF(x),

n = 0, ±1, ±2, ....

(1)

Hence f( t) is also determined by f( n ), n = 0, ± 1, . . . . Thus ( 1) will play the role of the characteristic function. We agree to call it the char-

340 IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

acteristic sequence of F(x). We denote by K the class of circular distribution functions, namely, the distribution functions F(x) such that F(x)=O, x ~ 0; F(x) = 1, x ~ 1. For convenience we suppose throughout that if F(x) E K, F(-:n:) = 0, F(:n:) = 1. (2) The above fact is no more than Theorem 3.4.1. Furthermore, by Theorem 4.2.1, the following formula holds:

F(x)- F(O)

1

=

lim ,.,_ N-+oo

~

e-i•z _

N

1

L -----;.,-----f(v) .. •--N -lV

(3)

The apparent difference between this formula and Section 4.2, Eq. (5) is based on the difference between the definition of Fourier-Stieltjes coefficients en [Section 3.2, Eq. (1)] and that of f(v). Suppose that a sequence Fn(x) E K. If Fn(x) converges weakly to a nondecreasing function F(x) in (-:n:,:n:), then Fn(x) (-oo < x < oo) is considered to converge weakly to the distribution function which is identical with F(x) in ( -:n:, :n:) and is 0 for x ~ 0; 1 for x ~ 1. With this in mind, we shall prove the following classical theorem of Caratheodory [1] (also see Zygmund [1 ], p. 82). (i) If a sequence {Fn(x)} of distribution functions of K converges weakly to a distribution F(x) of K as n- oo, then the characteristic sequence fn(v) of Fn(x) converges to the characteristic sequence f(v) of F(x) as n- oo for each v = 0, ±1, .... (ii) If the characteristic sequence fn(v) of Fn(x) converges for each v = 0, ± 1, ±2, ... , to f( v) as n - oo, then f( v) is the characteristic sequence of some distribution function F(x) E K and Fn(x) converges weakly to F(x).

Theorem 9.2.1.

Proof. Statement (i) is an immediate consequence of Theorem 9.1.2. Note that we assumed (2). (ii) Consider the sequence Fn(x) in the interval ( -:n:, :n:). Since it is uniformly bounded by 1, by the Helly theorem (Theorem 9.1.1), there is a subsequence {Fn"'(x)} of {Fn(x)} which converges weakly to a nondecreasing function F(x). Since fn.~:(v)- f(v) as k - oo by assumption and

lim k-+oo

J"

-n

ei•z

dF11 k(x) =

J"

-n

ei•z

dF(x),

by Theorem 9.1.2, f(v) is the characteristic sequence of F(x).

9.3.

341

CONVERGENCE OF DISTRIBUTION FUNCTIONS

Now suppose that Fn(x) does not converge weakly to F(x). Then there is a subsequence {Fmt(x)} of Fn(x) such that Fm"(x) converges weakly, but lim F ,.(E) -=1= F( E), k-+oo

where E is some continuity point of F(x), -n < E < n. Let lim Fm,(x) = G(x) weakly. Then again by Theorem 9.1.2, the characteristic sequence of G(x) isf(v), and hence

f(v)

= J~ ei•z dF(x) =

J:,. ei•z dG(x),

'JI

=

0, ±1,.. ..

(4)

But G(E) -=1= F(E). However (4) implies from the unicity theorem for Fourier-Stieltjes coefficients (Theorem 3.4.1), that G(E) = F(E) for the continuity E of F(x), which is a contradiction. This completes the proof. We now proceed to the case of the general distribution functions. Roughly speaking, we have similar results, which will be the topic of the subsequent sections.

9.3. Convergence of Distribution Functions Let F(x) andf(t) be a distribution function and its characteristic function, respectively:

f(t) =

J:

eitz dF(x).

(1)

In order to discuss the convergence of distribution functions in terms of their characteristic functions, it is convenient to introduce

L(x, a) = n- 1

J:

(sin 2 atfat 2 )f(t)e-itz dt,

(2)

where a> 0, -oo < x < oo. The convenience of L(x, a) is based on formula (3) in the following theorem.

Theorem 9.3.1. L(x, a), defined by (2) for any distribution function F(x), is expressed by L(x, a)= (2a)- 1

J

2cx

0

[F(x

+ u)- F(x- u)] du,

a> 0.

(3)

342 IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

Proof. L(x, a)

=

n- 1

= a- 1

J~oo (sin 2 atfat 2 )e-ixt dt J~oo

eitu

dF(u)

J~oo T(u - x, a) dF(u),

where

0, T(x, a) = { a+ (x/2), a- (x/2),

IX

I~

-2a 0

2a,

~X~ ~X~

0, 2a.

(4)

This is the formula obtained in Section 4.11, Eq. (8). So we have

f-oooo T(u, a) dF(u + x) = a- J211211 T(u, a) dF(u + x) = a- J T(u, a) d[F(x + u) - F(x - u)],

L(x, a)= a- 1

1

_

2<1

1

0

from which (4) is immediate after the application of integration by parts. Let {Fn(x)} be a sequence of distribution functions and let fn(t) be the characteristic function of Fn(x). Ln(x, a) is defined by (2) and Fn(x) to be (5)

If Fn(y) - Fn(x) converges to F(y) - F(x) at continuity points x and y of a function F(x) E M, we say that Fn(x) converges weakly to F(x) up to additive constants. This notion was used in statement (ii) of Theorem 9.1.3. We shall begin with the following theorem.

Theorem 9.3.2. A sequence {Fn(x), n

= 1, 2, ... } of distribution functions converges weakly to F(x) EM, up to additive constants if and only if Ln(x, a) converges for every a> 0 and x, -oo < x < oo. When Ln(x, a) converges, the limit function is written by L(x, a) with f(t) = J~oo ei:rJ dF(x).

9.3.

343

CONVERGENCE OF DISTRIBUTION FUNCTIONS

Proof. The "only if" part and the last statement are obvious from (3) with Fn(x), L 11 (X, a)= (Za)- 1

J

2«

0

(Fn(X +u)- Fn(x- u)] du,

and the limit should be

L(x, a) = (Za)- 1

J [F(x + u) 2«

0

F(x- u)] du,

the right side being written in the form (3 ). The converse is shown in the following way: The convergence of Ln(x, a) implies the convergence of

for every -00

<

~

> 0. Since this is a symmetric function, it converges for every < 00.

~

Since Fn(v) ED, by Theorem 9.1.1, {Fn(v)} contains a subsequence {Fnk( v), k = 1, 2, ... } which converges weakly to an F( v). If there are two convergent subsequences {Fnk(v)} and {Fmt(v)} with the limit functions F(v) and G(v), respectively, then

J:

[F(x

+ u) -

F(x- u)] du =

J:

[G(x

+ u) -

G(x- u)] du

for - oo < ~ < oo. This implies that for every x and almost every ~ (where the exceptional set Ex may depend on x), F(x + ~)- F(x- ~) = G(x + ~)- G(x- E). Choose continuity points a and {1 of F(v) and G(v) and x to be t({J +a) and let ~--+ !({1- a), ~ t; Ex. Then we see that F({J)- F(a) = G({J)- G(a). From this we conclude that Fnk(v) and F mk( v) converge weakly to the same function up to additive constants, which says that Fn(x) converges weakly up to additive constants. We are now interested in the convergence of a sequence of distribution functions to a distribution function. Before discussing this matter [using Ln(x, a)], we recall that for the Fejer integral of g(x), we have lim n- 1 a:-+oo

Joo

-oo

(sin 2 atfat 2 )g(t

+ x) dt =

g(x)

(6)

if g(x) is bounded and continuous at x. This is a particular case of the general convergence theorem (Theorem 5.2.1 or 5.2.4).

3# IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

This theorem implies the following corollary:

Corollary 9.3.1. If a sequence J,.(t) of characteristic functions converges to a function J(t) almost everywhere, then there is a function F(x) of M such that F( oo) - F(- oo) ;::::;; 1 and J(t) = J~oo eixt dF(x)

a.e.

Proof. Since J,.(t)--+ J(t) a.e., L,(x, a) of (5) converges for every x, - oo < x < oo, and every a > 0 to L(x, a)=

:n;- 1

J~oo (sin 2 atfat 2 )f(t)e-ixt dt.

On the other hand, by Theorem 9.3.2, there is an F(x) of M such that

L(x, a)

=

:n;- 1

J~oo (sin 2 atfat2 )r/>(t)e-ixt dt

with

rf>(t) = J~oo eixt dF(x). From the uniqueness of the Fourier transform, J(t) shows the corollary.

= rf>(t) a.e. This

We now show the following theorem (Andersen and Kawata [1]).

Theorem 9.3.3. A sequence {F (x)} of distribution functions converges weakly to a distribution function F(x), if and only if 11

(i) (ii)

L 11 (x, a) converges for every a> 0 and x, -oo < x < oo, and L,.(O, a)= L 11 (a) converges uniformly for all a ;::;: A for some

A> 0. Before proving this, we remark that if (ii) holds for some A > 0, then it holds for any positive number A. This is contained in the following proof.

Proof. Suppose F 11 (x) converges weakly to a distribution function F(x). Then from the preceding theorem, (i) holds and the limit function of L 11 (x, a) is written L(x, a) =

:n;- 1

J~oo (sin 2 (J.tfat 2 )j(t)e-itz dt,

(7)

9.3.

345

CONVERGENCE OF DISTRIBUTION FUNCTIONS

where J(t) is the characteristic function of F(x). From (6), if we write lim L 11 (0, a)

lim L 11 (a),

=

n-->oo

n-->oo

by L(a), then lim L(a)

= f(O) = 1.

cz-->-oo

L 11 (a) is a nondecreasing function of a, as is seen from the form

L,.(a) = (2a)- 1 and

L(a) = (2a)- 1

J [F (u) 2cz

0

f

b

0

Fn( -u)] du

11

[F11 (u) - F,.( -u)] du,

which are continuous for all (J. > 0. Moreover, Ln(+oo) =fn(O) = 1. In other words, a nondecreasing function L 11 (a) converges to a continuous function L(a) and Ln(oo)---+ L(oo), L 11 (A)---+ L(A) for any A > 0. Hence, by the remark given at the end of Section 9.1, the convergence of Ln(a) to L(a) is uniform over [A, oo]. Conversely, suppose (i) and (ii) hold. By Theorem 9.3.2, F,(x) converges to a function F(x) EM up to additive constants. From (6), lim L 11 (a)

= /

11

(0)

1

=

« ......~

for each n. From (ii), L 11 (a) converges uniformly to L(a), for any e > 0,

I L 11 (a) - L(a) I <

E

for n ~ N 0 , a ~ A, N 0 being independent of a ~ A. Taking the limit, letting a---+ oo, we get lim,._.."" L(a) = 1, which shows that

f(O) = F(oo)- F(-oo) = 1. This completes the proof.

Theorem 9.3.4. If a sequence {F,(x), n

1, 2, ... } of distribution functions converges to a distribution function F(x), then the characteristic function j,.(t) of F 11 (x) converges to the characteristic function J(t) of F(x) uniformly in every finite interval of t. =

346

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

Proof. Let us take any finite interval [- T, T] and any number e Choose continuity points -A and B of F(x) so that

1- F(B)

> 0.

+ F(-A) <E.

(8)

Once -A, B are chosen, they are fixed. Choose N 1 so large that

I Fn(B) for n

~

- F(B)

~

N 1 • Then for n

11- Fn(B)

I<

I Fn(-A)-

e,

F(-A)

I<

(9)

e,

N1,

+ Fn(-A) I <

+I F(B)- Fn(B) I +I Fn(-A)- F(-A) I+ F(-A)

1- F(B)

< 3e.

(10)

Now

Ifn(t)

- f(t)

I~

IrA eit:c dF,.(x) + [1

- Fn(B)

rA eit:c dF(x)

I

+ Fn( -A)] + [1 -

F(B)

+ F( -A)],

which by (9) and (10), gives us

Ifn(t)- f(t) I ~IrA for n

~

eit:c dFn(x)- rA eit:c dF(x)

I+

4e

(11)

N 1 • Using integration by parts,

IrA eit:c d[Fn(x) -

I

= [F11 (B) - it

r.

F(x)]

F(B)]eitB -

T rA

[Fn( -A) - F( -A)]ritA

x[FII(x) - F(x)] dx

~I Fn(B)- F(B)

+

I I

I+ I F,.(-A)-

I :c I I Fll(x) -

F(-A)

F(x) I dx.

I (12)

Since Fn(x) converges to F(x) except at discontinuities of F(x), the last integral converges to zero. Hence there is an N 2 such that

Together with (11 ),

lfn(t)- f(t) I < 6e

+ eT

for

n

~

N0•

This shows the theorem. Now we discuss the converse of Theorem 9.3.4. As a matter of fact, the converse is true; that is, if the characteristic function of Fn(x) converges to some function f(t) uniformly over every finite interval of t, then Fn(x) converges weakly to a distribution function F(x) and f(t) becomes the characteristic function of F(x). This is, together with Theorem 9.3.4, due to Levy [1] and will be shown in Theorem 9.3.7. However, under the weaker condition, the same conclusion can be obtained. Actually we do not need the uniform convergence of fn(t) over every finite interval. First of all we show the following general theorem of Schmetterer [1].

Theorem 9.3.5. Suppose that the characteristic function fn(t) of the distribution function Fn(x) converges to some function f(t) at every t. If the Fourier series of f(t) in ( -n, n) is (C, 1)-summable at t = 0 to 1, then Fn(x) converges weakly to a distribution function F(x) and the characteristic function of F(t) is identical with f(t). However this is a simple consequence of Theorems 9.3.2 and 9.3.3.

Proof. lf[n(t) ___. f(t) for every t, then from the definition (5) of Ln(x, a), it is obvious that Ln(x, a) converges to n- 1

J:

(sin 2 atfat 2 )f(t)e-itx dt.

Hence by Theorem 9.3.2, Fn(x) converges to some function F(x) up to additive constants, and

L(x, a)

=

n- 1

where

J*(t)

J: J:

(sin 2 atfat 2 )J*(t)e-itx dt,

=

eitx dF(x).

E

M

348

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

Therefore, n- 1

I~"" (sin 2 atfat 2 )J(t) dt = n- 1 I~"" (sin 2 atfat 2 )f""(t) dt.

Now the (C, 1)-summability of the Fourier series of f(t) to 1 at t is equivalent to n- 1

I

111<6

(sin 2 atfat 2 )f(t) dt---+ 1,

c5

(13)

=

0

> 0,

as a---+=, by the Remark in Section 4.6, Eq. (13). Therefore, (13) gtves us

L(a) = L(O, a) = n- 1 smce n- 1

I

111>6

I:

(sin 2 atfat 2 )f""(t) dt---+ 1

(sin 2 atfat 2 )f""(t) dt ---+ 0

as a-+=

as a---+=·

This implies that J'I'(O) = 1 [since j•(t) is a continuous function], that is, F(=)- F(-=) = 1 and F(x) is a distribution function. (This last step is just the same argument as that at the end of the proof of Theorem 9.3.3.) The last statement of the theorem is obvious because j*(t) now becomes the characteristic function of F(x) and from Theorem 9.3.4, fn(t) should converge to j•(t). [Actually f(t) = j•(t).] This completes the proof. Now since the Fourier series of a continuous function at a point is (C, 1)-summable there, the following theorem is the consequence of Theorem 9.3.4.

Theorem 9.3.6. If the characteristic function fn(t) of a distribution function Fn(x) converges to some function f(t) for every t and if f(t) is continuous at the origin, then Fn(x) converges weakly to a distribution function F(x) and f(t) is the characteristic function of F(x). A further consequence of this theorem is the following:

Theorem 9.3.7. If the characteristic function fn(t) of a distribution function Fn(x) converges to some function f(t) at every t and fn(t) converges uniformly to f(t) in a neighborhood of the origin, then Fn(x) converges weakly to a distribution function F(x) andf(t) is the characteristic function of F(x).

9.3.

349

CONVERGENCE OF DISTRIBUTION FUNCTIONS

In view of Theorem 9.3.4, we see that the uniform convergence of fn(t) in a neighborhood of the origin implies the uniform convergence in every finite interval. Theorem 9.3.7, with Theorem 9.3.4, is due to Levy [1] and is called the continuity theorem. Suppose now that the distribution functions are standardized. We now ask what can be said about the convergence of distribution functions if the corresponding characteristic functions converge uniformly to a function over the whole interval (- oo, oo ). Concerning this, we can show the following theorem which was given by Dyson [1].

Theorem 9.3.8. If a sequence Un(t)} of characteristic functions of standardized distribution functions Fn(x) converges to a function f(t) uniformly over the whole line ( -oo, oo), then Fn(x) converges uniformly to a distribution function over ( -oo, oo), and f(t) is the characteristic function of F(x). From Theorem 9.3.7, Fn(x) converges weakly to a distribution function F(x) and the last statement of the theorem holds. Furthermore, the theorem claims that even at discontinuities of F(x), Fn(x) converges to F(x) and the convergence is uniform over (-oo, oo). Proof. From Theorem 9.3.7, Fn(x) converges to a distribution function F(x) at every continuity point of F(x). Let F(x) be standardized. First we prove that at discontinuity point x 0 of F(x), Fn(x0 - 0) and Fn(x0 0) converge to F(x0 - 0) and F(x0 0), respectively, and hence Fn(x0 ) to F(x0 ), using the relation

+

n-1

+

J~oo (sin 2 atjat 2 )e-ixt[Jn(t) - f(t)] dt

= (Za)- 1

J:

{[F11 (x+u) - Fn(x-u)] -

[Fn(x+u) - F(x-u)]} du (14)

for a > 0. For any e, choose N such that I fn(t) - f(t) I < e for - oo < t < oo. N is independent of t because of the uniform convergence of fn(t) to f(t) over ( -oo, oo). Then (14) shows that

(Za)- 1

f

2ac

0

{[Fn(x+u) - Fn(x-u)] - [F(x+u) - F(x-u)]} du

for

n

~

N.

350 IX. Let a

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

0+ for each n. We then see that

--+

I [Fn(x + 0) - Fn(x- 0)] - [F(x

+ 0) -

F(x - 0)] I

~

e

(15)

for n ~ N (uniformly for x). Let x0 be any discontinuity point of F(x). Choose rJ > 0 such that x 0 ± rJ are continuity points of F(x). Then for n ~ N

Fn(x0

-

rJ) ;;;;; Fn(x0

0)

-

+ 0) - F(x 0) + F(x + 0)] + F(x 0) + [Fn(x + 0) - F(x + 0)] ;;;;; e + F(x 0) + Fn(x + rJ)- F(x + 0) ;;;;; e + F(x 0) + Fn(x + rJ)- F(x + rJ) + F(x + n)- F(x + 0). ;;;;; [Fn(X 0

0) - Fn(x0

-

0 -

0 -

0

0

0

0 -

0

0

0

+ rJ)- F(x + 0) <

Take rJ so small that F(x 0 -

Let n

--+

rJ)

~

Fn(X0

-

0

0 -

0

Fn(X0

0

0)

e. Then

0

~

2e + F(x0

0)

-

+ Fn(x + rJ) 0

F(x 0

+ rJ).

oo. Then we see that ft-+oo

;;;;; lim sup Fn(x0

0)

-

~

2e

+ F(x

0 -

0).

ft-+oo

Let 'YJ--+ 0+ in such a way that x 0 finally we see that

F(x0 - 0)

~lim

inf Fn(x0

-

0)

-

rJ remains a continuity of F(x). Then

~lim

n-+oo

sup Fn(x0

-

0)

~

Fn(x0

-

0) + 2e,

n-+oo

which shows that lim F11 (X 0

=

F(x 0

-

0).

(16)

+ 0) =

F(x0

+ 0).

(17)

-

0)

n-+oo

In a similar way we get lim Fn(x0 n-+oo

Now take, for any e, the continuity points -A and B of F(x) so that F(-A) < e,

1- F(B)

<e.

(18)

9.3.

351

CONVERGENCE OF DISTRIBUTION FUNCTIONS

Let N 1 be chosen such that

I F (-A)- F(-A) I < e,

I F 11 (B) - F(B) I <

11

for n

~

~

F(x)

0

~

Fn(x)

Therefore for x

~

~

F(-A)

~

-A, n

~

Similarly, we see that for x

~

~

0

~

F(-A)

x

- F 11 (x)

B, n

~

+ e ~ 2e.

~

I<

2e.

(20)

2e.

(21)

N1 ,

- Fn(x)

with the property that x - t5x, x ~

~

N1,

I F(x) With any point x, -A

< e,

F 11 (-A)

I F(x)

0

(19)

~-A

N 1 • Then for x

0

e

I<

B, we associate an interval

+ !5x are continuity points of F(x)

< e F(z) - F(x + 0) < e F(x- 0) - F(y)

for

x - !5x

for

x

<

7.

~

<

y < x, X

+ !5x,

and (22)

where e is an arbitrary preassigned number. Then by the Borel covering theorem, there is a finite number of intervals lx1 , j = 1, 2, ... , m such that U~ 1 lx1 ::J [-A, B]. For simplicity we shall write

Choose n 1 such that, for n

~

n1 ,

I Fn(xi ± t5i)- F(xi ± t5i) I < I Fn(xi ± 0)- F(xi ± 0) I <

e,

j = 1, 2, ... , m,

(23)

e,

j

(24)

= 1, 2, ... , m.

These inequalities are possible, because of (16) and (17). Take any point x in [-A, B]. Then there is at least one interval Ik which contains x. If xk- !5k < x < xk, then for n ~ n 1 , from (23) and (24), Fn(x) ~ Fn(xk - t5k) ~ F(xk - t5k) - e, Fn(x) ~ Fn(xk - 0) ~ F(xk - 0)

+ e.

352

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

Hence

On the other hand,

Thus

I Fn(x) - F(x) I ;:=;; F(xk - 0) - F(xk - 0) + e < 2e. Similarly, for n > n 1 , xk < x < xk

+ 15k>

I F(x) - Fn(x) I < If x

2e.

= xk, then by (24 ), whether or not xk is a discontinuity of F(x ),

+ 0) + Fn(Xk - 0)] [F(xk + 0)- F(xTt- 0)]} <

I Fn(xk) - F(xk} I ~ H[Fn(Xk -

e

for n > n 1 • This together with (20) and (21) gives

I Fn(x) - F(x) I < 2e for every - oo < x < oo. This completes the proof. REMARK.

The converse of Theorem 9.3.8 is also true.

Suppose Fn(x) is standardized and Fn(x) F(x) is also standardized. Then (2a)- 1

J

(2a)- 1

J [F(x + u) -

converges to

211

0

[Fn(x

+ u) -

211

0

---+-

F(x) uniformly, where

Fn(x- u)] du

F(x - u)] du

uniformly for a > 0. Hence, for any e > 0,

I-2n1 Joo

-oo

sin 2at e-ixlj, (t) dt - - 1 at 2 n 2n

Joo -oo

I

sin 2at e-ixlj(t) dt < e at 2

(25)

for n ~ n0 , where n0 can be chosen indt;pendently of a > 0, and f(t) is the characteristic function of F(x).

9.4.

353

A GENERAL INTEGRAL TRANSFORM

Let a

--+

oo in (25 ). Then we have, for n

lfn(x)- f(x) I

~

~

n0 ,

E,

which shows our remark.

9.4. Continuous Distribution Functions: A General Integral Transform of a Characteristic Function Let f(t) be the characteristic function of a distribution function F(x). We know, from Theorem 3.8.1, that

:.~ (2T)- 1

rT

f(t)e-ixt dt = F(x

+ 0) -

F(x - 0).

(1)

On the other hand, from Wiener's formula we have shown [Section 5.5, Eq. (21 )] that lim L(x, a) = lim n- 1 cz-+0+

cz-J-o+

Ioo

(sin 2 atfat 2 )f(t)e-itx dt

-oo

= lim (2T)- 1 IT J(t)e-ixt dt.

T-+oo

(2)

-T

But this relation is an immediate consequence of (1) and the Eq. (3) of Section 9.3:

L(x, a) = (2a)- 1

I

2cx

0

[F(x

+ u) -

F(x- u)] du.

(3)

We may write, from (1) and (3),

L(x, 0+) = lim L(x, a) = .4fe[J(t)e-ixt],

(4)

cx-+o+

where generally .4fe[g(t, s)] denotes lim (2T)- 1

T-+oo

IT g(t, x) dt -T

when it exists. If g(t, x) is a function of t only, we may drop the subscript t of .4fe and write .L(g). We show the following theorem.

Theorem 9.4.1. In order that a sequence {Fn(x)} of continuous distribution functions converge to a continuous distribution function, it is

354 IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

necessary and sufficient that Ln(x, a) converge uniformly for a over (0, oo) for every x, - oo < x < oo. Proof. Suppose that Ln(x, a) converges uniformly for a, 0
L(x, a) = .n- 1

J:

(sin 2 atfat 2 )f(t)e-izt dt,

f(t) being the characteristic function of F(x). Hence for any e > 0,

I Ln(x, a) - L(x, a) I < e

(5)

if n ~ N, N being independent of 0
J~ [F11 (x

+ u) -

Fn(x - u)] du

converges uniformly in a to (2a)- 1

J [F(x + u) 21Z

0

F(x - u)] du

(also uniformly in -oo < x < oo). This shows that Ln(x, a) converges to L(x, a) uniformly in a > 0. The last argument shows the following: The uniform convergence of L,.(x, a) in a > 0, for every x, implies the uniform convergence of L,.(x, a) in a> 0, -oo < x < oo. REMARK.

In the preceding section and in this section so far we have made use of the Fejer integral L(x, a). We remark here that ii we replace L(x, a) by the Poisson integral of f(t)ritx, the arguments still hold and similar results are obtained. More generally, suppose that a nonnegative, even function K(x) E U(- oo, oo) satisfies

(Z.n)- 1' 2 J~oo K(x) dx = 1.

(6)

9.4.

355

A GENERAL INTEGRAL TRANSFORM

Let k(t) be the Fourier transform of K(x), k(t) ~

Suppose that k(t) Form j(x, a)

= (Zn)- 112 J~oo

0, - oo

K(x)e-ixt dx.

(7)

< t < oo. Note that k(t) must be even.

= [a/(2:n:) 112] J~oo

a> 0,

K(at)f(t)e-ixt dt,

(8)

which plays the role of L(x, a), where f(t) is a characteristic function. We then see that ](x, a)

= J~oo k[(x

- u)Ja] dF(u),

(9)

where F(x) is the corresponding distribution function because j(x, a)

= [a/(Zn)] J~oo = =

J: J:

K(at) exp -ixt dt

dF(u) (2:n:)- 112

J:

J:

exp itu dF(u)

K(v) exp[ -i(x- u)vfa] dv

k[(x- u)Ja] dF(u).

We then see thatj(x, a) satisfies lim ](x, a)

=

F(x

+ 0) -

F(x - 0),

(10)

«-+o+

lim ](x, a)= 1.

(11)

«-+OO

Note that k(O) = 1, from (6), and is bounded. Since k(t) is the Fourier transform of a function of £1(-oo, oo), k(t) --+0 as It I-+ oo. Therefore, k[(x- u)Ja] --+ 0 as a--+ 0+ if u -::j::. x and k[(x- u)Ja] = 1 if x = u. Hence from (9), letting a--+ 0+, we have (10). If a--+ oo, then k[(x- u)Ja]--+ 1 for every pair x and u and (11) is obvious. If we suppose that k(x) (which is an even function) is a nondecreasing function of x > 0, then we have another property, that is, that ](x, a) is a nondecreasing function of a > 0. These properties of j(x, a) enable us to proceed in a way similar to that of the preceding section, but we shall not give the details here.

356

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

9.5. A Basic Theorem on Analytic Characteristic Functions Let f(t) be the characteristic function of a distribution function F(x). In association with this, we now consider the bilateral Laplace-Stieltjes transform

J1(s) = J~oo rsx dF(x), a,

T

s =a+

h:,

(1)

being real. Write / 1

-(s)

=

J:oo rsx dF(x).

(2)

Then insofar as / 1+, j 1- exist for some s. Since F(x) is bounded, / 1+(s) is analytic in a > 0 and moreover it is continuous on a = 0, - oo < T < oo. If the origin is the convergence abscissa, then from Theorem 8.4.1, it must be a singular point since F(x) is non decreasing. In other words if / 1+(s) is analytic at 0, then the convergence abscissa must be negative, that is, from Theorem 7.3.2, f 1+(s) is analytic in a> -TJ 2 for some 'f/z > 0. In the same way, ifj1-(s) is analytic at 0, then it is analytic in a < 1J 1 for some 'f/ 1 > 0. Therefore, f 1(s) is analytic in 'f/ 1 > a > -TJ 2 • Now suppose that f 1(s) is analytic at the origin; then / 1+(s) and J1-(s) both are analytic at the origin. Because since J1-(s) is always analytic in a < 0, if f 1(s) is analytic at the origin, then / 1+(s) must be analytic in some semicircle Is I < 15, a< 0. Butf1+(s) is analytic in a> 0. Hence, by Theorem 1.14.1,f1 +(s) is analytic in Is I< 15, and thenj1-(s) is also analytic in I s I < d. Now take

J(z)

= J~oo eizx dF(x).

(3)

Since by the transformation s = -iz from the s-plane to the z-plane, f(z) = / 1 (s) and the right half-plane of s is transformed onto the upper half-plane of z. The above result can be stated in the following form (see Lukacs [1, 3]).

Theorem 9.5.1. If the characteristic function f(t) is analytic at t = 0, then there is a strip 'f/ 1 > lm z > - 'f/z, 'f/ 1 > 0, 'f/z > 0 in the z-plane,

9.6.

CONTINUITY AND UNIQUENESS THEOREMS

357

and an analytic functiong(z) there such thatg(t) =f(t), -oo < t < oo, i-r. Furthermore, g(z) can be represented as where z = t

+

g(z)

= J~oo eizz dF(x)

in this strip. The last statement is obvious from the above consideration. We shall write f(z) for g(z) from now on. If 'f/ 1 > lm z > -'1} 2 is the largest strip in which f(z) is analytic, it is called the strip of analyticity off(t). It can be the whole plane; in this case, f(t) or f(z) is called an entire characteristic function. It is obvious that if 'f/ 1 > lm z > -'1} 2 is a strip of analyticity, then i'f} 1 and -i'f} 2 are singular points of f(z). Since

J~ e'z dF(x) < oo for every 0 < r < 'fJ is equivalent to the convergence off1 (s) in Res> -'f} and a similar statement is true for f 2 (s), transforming from f 1 (s) to f(z), we may make the following claim:

Theorem 9.5.2. The strip of analyticity of f(z)

is given by

-'f}

< lm z <

'fJ, 'fJ

J: for every r >

'fJ

=

J:

eitz dF(x)

> 0 if and only if

erlzl

dF(x) < oo

(4)

and not for r > "'·

9.6. Continuity Theorems on Intervals and Uniqueness Theorems Let {Fn(x), n = 1, 2, ... } be a sequence of distribution functions and let fn(t) be the characteristic function of Fn(x). Suppose that fn(t) converges to a function on some neighborhood (- U, U) of the origin. If we do not assume the convergence of fn(t) outside (- U, U), we can still sometimes derive the convergence of Fn(x) to a distribution function and accordingly the convergence offn(t) on the

358

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

whole interval (- oo, oo ). One of the simplest examples will be the following. If fn(t) converges to 1 in an interval (- U, U), U > 0, then fn(t) converges to 1 in every finite interval and F 11 (x) converges to a unit distribution function U(x) for x -=1= 0.

From Theorem 3.7.1 (i),

I!"(2t) - fn(t) I ~ 2ll 2 [1 - Re j (t)]I 12 • 11

Hence if fn(t) --+ 1, I t I < U, then /,,(2t) --+ 1 for I t I < U or fn(t) --+ 1 in I t I < 2U. Repeating this argument, we find thatj11 (t) --+ 1 for every t. Then from Theorem 9.3.5, F 11 (x) converges weakly to the unit distribution function, since 1 is the characteristic function of the unit distribution. We shall first of all prove a theorem.

Theorem 9.6.1. Let F 11 (x), n = 1, 2, ... , be distribution functions and letfn(t) be the characteristic function of F 11 (x). If F 11 (x) converges weakly to a function F(x) E M and j 11 (t) converges to a function f(t) in a neighborhood (- U, U) of the origin, and if either (i) the Fourier series of f(t) in (- U, U) is (C, 1)-summable to 1 at 0, or (ii) f(t) is continuous at t = 0, then F(x) should be a distribution function and its characteristic function should be identical with f(t) in (- U, U).

Proof. The proof is substantially included in that of Theorem 9.3.5 because by Theorem 9.3.2, L 11 (x, a) (defined in the proof of Theorem 9.3.2) converges to L(x, a) for every x >a, -oo < x < oo, which is the Fejer integral of e-itzJ*(t), where

f"'(t) =

roo

eixt

dF(x),

-oo

and if (i) is true, then as in the proof of Theorem 9.3.2, we obtain f"'(O) = 1. This shows that F(x) is a distribution function. Condition (ii) is a stronger condition than (i). The last statement of the theorem is obvious since f*(t) then becomes the characteristic function of F(x) and j 11 (t) converges to f*(t) everywhere (actually, uniformly in every finite interval).

9.6.

CONTINUITY AND UNIQUENESS THEOREMS

359

Now let Q be the class of characteristic functions, each f(t) of which satisfies the following conditions: there is a neighborhood of the origin ( -~.~)and a function g(z), z = t + i-r, which is analytic and bounded in It I ~ ~. 0 < T < r, such ~hat f(t) is the boundary function of g(z) on It I < ~ [that is, lim1-+o+g(t +iT)= g(t) and f(t) = g(t) a.e.].

Theorem 9.6.2. A characteristic function f(t) of Q is uniquely determined by the values of f(t) in a neighborhood of the origin. That is, if two characteristic functionsf1 (t) andj2 (t) of Q are identical on I t I < ~ for some ~ > 0, then j 1 (t) = j 2 (t) for - oo < t < oo. Proof. Let f 1(t) = g 1(t) a.e. in I t I < ~. and g 1(t) be the boundary function of g 1(z) which is bounded and analytic in I t I < ~. 0 < T < r, where z = t + i-r. Write

Jl,l(z) = J~ eizx dF(x), where

Then f 1,1(z) and f 1. 2(z) are analytic in T > 0, and T < 0, respectively. Then f 1,1(z)- g1(z) is analytic in It I < ~. 0 < T < r, and is seen to be bounded in the same rectangle. Since by assumption,f1,1(t) - g(t) = f 1. 2(t) a.e. on It I < ~.we obtain, by Theorem 1.14.1, f 1,1(z)- g 1(z) and f 1, 2(z) are analytic continuations of each other. Hence, f 1, 2 (z) is analytic at the origin and therefore by using Theorem 9.5.1, we see that J 1, 2 (z) is analytic everywhere on the real axis (there is an analytic function in a strip I t I < 'TJ for some 'YJ, which is identical with f 1, 2(z) on It I < ~). Then j 1(z)- g 1(z) is also analytic on - oo < t < oo. The same thing is true for f 2(z) - g 2(z), where f 2,1(z),f2, 2(z), and g 2(z) are counterparts of f 1, 1(z),f1, 2(z), and g1(z), and g 2(z) is supposed to be analytic in I t I < ~', 0 < t < r'. Then [j1(z)- g 1(z)]- [j2(z)- g 2(z)] is also analytic on ( -oo, oo). But since j 1 (t) = j 2 (t) on I t I < ~ 0 , ~ 0 being min(~. ~'), g 1 (z) = g 2 (z) in the common rectangle by Corollary 1.14.1. Therefore,

is zero on I t I < ~ 0 and is analytic on (- oo, oo ). This shows the theorem.

360

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

There is another class of characteristic functions having uniqueness. Let O(x) be a nonnegative, nondecreasing function for x > 1, satisfying

J~ [O(u)fu 2 ] du

=

(1)

oo.

Let Q 1 be the class of characteristic functions f(t) for which the corresponding distribution functions F(x) satisfy F(x) ~A exp -0(1 xI)

(2)

as x ---+- - oo, where O(x) is a fixed function independent ofF conditioned as above and A is some constant independent of F.

Theorem 9.6.3. A characteristic function f(t) of Q 1 is uniquely determined by the values of f(t) in a neighborhood of the origin. (See Note 6.) Before showing this theorem, we observe the following: Let G(x) be a bounded function of bounded variation over ( -oo, oo), and let g(t) be its Fourier-Stieltjes transform

g(t) =

r:

eixt dG(x).

g(t) is, of course, a bounded continuous function. Define ga(t) = g(t)(sin atfat). Then ga(t)

E

(3)

£2( -oo, oo).

Lemma 9.6.1. ga(t) has the Fourier transform in £2(-oo, oo): Ha(x) = (:n/2) 1i 2a- 1 [G(x Proof.

+ z)- G(x- a)].

It is sufficient to show that lim(2:n)- 112 T-+oo

JT-T ga(t)eizt dt =

Ha(x)

for almost all x. (2:n)- 112

rT

ga(t)e-izt dt = (2:n)- 112

rT

(sin atfat) dt

= (2:n)- 112 f~oo dG(u)

r:

(4)

eitlu-zl dG(u)

s:T (sin atfat) eillu-xl dt,

9.6.

361

CONTINUITY AND UNIQUENESS THEOREMS

which converges as T--+ oo, by Lemma 2.6.4, to

f

.C-Hl

(n/2) 1 ' 2a- 1 if x - a, x all x.

x-a

dG(u)

+ a are continuity points of G(u ). Hence (4) is true for almost

We remark that the characteristic function of the limit distribution function of distributions whose characteristic functions belong to Q 1 also belong to Q 1 •

Proof of Theorem 9.6.3. Letf1 {t) andf2 (t) be in Q 1 and letf1 {t) = f 2 (t) in I t I < !5 for some !5 > 0, where k

=

1, 2.

Write Then

g(t) = J~oo

eixt

dG(x) = 0

(5)

in It I < !5. Obviously, G(x) = O(exp -0(1 xI)) as x--+ -oo. Hence

Ha(x) = (n/2) 1' 2a- 1 [G(x +a)- G(x- a)]= O(exp -0(1 xI)) as x--+ -oo for a fixed a > 0; and, ga(t) in (3), which is the Fourier transform of Ha(x) by L~mma 9.6.1, is zero in I t I < !5 from (5). Therefore, by Theorem 8.3.2, ga(t) = 0 for -oo < t < oo. Thus g(t) = 0, -oo < t < oo. Since g(t) = f 1(t) - f 2 (t), this shows the theorem. Now we can show rather easily the following theorems:

Theorem 9.6.4. Let J., (t ), n = 1, 2, ... , be the characteristic function of a distribution function F11 (x). Suppose that J.,(t) converges in an interval (- !5, !5) to a function f(t). (i) If f(t) is the boundary function in ( -!5, £5) of a function f(z) which is analytic and bounded in I t! ; : :; !5, 0 < T ~ r (r > 0), z = t+iT, and (ii) If f(t) is continuous at t = 0, or more generally, the Fourier series off( t) is (C, 1)-summable to 1 at t = 0,

362

IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

then F 11 (x) converges to a distribution function F(.x) andf(t) is uniquely extended to the characteristic function of F(x) on (- oo, oo ).

Theorem 9.6.5. Let.f;,(t), n = 1, 2, ... , be the characteristic functions of Q 1 • If f 11 (t) converges in some (- <5, <5) to a function f(t) which is continuous at t = 0 and f(O) = 1, or more generally, the Fourier series of which is (C, 1)-summable at t = 0 to 1, then F11 (x) converges to a distribution function F(x) andf(t) is uniquely extended to the characteristic function of F(x) on (- oo, oo ). Proof of Theorem 9.6.4. By the first Helly theorem (Theorem 9.1.1 ), the sequence {F,(x)} contains a convergent subsequence {F11 k(x)}. From Theorem 9.6.1, the limit function is a distribution function. The characteristic function f 1 (t) of the limit distribution function is identical with f(t) in ( -<5, <5). Hence f 1(t) E Q. From Theorem 9.6.2, f 1 (t) is uniquely determined by the values of f(t) in ( -<5, b). This shows that every {Fnk(x)} has the same limit characteristic function. This completes the proof. Proof of Theorem 9.6.5. The proof is just the same as that of Theorem 9.6.4 if we note the remark given after the proof of Lemma 9.6.1 that Q 1 is closed.

9.7. The Compact Set of Characteristic Functions Consider a class of characteristic functions which is said to be compact if any sequence of characteristic functions of the class contains a subsequence which converges uniformly in every finite interval. Needless to say, the limit of the locally uniform convergent subsequence is again the characteristic function by Theorem 9.3.7. We agree to say that a sequence of functions converges locally uniformly if it does so uniformly in every finite interval. Let {¢>n(t), n = 1, 2, ... } be a sequence of functions defined over [a, b] (finite or infinite). If, for any e > 0, there exist n0 = n0 (e) and h0 = h0 ( e), depending only on e, such that

I tPn(t +h)- tPn(t) I < for n ~ n0 , I hI tinuous in [a, b].

~

e

(1)

h0 , t, t +hE [a, b], then {t/>n(t)} is called equicon-

9.7.

THE COMPACT SET OF CHARACTERISTIC FUNCTIONS

363

Let t 0 be a point of (a, b). If for any e > 0, there exist n 0 = n0 (e) and lz 0 = h0 ( e} such that

I c/J (to 11

+ 1z} -

c/J 11 {to}

I< e

(2)

for n;;;:; n0 , I hI ;;;:;; h0 , t 0 +hE (a, b), then {cp,(t)} is called equicontinuous at t = t 0 • If any sequence from a class of functions is equicontinuous in [a, b] or at t = t 0 , the class of functions is called equicontinuous in [a, b] or at t = t 0 , respectively. We shall show the following theorem.

Theorem 9.7.1. A class of characteristic functions is compact if and only if it is equicontinuous at the origin. This was given by Raikov [1]. Proof. The "if" part is a consequence of the inequality for the characteristic function l(t),

ll(t +h)- l(t)

1 ;;;:;;

2112[1- Rel(h)]ll2

(3)

[Theorem 3.7.1 (i)] and Arzela's theorem (Hobson [1], Vol. 2, p. 168) that a uniformly bounded class of equicontinuous functions over a finite interval contains a sequence which converges uniformly. In fact, if the class is the equicontinuous class at the origin, then from (3 ), it is the class of equicontinuous functions over (- oo, oo ), and since the characteristic function is bounded by 1 in absolute value, we have the compactness of the class from the above theorem of Arzela. "Only if" part. If the class is not equicontinuous at the origin, then there are an e > 0 and hk = hk( e) converging to zero and nk = nk( e), nk - oo as k - oo, such that

k

=

1, 2, ....

(4)

Then the sequence lnk(x) does not contain a sequence converging uniformly in an interval containing the origin, because if it does, and Ink (x) converges uniformly in [a, p], a < 0 < p, then, writing the limit of In" (x) by l(x), we have 1(0) = 1 and

•

taking k. large enough, which contradicts (4 ).

364 IX.

CONVERGENCE OF DISTRIBUTION AND CHARACTERISTIC FUNCTIONS

The "only if" part of Theorem 9.7.1 is slightly changed in the following form, which is also due to Raikov [1].

Theorem 9.7.2. Suppose that a sequence of characteristic functions is not equicontinuous at the origin. Then there is no interval (a, {3) containing the origin such that the sequence converges in (a, {3) to a function which is continuous at the origin.

Proof.

If a sequence Un(t)} of characteristic functions is not equicontinuous at the origin, then as in (4), there are an e > 0, hk- 0, and nk such that (4) holds. (4) implies that

Re fnk(hk)

< 1-

E

cos 0,

where 0 is the angle between the segment (0, 1) and the segment joining 1 and the intersection of the unit circle and I z - 1 I = e in the z-plane (Fig. 9.1). Write 1 - E COS () = t' < 1. Then

k = 1, 2, ....

(5)

Suppose that fn(t) converges in (-a, a), a > 0 to f(t) continuous at the origin. Sincef(O) = limn_,.oofn(O) = 1,

I(2h)-1 s:h f(t) dt I> [(1 + -r)/2)1'2 for a sufficiently small h (
FIGURE

9.1.

l~k-

(6)

h. Then we see that

365

NOTES

which is not greater than

because of Theorem 3.7.1 (iv). The right-hand side of this last equation is equal to or less than [(1 + -r)/2] 112 in view of (5), that is, (7) Since l1,A--+ h and fnk(t) converges to f(t) boundedly, we see from (7) that 1 (2h)- 1

I~k f(t) dt 1 ~

[(1

+ -r)/2]11

2•

This contradicts (5 ). NOTES 1.

For more details about Theorem 9.1.5, see Krickeberg [1].

2. In the proof of Theorem 9.3.2, "if part," we may avoid direct use of Theorem 9.1.1 (Helly's first theorem). This is because J~ Fn(v) dv is a convex function of P and has a limiting function J(P) which is also convex and continuous. But a convex function has right and left derivatives at every point and they are equal almost everywhere, the exceptional set being at most countable. The convergence of Fn(P) follows for all P except those in this set. 3. In Theorem 9.3.8, Fn(x) is supposed to be standardized. This assumption is replaced by the fact that Fn (x) is left-continuous for all n or right-continuous for all n. 4. In connection with Theorem 9.4.1, we note that if the distribution functions Fn(x) converge to a continuous distribution function, then .4'(1/n I")--+ 0, where fn is the characteristic function of Fn(x). The converse is not true. (See Kawata and Udagawa [1].) 5. Section 9.4, Eqs. (10) and (11) are the consequences of Theorems 5.2.2 and 5.5.2, respectively. But because of the fact thatf(t) is a characteristic function, we do not need the additional conditions on K(x) that are required in those theorems. 6. Theorem 9.6.3 is stated in a more general form: If / 1 , /2 e Q1 and / 1 (t) = / 2 (t) in some interval (a, b) (not necessarily a neighborhood of the origin), then f 1 (t) = / 2{t), -oo < t < oo. This is seen if Theorem 8.3.2 is applied to the interval (a, b). 7. Theorems 9.6.1, 9.6.3, and 9.6.5 do not seem to be stated explicitly so far in the literature.

X Some Properties of Characteristic Functions

1 0.1. Characteristic Properties of Fourier Coefficients The purpose of these few sections is to discuss the characteristic properties of a characteristic function of a distribution function and some related topics. However, it would be more convenient, for a better understanding, to begin with similar problems concerning Fourier coefficients or Fourier transforms, although it is not intended to go into great detail. Suppose that a sequence {en, n = 0, ± 1, ±2, ... } of complex numbers or a sequence {an, n = 0, 1, 2, ... ; bn, n = 1, 2, ... } be given [2cn =an- ibn,

2c-n =an+ ibn,

n

= 0, 1,...

(b 0

= 0)].

We ask when {en} or {an, bn} is the sequence of the Fourier coefficients of a functi9n of specified class. Let us form the formal trigonometric series ~

la 0

~

+ L (an cos nx + bn sin nx) = n=--oo L cneinz n-1

366

(1)

10.1.

367

PROPERTIES OF FOURIER COEFFICIENTS

and the (C, 1)-means of it, that is,

un(x) = (n

n

n

+ 1)- k=O L sk(x) = k=-n L {1 - [I k I/(n + 1)] }ckeikz, 1

(2)

where

sk(x) = !a0

k

k

j=l

j--k

+ L (ai cosjx + bj sinjx) = L cieiiz.

[See Section 4.5, Eq. (15).] We first show the following theorem.

Theorem 10.1.1. A necessary and sufficient condition that a sequence {en, n = 0, ±1, ... } be a sequence of Fourier-Stieltjes coefficients of a function of V( -:n;, :n;), is that

J:,. I Un{x) I dx ~ K

(3)

for some constant K. Proof. Suppose that en is a Fourier-Stieltjes coefficient of F(x) e V( -:n;, :n:), that is,

Cn

=

{Z:n:)-1 J~

e-inz

dF(x),

n=0,±1, ....

Then, as in Section 4.6, (4)

where

Kn(u) = l(n

+ 1)- [sin(n + 1) !ufsin !u)2, 1

(5)

Kn(u) being the Fejer kernel and :n;- 1

J:,. Kn(u) du = 1.

We then see that

J:,. I un(x) I dx ~ J:,. dx J:,. Kn(x- u) I dF(u) I J:,. I dF(u) I J:,. Kn(x- u) dx J:,. I dF(u) I K. :n;- 1

=

:n;- 1

=

=

(6)

X.

368

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

Next we shall show the sufficiency of (3). Write

Then from (3),

J:,. I ndF(x) I J:,. I an(x) I dx ~ K, =

that is, F11 (x) is uniformly of bounded variation. F,.(x) can be written by G,.(x) - Hn(x), where G 11 (x) and H 11 (x) are nondecreasing and are chosen so that they are uniformly bounded. Then by Theorem 9.1.1, we see that F11 (x) contains a subsequence Fnrc(x) which converges to a function F(x) of bounded variation at continuity points of F(x). If we consider IP11 (x) = F 11 (x) - CoX, then IP11 (x) is periodic outside ( -:n, n) since F 11 ( -n) = 0, F 11 (n) = 2nc0 , and I'Pn (x) converges to some IP(x), even at x = -:n, :n when IP(x) is supposed to be standardized at ±n. Hence Fnrc(x) may be supposed to converge at x = -:n and :n if F(x) is defined to be standardized. Now let nk > j for a given j. Then

(2n)- 1

J:, e-iiz dF rc(x) = (2n)- J:, e-iizanrc(x) dx (2n)- f" e-iiz ~ [1 1

11

=

1

-n

= [1Let

n~o:-+

(I , 1/n~o:)]c,ei·x dx

v--nk

(ljlfn~o:)]ci.

oo, j being fixed. Then using Theorem 9.1.2

This proves the theorem.

Theorem 10.1.2. A necessary and sufficient condition that {c.,, n

=

0,

± 1, ... } be a sequence of Fourier-Stieltjes coefficients of a bounded nondecreasing function in [-n, :n], is that an(x);::;; 0, n = 0, 1, 2, .... Exactly the same method of proof as that used in Theorem 10.1.1 is applicable. In fact, (6) follows if a11 (x);;;;; 0 since Kn ;::;; 0. The other parts of the proof do not need to be changed except that the functions of V are replaced by the bounded nondecreasing functions.

10.1.

369

PROPERTIES OF FOURIER COEFFICIENTS

Theorems 10.1.1 and 10.1.2 are true if a.. (x) is replaced by Abel's mean:

f(r, x) =

00

2:

c,.einxrlnl

n--oo 00

= iao

+ L (an cos nx + b.. sin nx)rn,

O~r
n=l

The condition in Theorem 10.1.1 turns out to be such that

J: IJ(r,

x) I dx

~ K,

0

~

r < 1,

(7)

where K is a constant. The condition a.. (x) ;;;; 0 in Theorem 10.1.2 turns out to be

f(r, x) ;;;; 0,

0

~r

< 1,

XE

[-:n,:n).

(8)

In terms of the probability distribution function, we may state (from Theorem 10.1.2) the following theorem:

Theorem 10.1.3. A necessary and sufficient condition that {en, n = 0,

±1, ±2, ... } be a characteristic sequence of a distribution function with bounded spectra contained in [ -:n, :n] is that an(x) ;;;; 0 and c0 = 1. Now we discuss the condition for c.. to be the Fourier coefficient of a function belonging to L( -:n, :n).

Theorem 10.1.4. A necessary and sufficient condition for {c,, n = 0,

±1, ... } to be a sequence of Fourier coefficients of a function belonging to £1( -:n, :n) is that

J:, I O'm(x) -

a..(x) I dx- 0

(9)

as m, n- oo.

Proof. Necessity. Suppose that c,. is the Fourier coefficient of a function f(x) E £1( -:n, :n). Then

X.

370

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

where Kn(x) is the Fejer kernel given by (5) and f(x) is periodically continued. We then see that

J:,. [f(x- t) - f(x)]Kn(t) dt, f(x) I dx ~ n- J:,. dx J:,. I f(x- t) - f(x) I Kn(t) dt = J:,. Kn(t)(t) dt,

an(x) - f(x) = n-1

J:,. I an(x) -

1

where

(t) =

J:,. If(x -

t) - f(x) I dx.

Since (t) is continuous and (0) = 0, the above integral converges to zero as n ---+ oo, from Theorem 4.6.1. Hence

J:,. I an(x) -

f(x) I dx---+ 0,

from which (9) follows. Sufficiency. From (9) it follows that

J:,. I an(x) I dx = 0(1) as n---+ oo. Therefore, by Theorem 10.1.1, en, n = 0, ±1, ... are the Fourier coefficients of a function F(x) of bounded variation. It is sufficient to show that F(x) is absolutely continuous. In order to do this, it is enough to show that

Fn(x) =

J:,. an(u) du

is uniformly absolutely continuous, that is, for any e > 0, there exist a !5 and an N (both depending only on e) such that, for any nonoverlapping intervals (ai, hi), i = 1, 2, ... , m, with L~ 1 (hi - ai) < !5, we have m

L I Fn(hi) i-1

Fn(a;)

I ~ e,

n

~N.

(10)

This is so, because, choosing a subsequence Fnt(x) converging weakly to F(x), we obtain from (10), m

L I F(hi) i-1

F(a;)

I~e

(11)

10.1.

371

PROPERTIES OF FOURIER COEFFICIENTS

if a,, b,, i = 1, ... , m, are all continuity points of F(x). Therefore, if F(x) is defined by F(x- 0), then (11) remains true for F(x) at every set a,, b;, i = 1, 2, ... , m, and F(x) becomes absolutely continuous. From Theorem 10.1.1, Cj

= =

J:,. e-iix dF(x) = (2n)- J:,. e-iix dF(x) (Zn)-1 J:,. e-iizJ(x) dx (2n)- 1

1

if we write F(x)

=

J:,.J(u) du.

(Note that nondecreasing functions which are identical except at their discontinuities generate the same S"tieltjes integral.) Thus it is sufficient to show (10). Choose N so large that

J:,. I an(x) for n

~

a.v(x) I dx

<

e/2

(12)

N. Fix N. Let m

S

=

U (ai, b,), i-1

(ai, bi) being nonoverlapping, and choose <5 so small that (13) if the measure of Sis less than <5. This is obviously possible since aN(x) is bounded for a fixed N. Then we see that, for n ~ N, m

L I Fn(bi) i-1 =

F(ai)

I

i~J IJ:: an(x) dx I

fs I an(x) I dx ~ fs I an(x) - a.v(x) I dx + fs I aN(x) I dx ~ J:,. I an(x) - a.v(x) I dx + fs I a.v(x) I dx ~

~

!e + !e = e

by (12) and (13). This shows the theorem.

X.

372

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

10.2. Basic Theorems on Characterization of a Characteristic Function The results in the preceding section can be carried over to the Fourier transform and the characteristic function, which gives us the characteristic property of, for instance, the characteristic function. However, we make a different approach to the same problem. We recall first that a sequence of functions over (- =, =) is said to converge locally uniformly if it converges uniformly in every finite interval. In this case we call the limit the locally uniform limit. We begin with the following lemma: If a series :L~ an 1s (C, 1)-summable to sand 0, then it converges to s. (ii) If, for a locally integrable function g(t), r:Tg(t) dt is (C, 1)summable to s as T--+ =and g(t) ~ 0, then it converges to s as T--+ =·

Lemma 10.2.1. (i) an

~

[For (C, 1)-summability of a series or an integral, see Sections 4.5 and 4.8.] Proof. We prove (i); the proof of (ii) is done the same way. From Section 4.5, Eq. (15), the (C, 1)-summability of an to s is equivalent to

as Now since an

~

n--+

=·

0,

where [x] is the integral part of x. This shows the convergence of :L an. It is then obvious that the sums must be the same. We shall show a basic theorem:

Theorem 10.2.1. Suppose that f(t) is a bounded continuous function with /(0)

= 1.

If there is an R 0 such that for every R

J

R

-R

j(t)e-i:zt dt ~ 0

~

R0 , (1)

10.2.

CHARACTERIZATION OF A CHARACTERISTIC FUNCTION

373

for every -oo < x < oo, or f(t) is a locally uniform limit function of a sequence of such functions, then f(t) is the characteristic function of a distribution function. Conversely, for any characteristic function, either there is an R 0 such that for every R ~ R 0 , ( 1) holds, or it is a locally uniform limit of a sequence of functions for which such R 0 exists. The following remark is obvious from the proof of this theorem given below: REMARK. If we drop the condition thatf(O) = 1, then Theorem 10.2.1 holds with the Fourier-Stieltjes transform of a bounded nondecreasing function in place of the characteristic function.

The integral

R f(t)e-ixt dt J -R in (1) is the Fourier transform (up to a constant factor) of the truncated function / 11 (t) = f(t), I t I ~ R, / 11 (t) = 0, I t I ~ R. The theorem is also true if we consider the Fourier series of fR(t) or the Fourier series of

111( t ) --

{ /(t), f(t + 2R),

-R -oo

t < R, < t < oo.

~

(2)

Let us write the Fourier series of f 11 (t) as

JR(t) ,....,

L

cn(R)einnttR,

(3)

n=-oo

where

Cu(R) = (2R)-1 JR j(t)e-innt!R dt. -R

(4)

We may claim the following theorem.

Theorem IO.Z.Z. Suppose that f(t) is a bounded continuous function with f(O) = 1. If, for f(t), there is an increasing sequence {Ri} with 0 < Ri ---+ oo such that n

= 0, ±1, ±2, ... , j = 1, 2, ... ,

(5)

or f(t) is the locally uniform limit of a sequence of such functions, then f(t) is the characteristic function of a distribution function. Conversely for any characteristic function, either there is an increasing sequence {Ri} with 0 < Ri ---+ oo such that (5) holds, or it is the locally uniform limit of a sequence of functions for which such {Ri} exists.

X.

374

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

We shall prove here Theorem 10.2.1.

Proof of Theorem 10.2.1. First part. Letfn(t) be the function obtained by the truncation of f(t) at I t I = R. Then from Theorem 4.8.2, fn(t) for

=

~~ (Zn)-1 (T

I t I < R. From condition

J:

(J v 1/T)]eitv dv J~00 fn(u)e-ivu du

[1 -

(1)

f n(u)eivu du

for R ;?; R 0 • Form (6) with t

(6)

=

J~R f(u)e-ivu du

;?; 0

= 0 and Lemma 10.2.1 (ii),

J~oofR(u)e-ivu duE V(-oo, oo), and hence (6) can be written (7)

Now let us write

tfon(v)

=

J:fn(u)e-iuv du,

Then t/Jn(v) ;?; 0, R ;?; R 0 , and

F R(x) = (2n)- 1 J:oo tfon(v) dv is a distribution function since it is nondecreasing, and from (6) with t = 0,

1 =fn(O) = (Zn)- 1 J~oo tfon(v)dv = F(oo). Therefore we see that (6) turns out to be

If there is an R(;?; R 0 ) such that f(t) = fn(t), then f(t) is the characteristic function of FR(x). Otherwise, limn-+oofn(t) = f(t) holds locally uniformly. Therefore, from Theorem 9.3.7, FR(x) converges weakly to some distribution function F(x) and

f(t)

=

J~oo eixt dF(x).

10.2.

CHARACTERIZATION OF A CHARACTERISTIC FUNCTION

The converse part.

Let

f(t)

=

J:

375

eizt dF(x)

with a distribution function F(x). Set

ifJA(t) = { ~- (It

It I ~A, It I >A.

for for

1/A)

We then have

ifJA(t) = n- 1 J~oo [sin 2 (Auf2)/(Au 2 /2)]eiut du

= J~oo eiut daA(u), where

C1.tt(u)

= n-1

J:oo [sin (Au/2)/(Au f2)] du. 2

2

We know a.4 (oo) = 1, and then C1.tt(u) is a distribution function. Now from Theorem 3.10.1,

that is, ifJA(t)f(t) is a characteristic function. Since as A---. oo, ifJA(t)f(t) converges locally uniformly to f(t), it suffices to prove that -oo

for R

~A.

<X<

oo,

In fact,

f~R e-iztq,A(t)j(t) dt =

rA e-izt[l -

(I t l/A)]f(t) dt

= rA e-i:tl[l -
J:

I:

eitu dF(u)

dF(u) r.tt [1 - (It 1/A)]e-itc:z:-u) dt

= 2 J~oo {sin 2 [A(x- u)/2]/[A(x- u) 2 /2]} dF(u) ~ 0. This completes the proof.

376

X.

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

The proof of Theorem 10.2.2 is done in a similar way, and will be left to the reader.

10.3. Characteristic Properties of Characteristic Functions We shall derive from Theorem 10.2.1 several important theorems on characterization of a characteristic function. A continuous function f(t), - oo < t < oo, is called nonnegative definite if, for any measurable bounded function e(x) vanishing outside some finite interval (that is, with bounded support),

J:

J:f(t- s)~(t)~(s) dt ds ~ 0.

(1)

The equivalent definition of nonnegative definiteness is the following: A continuous function is said to be nonnegative definite, if for any finite sequence {tk> k = 1, 2, ... , n} of real numbers and any finite sequence {~ k> k = 1, 2, ... , n} of complex numbers, (2) The equivalence of (1) and (2) is shown in the following way. Suppose that (1) holds. For a given {~k> k = 1, 2, ... , n} of complex numbers and a given {tk, k = 1, 2, ... , n} of real numbers, define

~(t) = { (2e )-l~k> 0,

for I t - tk I < e, elsewhere,

where e > 0 is chosen so small that no intervals (tk- e, tk + e), k = 1, 2, ... , n, overlap (t 1 ,. •• , tn are supposed to be distinct). Then (1) turns out to be

This becomes (2) when e __. 0+. Conversely, suppose that (2) holds. In order to prove (1 ), it is sufficient to suppose that ~(t) is continuous in a finite interval and vanishes outside it because the class of continuous functions is dense in V norm in the class of bounded functions. Then, by the definition of the Riemann in-

10.3.

377

PROPERTIES OF CHARACTERISTIC FUNCTIONS

tegral, the integral on the left-hand side of ( 1) is the limit of

where t 0 < t 1 < · · · < t,. is a division of the interval outside which ~(t) = 0. (3) is nonnegative in view of (2). Hence, as its limit, the integral on the left-hand side of (1) is nonnegative. This shows the required proof. Nonnegative definiteness is also defined on an interval (a, b). A continuous function f(t), defined on (a, b), is called nonnegative definite if (2) holds for every sequence {~k> k = 1, 2, ... , n} of complex numbers and for any sequence {tk, k = 1, 2, ... , n} such that ti - tk E (a, b),j, k = 1, 2, ... , n. In this definition, (a, b) may be replaced by a more general set. The corresponding definitions of integral form are also made. However, we agree that if we say that a function is nonnegative definite, then it means that it is so in (- =, = ). We now prove an extremely important theorem given by Bochner [5].

Theorem 10.3.1. A continuous function f(t) with /(0) = 1 is a characteristic function if and only if it is nonnegative definite.

Proof. Suppose that f(t) is a characteristic function J:'oo eizt dF(x). Then, for a bounded measurable function ~(t) vanishing outside (-A, A),

f: f:

f(t - s)~(t)~(s) dt ds

=

A fA-A f -A

-

=

f: ItA ~(t)eitx r

~(t)~(s)

foo

dt ds dt

-oo

ei
dF(x)

~ 0.

Hence a characteristic function is nonnegative definite. Next suppose that a continuous function f(t) with /(0) negative definite. Then ( 1) holds. Take

A -ll2e-iut

~(t) = { 0

for 0 ~ t elsewhere,

,

=

1 1s non-

~A,

where u is a any fixed real number. From (1 ), 0 ~ A- 1

fA0 fA0 J(t -

x)e-i!t-slu dt ds

=

fA

~

[1 -

(J v J/A)]J(v)e-ivu dv.

X.

378

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

Therefore, if we define the function

f(t, A) = { {(t)[1 - (I t 1/A)] then for every R

f

~

R

-R

for for

It I ~A, It I> A,

A,

f(t, s)e-iut dt ~ 0,

-00

< u < oo,

and f(t, A) converges to f(t) locally uniformly as A---+ oo. Hence from Theorem 10.2.1, f(t) is a characteristic function. We next show the Khinchin criterion for a characteristic function (Khinchin [4 ]).

Theorem 10.3.2. A function of the form f(t)

= c- 1

J~oo ifJ(x + t)ifJ(x) dx,

(4)

where ifJ(x) is any function of L 2 (-oo, oo) and

c=

(f oo

-oo

I ifJ(x) 12dx

)112

'

or the locally uniform limit of such functions is a characteristic function. The converse is also true. Proof. The first part. From Theorem 9.3.7, it is sufficient to show that (4) is a characteristic function. Thatf(O) = 1 is obvious. The functionf(t) is continuous since 1f(t

+h)- f(t) 1 = c- 1 f~oo 1 ifJ(x + t +h)- ifJ(x + t) ~ c- 1

[f oo

-oo

I

ifJ(u +h)- ifJ(u)

12

du

]1/2 [foo

-oo

I

ifJ(x)

and the first factor converges to zero as h ---+ 0. Consider

ifJA(x) = { ifJ(x), 0, We then see that

lxl ~A, lxl >A.

1 1 ifJ(x) 1 dx

12

dx

]112

10.3.

379

PROPERTIES OF CHARACTERISTIC FUNCTIONS

< x < oo because

holds uniformly for - oo

IJ~oo 1/JA(x + t)ifJA(x) dx- J~oo 1/J(x + t)t/J(x) dx I ~I J~oo ifJA(x + t)[ifJA(x)- 1/J(x)] dx

+ IJ~oo 1/J(x) [1/JA(x + t)- 1/J(x + t)] dx I ~ [f~oo I ifJA(x + t) 1

2

+ [f~oo I 1/J(x) 1

2

~ 2 [f: I 1/J(x) 1

2

dx

2

[f~oo I 1/JA(x)- 1/J(x) 1

r [f~oo r [J~oo 2

dx

dx

r

2

I

2

I

1/J(Ax

dx

r 2

+ t)- 1/J(x + t) 1

1/JA(x)- 1/J(x)

12

which does not contain t and converges to zero as A sufficient to show that

dx ---+

r

2

dx

r

2

2 •

oo. Hence it is

where

is a characteristic function. fA(O) = 1, and the continuity of fA(x) 1s obvious as before. Let R be R ;?; 2A. Then

J~R JA(t)e-izt dx =

JR-R e-izt dt Joo 1/J.t(X + t)t/JA(u) du oo -ifJA(u)eiuz du fR ifJA(u + t)e-i
~

= C,d 1

=

Noting that

C,dl

-R

J~oo 1/JA(u)eiuz du J~oo 1/JA(u + t)e-i
I u + t I >A for I u I
It

I > R, the last member is

IJ~oo 1/JA(u)e-iuz du r ;?; 0.

Hence Theorem 10.2.1 shows what we wanted to have. The converse part is shown in the following way. Suppose that a given function is a characteristic function. From Theorem 10.2.1, it is a locally

X.

380

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

uniform limit of a sequence of functions f(t), for which

J-RR J(t)e-ixt dt ~ 0,

-CXl

<X<

CXJ,

(6)

for R ~ R 0 , R 0 being some constant. Since fR(t), defined to be f(t) for I t I < R and 0 for I t I > R, converges locally uniformly to f(t), it is sufficient to show thatfR(t) is of the form (4). From Section 10.2, Eq. (7) in the proof of Theorem 10.2.1, we have (7)

Since

we write

'PR(t) = Then 'PR(t)

E

oo [(2n)-U2 J_oofR(u)e-itu du ]U2.

£2(- =, =) and (7) turns out to be fR(t) =

J:

'PR2(v)eitv dv.

Let 'PR(v) be the Fourier transform of tPR(u) in £2(-=,±). Then the Fourier transform of tPR(t + u) (as a function of u) is 'PR(v)eivt, and the Fourier transform of 'PR(v) = 'PR(v) is tfo(u). Hence by the Parseval relation [Section 6.2, Eq. (13)],

J:

'PR2(v)eitv dv =

Therefore fR(t) (R theorem.

~

J:

tPR(u)tfJR(t

R 0 ) has the form (4) (c

+ u) du.

= 1). This proves the

We shall give another theorem due to Cramer [2] and Domingnez [1], in which the criterion for a characteristic function is given in terms of a function similar to ](x, a) defined by Eq. (8), Section 9.4.

Theorem 10.3.3. Suppose that K(x) is any function satisfying the following conditions:

J~oo I K(x) I dx < K(x)

= (2n)- 1' 2

J:

CXJ,

k(t)eixt dt

(8)

(9)

10.3.

381

PROPERTIES OF CHARACTERISTIC FUNCTIONS

for some k(x)

E

V( -oo, oo) with k(x) ;;;;; 0, -oo < x < oo and K(O) = (231:)- 1' 2

J:

k(t) dt = 1.

(10)

Let

J(x, a) = (2:n:)- 1' 2 J~oo e-itzK(at)J(t) dt for a continuous bounded function f(t). Then f(t) is a characteristic function if and only if f(O) = 1 and ](x, a) ;;;;; 0 for 0
Proof. Suppose that for 0
For every A > 0, :n:-1 f~oo {sin 2 A(x- u)/[sin 2 (x- u)/2]} dx

r:

e-itxK(at)J(t) dt ;;;;; 0,

which is written by

rA

[1 - (It 1/A)]K(at)f(t)e-itu dt ;;;;; 0.

(11)

Let A > 1 and a= 1/A. Then (11) becomes

f~A

[1 -

(I t 1/A)]K(t/A)f(t)e-iut dt

;;;;; 0,

-00


00.

(12)

Write

f(t, A) = { ~~ - (I t 1/A)]K(t/A)J(t),

I tl ~A. I tl >A.

Then (12) is written, for R > A, as

fR J(t, A)e-iut dt ;;;;; 0,

-00


oo,

and f(t, A) converges locally uniformly to f(t) as A-+ oo. Thus by Theorem 10.2.1 (the first part), f(t) must be a characteristic function.

X.

382

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

The converse part is simple to show and, in fact,

J: e-itzK(at)f(t) dt =

J~oo e-itzK(at) dt

J: eitu dF(u),

where F(u) is a distribution function. This is equal to

J:

J~"" dF(u)

K(at)e-it
=

a- 1

J:

k[(x- u)fa] dF(u) ~ 0.

We shall give one more theorem.

Theorem 10.3.4. For a bounded continuous function f(t) to be a characteristic function, it is necessary and sufficient that, /(0) = 1 and for every l/J(x) e Ll(-oo, oo) for which the Fourier transform ~(t) of l/J(x) is nonnegative for -oo < t < oo,

J:

Proof.

l/J(x)f(x) dx ~ 0.

As is easily seen, if

J:

f(t)

=

J~oo eitu dF(u),

l/J(x)f(x) dx = (2n)11 2

J:

~(u) dF(u) ~ 0.

The necessity is shown with f(O) = 1. For proof of the sufficiency, take

l/J(t)

= {

~1 - (I x 1/A)]K(x/A)e-iut

I xI~ A, I xI> A,

for for

for any fixed u, where K(x) is the function in Theorem 10.3.3.

~(t) =

(2n)112 J~A

(1 _

I~~ ) K( ~

)

e-i
dx

J""-oo _s"'""'in_2::-:A~(x __u_....,..-:-:t),_ dx J"" e-ivzK(-'V-) dv sin 2(x - u - t)/2 -oo A ---;:=-A-=-:-:=- J"" si.n2 A(x- u- t) k(Ax) dx ~ 0, n(2n)11 2 -oo sm 2(x- u- t)/2

----;:=-1-=-:-:::-

n(2n)112

where k(t) is the function in Theorem 10.3.3. Then (13) turns out to be

t.4. [1 -

(I x 1/A)]K(x/A)f(x)e-iuz dx ~ 0,

-oo

<X<

00.

This is no more than (12) in the proof of Theorem 10.3.3. This gives the required conclusion by the same reasoning used there.

10.4.

383

FUNCTIONS OF THE WIENER CLASS

10.4. Functions of the Wiener Class Let us denote by S the class offunctions cf>(t) on - oo < t < oo such that

(1) exists for every x and by S' the class of functions of S for which f(t) is continuous. Wiener studied the properties of functions of S and S' (Wiener [5, 7]). S' is a proper subclass of S. In other words, there is a function f(t) defined by (1) which is not continuous. For example, take cf>(x) = exp ix2 (Wiener [7], p. 151). Then

f(t) = lim (2T)- 1 JT exp i[(x + t) 2 - x2] dx T-+oo

-T

= lim(exp it2f2T) T-+oo

JT-T exp 2ixt dx = { 01•,

t = 0, t =F 0.

If cf>(t) is thought of as a function of time, then (1) is the average of the product of the values off at the time x and the time x + t and is called the sample covariance function. If cf>(t) e S, then

If(t) I ~f(O) = ;~ (2T)-l s:T I cf>(x) 12 dx

(2)

and hence f(t) is bounded. If f(t) with cf> E S is continuous at t = 0, then it is continuous everywhere, -oo < t < oo, and hence cf>(t) e S' because

lf
+ h) =

~

J(t) 1

llim(2T)- 1

s:T [cf>(x + t + h) -

lim [(2T)-1 JT

T-+oo

-T

cf>(x + t)]cf>(x) dx

I

I cf>(x + t +h)- cf>(x + t) 12 dx] 112

r

s:T I cf>(x) 12 dx 2 = fl' 2(0) {;~[ (2T)-1 s:T I cf>(x + t +h) 12 dx + (2T)-l s:T I cf>(x + t) 12 dx - 2 Re(2T)- 1 s:T cf>(x + t + h)cf>(x + t) dx ]}112 . ( (2T)-l

= 2.f'2(0)[f(O)- Ref(h))1'2.

(3)

X.

384

Furthermore when 1/J(t)

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

E

S',

(2T)- 1 J:T t!J(x

+ t)t!J(x) dx

converges to f(t) locally unzformly fort, as will be seen from the following theorem and what follows it.

Theorem 10.4.1. If 1/J(t) E S' and /(0)

=

1, then f(t) is the character-

istic function of a distribution function. Proof. This is immediate by the application of Bochner's theorem (Theorem 10.3.1) because

L Li

f(t,- t;)~i~j

j

t

=

!~(2T)- 1

=

Lj Li T-+oo lim(2T)- 1 J

=

t ~~~(2T)- 1 J:T 1/J(x + t;)t/J(x + t;) dx ~i~i

=

lim(2T)- 1 JT T-+oo -T

~ J:T

I

+ t ; - t;)?J[X} dx ~i~i

1/J(x

T~I -T~I

1/J(x + t;)t/J(x

+ t;) dx ~i~i

Li 1/J(x + t;)~; 1 dx ~ 0. 2

This shows the theorem. Now consider t!JA(x) = 1/J(x), I xI ~A, t!JA(x) = 0, I xI >A and define

fA(t)

=

(2A)-I f~oo 1/JA(x

Then it is easy to show that lfA(t)

+ t)t/JA(x) dx.

I ~ 1/A(O) I and

fA(t) = (2A)-I tA 1/J(x

+ t)t/J(x) dx- 1

(4)

1,

where 11 = (2A)- 1 JA

A-t

fort~

1/J(x + t)t/J(x) dx

0 and I

I1

I

~

(2A)-l

[t~

~

(ZA)- 1

t:

I

I

1/J(x) 12 dx

1/J(x) 12 dx,

r·

[f~_,

I

1/J(x)

12

dx

ra

10.5.

385

SUFFICIENT CRITERIA FOR CHARACTERISTIC FUNCTIONS

which converges to 0 as A --+- oo in every finite interval in t easy to see). The same is true fort ~ 0. Hence

~

0 (this is

JA.(t) - f(t)

(5)

locally uniformly. This gives us the statement immediately following (3). We remark that f.,t(t) is also seen to be nonnegative definite and thus fA(t)ffA(O) is a characteristic function. Now from Theorem 10.4.1, for ifJ(x) e S',

f(t) =

J:

eixt dF(x),

(6)

F(x) being a bounded nondecreasing function, and by the inversion formula, F(x)- F(O) = lim (Zn)- 1' 2 A-+oo

fA

[(e-ixt- 1)/-it]f(t) dt

-A

at the continuity points x of F(x).

10.5. Some Sufficient Criteria for Characteristic Functions To begin with, we mention the following result on cosine series which is due to Young [1] (see Zygmund [1], p. 109). A sequence {an, n = 0, 1, 2, ... } of real numbers is called convex if (1)

for n = 0, 1, 2, ... , where .dan= an- an+I·

Theorem 10.5.1. If an--+- 0 and {an} is convex, then 00

lao

+ n-1 La

11

cos nx

(2)

converges except for x = 0 to a nonnegative function g(x) and (2) is the Fourier series of a functionf(x) of Ll( -:n, :n). The conclusion that (2) is the Fourier series remains true if {an} is convex for n ~ N, that is, .1 2an ~ 0, n ~ N, for some integer N. This is obvious.

X.

386

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

Proof. Let sn(x), n = 0, 1, 2, ... be the partial sum of the series (2). We use Abel's transformation n

n-1

L u,.;vk = k-m L ultvk lt-m

"m-t'Vm + UnVn'

(3)

where ""' = I;f=o u;, and where if m = 0, "m-1 is interpreted to be zero. Applying this formula to (2), we have n

L L1ak D~~:(x) + Dn(x)an, 11:-o

sn(x) = where D~~:(x)

=

(4)

k

l

+ j-1 L cosjx = sin(k + i)x/2 sin ix

[see Section 4.1, Eq. (6)] is the Dirichlet kernel. Applying formula (3) again, n

sn(x) =

L L1 2a~~:(k + 1)Kk(x) + (n + 1)Kn(x) L1an+t + Dn(x)an+I•

(5)

11:-o

where K~~:(x) =

i(k

+ 1)- [sin i(k + 1)x/sin lx)2 1

is the Fejer kernel [see Section 4.6, Eq. (3)]. Since an--+- 0 as n--+- oo, the last two terms on the right-hand side of (5) converge to zero if x -=1= 0 and 0 ~

n

L (k + 1) L1 a"' K~~:(x) 2

~ l(sin ix)-2

11:-o

n

L L1 2alt

lt-o

(by the convexity of {an}). The last member is n

i(sin lx)- 2

L L1 2 a~~: =

k-o

n

l(sin ix)- 2

L [L1a"' -

k-o

+

L1ak+I]

Hence L~o (k 1) L1 2ak K~~:(x) converges when x =I= 0. Therefore, sn(x) --+- f(x), x =I= 0, where 00

f(x) =

L (k + 1) L1 akKk(x) 2

11:-o

Let

~ 0.

10.5.

SUFFICIENT CRITERIA FOR CHARACTERISTIC FUNCTIONS

387

JN(x) converges increasingly to f(x), and (6) Since L:J2an ~ 0 and an converges to zero, we see that Llan ~ 0. Therefore, from (6),

This shows, by Fubini's theorem, thatf(x) e LI(-oo, oo). Furthermore,

J:,. I/.v(x)- f(x) I dx--+ 0

as N--+ oo

(7)

since the integral is 00

n

L

k-N+l

(k

+ 1) Ll ak. 2

We mention here that, from (7), as n > m, and since

I,.

m --+ oo,

n

Un(x) - fm(x)] dx = n

-n

L Llak- nn Llan+l, k-m+1

we have as

n--+ oo.

(8)

We now prove that (2) is the Fourier series ofj(x). Consider, for N ~ n, :n- 1

J:,. f(x) cos nx dx =

:n- 1

J:,. [f(x)- f.v(x)] cos nx dx J:,. f.v(x) cos nx dx

+ :n/1 + lz, 1

=

(9)

say. Then

I I1 I ~ :n-1 J:,. I f(x)

- f.v(x) I dx-

o

as N--+oo

X.

388

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

for all n and

where K,.,(x) can be written k

(1/2)

+ L {1 -

[j/(k

+ 1)]} cosjx

j-1

for k

~

/ 2

1, and K 0 (x)

=

= l [Section 4.5, Eq. (15)]. Therefore,

N

L (k + 1) A ad1 2

N

[nf(k

+ 1)]} = L A a,.,(k + 1 2

k-

n)

k-

N

=

L Aak- N k-n.

AaN+l =an- aN+l- N AaN-1•

which converges to an as N---+ oo in view of (8) and aN +I---+ 0. Hence, going back to (9), n

=

0, 1, 2, ... ,

which says that (2) is the Fourier series of f(x). Suppose that an (n = 0, 1, -1, ... ) are given and that an are real numbers with a_n = an, A 2an ~ 0, n = 0, 1, 2, .... Then by Theorem 10.5.1, an are the Fourier coefficients of some nonnegative function g(x) which is an even function, since the Fourier series is the cosine series. Therefore we may write

n= ±1, ... , Hence, if we write

then we may say that if {en, n = ... , -2, 1, 0, 1, 2, ... } is a sequence of real numbers with en = c_n, {en}, n ~ 0, is a convex sequence, en---+ 0 as n---+ oo, and c0 = 1, then {en} is the characteristic sequence of an absolutely continuous distribution with bounded spectra contained in [ -n, n] and the probability density is an even function.

10.5.

SUFFICIENT CRITERIA FOR CHARACTERISTIC FUNCTIONS

389

The above result is true for a function over (- oo, oo ).

Theorem 10.5.2. A real-valued, continuous functionf(t), -oo< t
= 1, f(t) = f( -t),

f(O)

f(t) is convex, for 0 < t, f(t) ---+- 0 as t---+- oo,

is the characteristic function of an absolutely continuous distribution function and the probability density is an even function. This theorem was given by P6lya [1]. The necessary and sufficient conditions for a characteristic function as given in the preceding section are sometimes not readily applicable when we want to construct characteristic functions or to judge whether a given function is a characteristic function. Theorem 10.5.2 gives a simple criterion in many cases even though it deals with a rather specific kind of characteristic function. We shall give some examples illustrating this later on. We shall prove the theorem. The proof of Theorem 10.5.1 is carried over if J"(t) > 0, t > 0. Even if J"(t) does not exist, still the same reasoning applies to show the theorem.

Proof of Theorem 10.5.2.

Define

sT(x) = (2n)- 1 I~T e-itx_f(t) dt.

(10)

Because of Theorem 1.11.1(ii) and (10),

sT(x) = n- 1 I: cos tx f(t) dt = (sin Txfnx)f(T)- (nx)- 1 I: sin tx D+f(t) dt.

(11)

From (iv) and Theorem 1.1l.l(iv), f(O)

= -

I

->+oo

0

(12)

D+f(t) dt.

Since D-t-f(t) is nondecreasing, D+f(t)

E

V(O, oo) and

as

t---+- oo.

(13)

X.

390

Hence from (11), for

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

xi= 0,

s(x) =lim sr(x) = -(nx)- 1

I"" sin tx D+f(t) dt.

(14)

0

T-+oo

Now we shall show that s(x) ;;;; 0 for all x > 0. Let x > 0. Then

-(nx)- 1 I

00

s(x)

=

I
D+f(t) sin tx dt

kn/z

k=O

Since 00

I

(-1)kD+f[u

+ (knfx)]

0

is, from (13), an alternating series with decreasing terms in absolute values, it is boundedly convergent and since the first term is

we have

s(x) ;;;; 0

for

x > 0.

(15)

From (14),

I0Xs(x) cos tx dx = Letting X

--+

-n- 1

I""0 D+f(u) du IX0 (sin ux cos txfx) dx.

oo, we have from Lemma 2.6.4,

I;"" s(x) cos tx dx =

-!I~ D+f(u) du.

Since from (13),

f(t) =I~ D+f(u) du, this is 2

I

-+00

0

s(x) cos tx dx

=

f(t).

Since this is true for t = 0 and s(x) ;;;; 0, the left integral is absolutely convergent and we may write

f(t) =

I:

s(x)eitz dx

(16)

if s( -x) is defined to be s(x) for x > 0. This with (10) shows our conclusion.

10.5.

SUFFICIENT CRITERIA FOR CHARACTERISTIC FUNCTIONS

391

One of the typical examples of Theorem 10.5.1 is that 00

L cos nx flog n

(17)

n-2

is a Fourier series. The corresponding example is that

f(t) =

{

1 - (tf2e), (1/Zlog t),

0 < t t > e,

~

e,

(18)

withf(t) = f( -t), the characteristic function of an absolutely continuous distribution function. We now give another criterion due to Dugue and Girault (see Dugue [4], p. 18 and also Dugue and Girault [1] and Lukacs [3]).

Theorem 10.5.3. Let f(t) be a real-valued continuous function which satifies the following conditions: (i) (ii) (iii) (iv) (v)

J(O) = 1, f(-t) =f(t), f(t) is convex in {0, r], f(t) is periodic with period 2r, and f(r) = 0.

Then f(t) is the characteristic function of a lattice distribution function.

Proof.

Define

It I< r, It I> r.

fl(t) = { f(t), 0,

Then / 1{t) satisfies the conditions of Theorem 10.5.2 and the Fourier transform of the probability density s(x) ~ 0 [in (16)] up to a constant multiplier. Since f 1 {t) E £1(- =, = ), s(x) is the inverse transform of / 1 (t) and thus

Hence the Fourier series of f(t) 00

J(t) ,....,

L

(19)

Cneint,

n""'-oo

has the coefficient Cn

=

(Zr)-1 fJ(t)e-n"ittr dt ~ 0,

n = 0, ±1, ....

X.

392

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

Since f(t) is convex and continuous [0, r], it is of bounded variation [Theorem 1.11.1(iii)] and therefore the Fourier series (19) converges tof(t). This shows thatf(t) is a characteristic function of a lattice distribution with point spectra contained in the set {nnfr, n = 0, ±1, ... }. The two characteristic functions/( t) and / 1 ( t) in the proof of this theorem are equal in the interval ( -r, r) but they are different of course and correspond to different distribution functions. REMARK.

This remark is of interest if the class Q or Q1 of characteristic functions considered in Section 9.6 is taken into account. For the characteristic functions in Q or Q1 , such a situation never takes place.

10.6. More Criteria for Characteristic Functions We shall study functions of characteristic functions, which are themselves characteristic functions. We begin with the following result (see for instance Lukacs [2]).

Theorem 10.6.1. Let {av, v = 1, 2, ... } be a sequence of real numbers. The necessary and sufficient condition that 00

f(t) =

L a,ifJ,(t)

(1)

v-1

should be a characteristic function for every sequence {ifJ,(t)} of characteristic functions is that

a,

~o.

(2)

Suppose (2) is satisfied and {ifJv(t)} is a sequence of characteristic functions. From (2), the series in (2) is uniformly convergent. We may suppose without loss of generality that a 1 > 0 (otherwise we may work with {am, am+I, ... }, where a,11 is the first member, which is positive). Let Proof.

Then gv(t) is a characteristic function. Therefore

10.6.

393

MORE CRITERIA FOR CHARACTERISTIC FUNCTIONS

where

cp,.(t)

I:

=

eizt dFv(x),

and ~ avFv(x) is clearly a distribution function. Since gn(t) converges uniformly to f(t), the latter function is a characteristic function, from Theorem 9.3.7. In order to show the converse, let us take cf>v(t) = eivz for v = 1, 2, ... , n, where n is a positive integer and cf>v(t) = 1, v = n 1, n 2, .... Then

+

J(t) =

L aveivz +v-n+l L av, v-l "

00

+

(3)

where the series L av must converge. Write

f(t)

=

I:

eitz dF(x),

F(x) being a distribution function. From Theorem 3.8.1, lim (2T)- 1 T-+oo

IT

f(t)e-itv dt = F(v

-T

On the other hand, from (3 ), if 1 lim (2T)- 1 T-+oo

IT

~

v

~

+ 0)- F(v- 0).

(4)

n,

j(t)e-ivt dt = av.

-T

Hence from (4), av ~ 0, for v = 1, 2, ... , n. Since n is arbitrary, av ~ 0 for all v. Letting t = 0, in (3 ), we have L:t av = 1. This shows the theorem. A simple consequence of this theorem is:

Theorem 10.6.2. If 00

g(z)

converges at z

=

L a,zn,

n-o

an

~

0,

n

= 0, 1, 2, ... ,

= 1, then f(t) = g[cf>(t)]/g(1)

(5)

is a characteristic function whenever cf>(t) is a characteristic function.

X.

394

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

This is immediate from the foregoing theorem, since rf>n(t) is the characteristic function of the n-fold convolution of F(x) with itself, F(x) being the distribution function whose characteristic function is rf>(t). EXAMPLE 1. Let g(z) = eAz Then g(1) = eA and hence

= :L:'=o (A.nfn!)zn, A. > 0 and rf>(t) = eit.

f(t) = exp A.(eit - 1)

(6)

is a characteristic function. Actually, it is the characteristic function of the Poisson distribution. EXAMPLE 2. Let rf>(t) = exp -at 2, a > 0, which is the characteristic function of a normal distribution function. Then

f(t) = exp(e-"'11

-

1)

is a characteristic function. In the same way, it is seen that exp[exp(A. exp it) -exp A.]

and

exp(exp[A. exp(-at2)] -exp A.)

are characteristic functions. Further examples are obtained by replacing g(z) by exp(exp yz), y > 0, and so on. (See Linnik [1].) EXAMPLE

3.

In a similar way we have, for A. > 0,

f(t) = exp A.[rf>(t)- 1]

(7)

is a characteristic function whenever rf>(t) is a characteristic function (De Finetti, see Lukacs [3], p. 83). Another important example is the following. EXAMPLE 4 (Luckacs [2]). and p > 1,

Whenever rf>(t) is a characteristic function

f(t) = (p - 1)/[P - rf>(t)]

(8)

is a characteristic function. In order to show this, take

g(z) = [(p- 1)/(P- z)]IIn, where the principal value of the power is adopted. This is written 1

g(z) = [(p- 1)/p]Itn{ [1- (zfp)]

}1/n

•

10.6.

MORE CRITERIA FOR CHARACTERISTIC FUNCTIONS

395

The binomial expansion gives us g(z)

= [(p - 1)/p ptn x[1

+ k-1 ~ {(1 + n)(1 + 2n)···[1 + (k-

1)n]f(np)kk!}zk],

which has positive coefficients in the power series. EXAMPLE 5. Let V(x) be any nondecreasing function on (0, M), 0 < M < oo, and let

g(z)

=

f(t)

=

f

.tf

0

ezz dV(x).

Then (9)

g[tP(t)]/g(O)

is a characteristic function whenever
f(t) = exp -It I",

0


2,

(10)

is a characteristic function. This function satisfies (11)

where b1 , b2 are arbitrary positive constants and b is a positive constant. More generally, the distribution function for which the corresponding characteristic function satisfies

(12) for every b1 , b2 > 0 and for some positive b and real constant y, is called a stable distribution function. In terms of distribution functions F( x ), (12) is expressed as

F[(x- c1 )/b 1 ]

•

F[(x- c2 )/b 2]

=

F[(x- c)fb]

(13)

X.

396

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

with some positive b > 0 and real c > 0 for every b1 , b2 > 0 and every real c1 , c2 . Now suppose f(t) is given by (10). If 0 < a ~ 1, then e-1" is convex for t > 0 (f"(t) > 0), and from Theorem 10.5.2, it is a characteristic function of an absolutely continuous distribution function. If a = 2, (10) is the characteristic function of a normal distribution function Next we shall show that for 1 < a < 2, (10) is a characteristic function. (Here we give a proof by Linnik [1 ], Chapter 1.) Set

ifJ(t)

=

1/(1

+ It 1"),

1
(14)

This is convex fort > 0 [ifJ"(t) > 0]. Hence, again from Theorem 10.5.2,

ifJ(t) = f : eitx v(x) dx,

(15)

where v(x) E £1(-oo, oo) and is an even function since ifJ(t) is real. Since ifJ(t) E £1(- oo, oo ), we have

v(x)

=

(2n)- 1 f: [e-itz/(1

+ ltl")] dt =

n- 1 Re f~[eitx/(1

+ t")] dt

(16) by the inversion of the Fourier transform. Consider the function 1/(1 + z") in the complex z plane. A zero z 0 of 1 z" is the value of z such that z = (-1 )1'"', that is, I z 0 I = 1, arg z 0 = (2k + 1 )nfa. For no integer k does (2k + 1 )nfa lie in the first quadrant; 0 <' (2k 1 )n/a < n/2, because 1 < a < 2. Hence the sinz") are not in the first quadrant. Consider the branch gularities of 1/(1 of z" : z"' = r"eirzo (z = xei6 ) and make the cut along the positive real axis. Then by the Cauchy integral formula,

+

+

f A exp(ixt) dt o 1 + t"

+

= i fA exp( -:xu) du _ A f"'2 exp(ixAe.iO) eio dO. o

1

+ z"'u"'

o

1

+ A "e' 6"

The second integral on the right is, in absolute value, not greater than

A f"'2 exp( -xA sin 0) d() o A"- 1 which converges to zero as A

f oo exp ixt dt o 1+t"

=

-+

for

A> 1,

oo for x > 0. Hence

i foo exp -:xu du o 1+z"u"

for

x > 0,

397

NOTES

which 1s

. Joo t

+ exp( -niaf2)u"] + exp(niaf2)u~ 1

exp( -xu)[1

11

o

2

du.

Therefore, from (16),

v(x)

=

n1 sin 2na Joo 0

exp(-xu)u" 1 + exp(nia/2) ux

d 12

u>

0

(17)

for x > 0. Since v(x) is even, from (15), we obtain the fact that rfo(t) in (14) is a characteristic function of a distribution with density v(x). Then rfo(t/B) is also a characteristic function for any B > 0. Let n be an integer and choose B so that Bx = n. Then [rfo(t/B)]n is also a characteristic function and

[r/J(t/B)]n = [1/(1

+ I t l"n- )]n 1

converges to a continuous function exp -I t 1~ as n ->- oo. Hence exp -I t I, is a characteristic function and it belongs to £1 (- oo, oo ), and the corresponding distribution function is absolutely continuous (Corollary 4.8.1). Thus we have shown the following theorem.

Theorem 10.6.3. exp -I t 1", 0 2 is not a characteristic function [see Section 3.7, Item (b), p. 97]. There are further results on forming characteristic functions, for which · the reader should see Lukacs [3]. NOTES 1. In connection with Theorems 10.1.1-10.1.4, we mention the following theorem. Let u,.(x) be defined by Section 10.1, Eq. (2). A necessary and sufficient condition for {c,., n = 0, ±1, ... } to be the Fourier coefficients of a function L,, p > 1, is that I u,.(x) lp dx be bounded (Young and Young [1]). For {c,.} to be the Fourier coefficients of a continuous function, it is necessary and sufficient that u,.(x) converge uniformly. For {c,.} to be the Fourier coefficients of a bounded function, it is necessary and sufficient that u,.(x) be uniformly bounded (Fekete [1]). For details see Zygmund [1], pp. 79-88. Theorem 10.1.1 is due to Young [1], and Theorem 10.1.4 to Steinhaus [1] and Gross [1]. In the above, u,.(x) may be replaced by Abel's mean.

r ..

X.

398

SOME PROPERTIES OF CHARACTERISTIC FUNCTIONS

2. Theorems 10.2.1 and 10.2.2 and the derivations of the theorems in Section 10.3 are due to Kawata [6]. 3. Theorem 10.3.4 is substantially due to Bochner [1] and explicitly to Andersen and Kawata [1].

4. Analogs of the theorems in Section 10.1 to Fourier transforms were also studied by Offord [2-4]. 5. The analogs or the generalizations of Theorems 10.1.1-10.1.4 for the characterizations of Fourier-Stieltjes transforms or Fourier transforms have been given by sever.tl authors. Here we mention several theorems of this type:

Theorem A (Cramer). Let K(x) andj(x, a) be the functions in Theorem 10.3.3. The necessary and sufficient condition that a bounded continuous function f(t) should be a Fourier-Stieltjes transform of a function F(x) of bounded variation over (-oo, oo) with J:.O I dF(x) I ~ M, is that e"oo I ](x, a) I dx ~ M for every 0 < a < 1 (Cramer [2]).

Theorem B (Bochner). The necessary and sufficient condition that a bounded continuous function f(t) should be a Fourier-Stieltjes transform of a function F(x) of bounded variation over (- oo, oo) with J:.O I dF(x) I ~ M, is that for any function rf>(x) e £1(-oo, oo),

Ifoo

-oo

rf>(t)f(t) dt

I~

M

sup -oo<:t
IJoo

r/>(t)etzl dtl.

-oo

Actually, Bochner proved the discrete analog of this theorem. (See Bochner [1] and also see Schoenberg [1].)

Theorem C (Cramer). Let K(x) be the function in Theorem 10.3.3 with additional conditions that K(x) be even and be O(l x 1-l-11), fJ > 0 as I x I -+ oo. The necessary and sufficient condition that a bounded continuous functionf(t) be a Fourier transform of a function g(x) e £1( -oo, oo), with J:C, I g(x) I dx ~ M, is that there correspond a «5 = d(e) such that for any measurable setS with mS < «5, 8 IJ(a, x) I dx ~ «5, where mS is the Lebesgue measure of S and ](a, x) is the function in Theorem 10.3.3 (Cramer [2]).

J

6.

For nonnegative definite functions, also see Akhiezer and Krein [1], Chapter 2 and Akhiezer [1], Chapter 5.

7. In Theorem 10.3.1, if the condition thatf(t) be continuous withf(O) = 1 is dropped, the conclusion is still true in the form thatf(t) is almost everywhere equal to the FourierStieltjes transform of a bounded nondecreasing function.

8.

For further studies on class S and S', see Wiener [5, 7] and also Masani [1] and Birkhoff [1].

*

Suppose that for f(t) e S', f(O) 0, and f(t) = 0 for t < 0. Let L(u + iT) be the Laplace transform of f(t) (the existence of the Laplace transform for u > 0 is easily

9.

399

NOTES

seen). Then f(t) =

lim (u/2n) a-+0+

foo-oo e""'

I

L(u

+ iT) 12 dT,

and F(x) in Eq. (6) of Section 10.4 is given by F(x)

=

lim (u/2n) a-+0+

J"'

I L(u

+ iT) 12 dT.

-oo

This was given by Bertrandias [1J. 10. In connection with Theorem 10.6.3, we mention that e- 1"' (t > 0) can be represented as the Laplace transform e- 1"' = f:;" e-"'1t/>,.(x) dx, 0 ,.(x)

=

.n- 1 Im[

J~ e-"'" exp( -e-i"u)"' du]

=

.n- 1 lm[

~0 [( -1)kfk!] exp( -i.nak) [F(ak + 1)/x«k+l]].

XI Distribution Functions and Their Characteristic Functions

11.1. Moments, Basic Properties In this chapter we shall study the relationships between the properties of distribution functions and their characteristic functions. We shall discuss, in particular, the connection between the existence of moments and the smoothness of their characteristic functions. But first it will be convenient to give some familiar elementary properties of moments. Let X(w) be a random variable on (D,9J, P) with the distribution function F(x). Then (1) IS

the rth moment of X, when

WI 400

X

I'

exists, where r

IS

a non-

11.1.

401

MOMENTS, BASIC PROPERTIES

negative integer

is the 11th absolute moment, 11 being any nonnegative real number. The following theorem is immediate from Holder's inequality and Minkowski's inequality.

Theorem 11.1.1. Let X and Y be two random variables. (i) then

If r, s > 1, (1/r) g'

(ii)

+ (1/s) =

I XY I ~ (WI

If r;;:;; 1 and

WI

X

I'<

(WI X+ Y 1')1/r (iii)

1, and X

X

12< oo, WI

Y

I'< oo,

l')li'(W I Y 1')11•. WI

oo,

~(WI

If 1 ;;:;; r > 0 and W I X

WI

X

I' <

Y

I'< oo,

(3)

then

1')11' + W(l Y 1')1/r.

oo, W

I Y I' <

(4)

oo, then (5)

(iv)

If W I X

I' <

oo, then W I X

(W I X 1')118

~

I' <

oo for 0 ~ s ~ r and

(W I X 1')1''·

(6)

The following theorem is also easy to show.

Theorem 11.1.2. Let {J be the rth absolute moment of a random variable. Then (i) {J./{J.+I is a nondecreasing function of 11 ;;:;; 0, and (ii) fJ!'• is a nondecreasing function of 11 ;;:;; 0, insofar as they exist. In fact, if {J.+I < oo, then taking I X Y in (3) with r = s = 2,

I1•-1112

for X and

w I X I• ~ (WI X I•-1)112(W I X 1•+1)112, which is This is (i).

I X I1•+1112 for

402

v'

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

If we take I X 1• for X and 1 for Y and r > v ~ 0 in (3), then we have

which is just P~2 ~

= v' fv, s = v' f(v'- v),

PW'.

We now prove a theorem which gives the estimate of P(l X (See Loeve [1], p. 157.)

I ~ a).

Theorem 11.1.3. Let g(x) be a nonnegative Borel function over (- oo, oo) and be even and nondecreasing in (0, oo ). Then for a random variable X such that 8"g(X) < oo, [Wg(X)- g(a)]fsup g(X) ~ P(l X

I ~a)

~ 8"g(X)fg(a),

(7)

where a is any positive number and the left-hand side is interpreted to be zero if sup g(X) = oo. Proof.

Let S = {ro: I X(ro) I ~a}. Then

8"g(X)

=

fog[X(w)] dP

=

f g[X(w)] dP + f.s.,g[X(w)] dP. 8

On S, g(l X I)~ g(a) and hence, noting g(X) = g(l X I) and g(X) ~ 0, Wg(X)

~

f g[X(w)] dP ~g(a)P(S) =g(a)P(l Xi~ a). 8

This is the right-hand inequality of (7). On sc, g(X) ~ g(a). If sup g(X) < oo, then 8"g(X)

~sup g(X)

f

8

dP

+ g(a) fsc dP ~sup g(X) P(S) + g(a),

which is the left-hand inequality of (7). If sup g(X) nothing to show.

=

oo, then there is

Corollary 11.1.1. P(l X

I ~ a)

~

a-r g' I X lr,

This is called Markoo's inequality.

r > 0, a> 0.

(8)

11.1.

403

MOMENTS, BASIC PROPERTIES

Corollary 11.1.2. P(l X - W X

I ~ a)

~

a> 0.

a- 2 var X,

(9)

This is Chebyshev's inequality. Let X( w) be a random variable such that P[X(w)

= j] = Pi•

j = 0, 1, 2, ... ,

00

Q(t)

=

W(tX)

L Piti,

=

It I~

(10)

1,

j=O

is called the (probability) moment generating Junction of a discrete random variable X. The defining series (10) is absolutely and uniformly convergent for I t I ~ 1 and Q(t) is a continuous function. Also, from the basic results on power series, Q(t) is differentiable as many times as we wish, for It I < t. Writing t = e-•, 1 ~ t ~ 0, 0 ~ s < oo, Q(t) = Q 1 (s)

= J~ e-sx dF(x),

0

~

s < oo,

1

~

t

~

0,

(11)

which is the Laplace-Stieltjes transform of F(x), where F(x) is the distribution function of X: x ~o.

F(x) = { 0 •

j <X

Po+···+ Pi•

~j

+ 1,

j

= 0, 1, 2, ....

(12)

From (10), we have g
=

00

L Pj[i!JU- k)!]tj-k.

j=k

(13)

In particular,

Q
k

=

0, 1, ....

(14)

In other words, Q(t) determines pj uniquely. The relationship between the generating function and the moments is derived from (11). In so doing, we give a simple lemma.

404

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Lemma 11.1.1. If J: xk dF(x) < oo, F(x) being a distribution function with F(O) = 0 and k being a nonnegative integer, then lim s-+o+

Joo e-•z:x;k dF(x) = Joo xk dF(x). 0

(15)

0

This is immediate from Theorem 8.5.1 if it is noted that

s > 0. Now, from (11) and Theorem 7.3.2,

s > 0. Hence, from Lemma 11.1.1,

Jooo xk dF(x) = Since Q 1 (s)

=

lim(-1)" d"Q 1 (s)fdsk.

(16)

s-+o+

Q(t) and from e-•

=

t,

dQ 1(s)fds = [dQ(t)fdt]( -t) or, more generally,

where the left-hand side means

Therefore from (16) we have the following theorem. Theorem 11.1.4. If X is a discrete random variable with P(X = j) = P; ,j = 0, 1, 2, ... , L;;" Pi= 1, and WX"< oo for some integer k ~ 0, then the right-hand side of the following equation exists and

W X"= lim (t dd )"Q(t), 1-+1-0 t where Q(t) is the moment generating function.

(17)

11.1.

405

MOMENTS, BASIC PROPERTIES

From (17) we get

wx =

wX

Q'(1- 0),

2

= Q"(1 - 0)

+ Q'(1 -

0),

where Q
=

W(e-tX)

=

J:

e-tz dF(x),

(18a)

where the integral is supposed to exist for some interval a < t < b, a < 0, b > 0. This is no more than the bilateral Laplace-Stieltjes transform. It is obvious that (18b)

Finally we shall talk about the so-called "moment problem," although we do not intend to go into details since it is beyond the scope of this book. (See Widder [1], Akhiezer [1], Akhiezer and Krein [1], and Shohat and Tamarkin [1].) Suppose that an infinite sequence of complex numbers 1-'i (j = 0, 1, 2, ... ) is given. The moment problems deal with the question of how to find the necessary and sufficient conditions for which there exists a distribution function F( x) such that 1-'i

= J~oo

xi dF(x),

j = 1, 2, ....

(19)

If the solution F(x) of (19) is uniquely determined (except at discontinuities), the problem is called determined. Then another question is how to find the conditions under which the moment problem can be determined. When (19) is replaced by 1-'i =

J8 xi dF(x),

j= 1, 2, ... ,

(20)

where S is a given closed Borel set, the moment problems are called S-moment problems. The moment problem described above with (19) is called the Hamburger moment problem and when S is [0, oo) or [0, 1] it is called the Stieltjes moment problem or the Hausdorff moment problem, respectively. Here we deal only with the Hamburger problem and give the main results without proof.

406

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Theorem 11.1.5. In order that, for a given sequence {p; ,j = 1, 2, ... }, Eq. (19) have a solution F(x), it is necessary and sufficient that the quadratic form of x 1 , ••• , Xn n

Qn(x) =

L f-'i+;X;X; i,j-o

(21)

be nonnegative for all real X; , for all n = 0, 1, 2, .... This condition is equivalent to

Lin=

I 1-'i+i lf.;-o ~ 0,

where I 1-'i+i lf.;-o is the (n ments 1-'i+i·

+ 1) X

n = 0, 1, 2, ... ,

(n

+ 1) determinant with

(22) (i, j) ele-

Theorem 11.1.6. (i) In order that there should exist a solution of (19) whose spectrum is not reduced to a finite set of points, it is necessary and sufficient that n = 0, 1, 2, .... (23) This condition is equivalent to the condition that Qn(x) of (21) be positive definite; that is, Qn(x) > 0 for all n and for all x0 , ••• , Xn, except when x 0 = · · · = Xn = 0. (ii) In order that there should exist a solution of (19) whose spectrum 1) distinct points, it is necessary and sufficient consists of exactly (k that 'JI = 0, 1, ... , k, Ll• > 0, (24) Ll• = 0, 'JI = k 1, k 2, ....

+

+

+

We now give examples of different distribution functions that have the same moments. From Section 7.6, Eq. (19), for a> 0, a> 0, -oo < t < oo,

J~ e-az: sin tx ,x«-1 dx

= [F(a)/(a 2

+t

2 )"'' 2 ]

sin[a arctan(t/a)].

Let 0 < A. < l, and write xA for x in this integral. We then get

J~ exp( -axA) sin(txA) xA"'- 1 dx

= [F(a)/A.(a 2

+t

2 )"'' 2 ]

sin[a arctan(t/a)].

For any nonnegative integer n, choose a so that aA. = n

t

= a

tan A.n.

+ 1, and choose (25)

11.1.

407

MOMENTS, BASIC PROPERTIES

Then sin[a arctan(t/a)]

=

sin(n

+ 1)n = 0

and

J~ exp( -axA) sin(txA) xn dx = 0, Define

F(x) =

n

= 0, 1, 2, ....

J C J: exp( -axA)[1 + e sin(tx)A] dx,

10,

X

;?; 0,

(26)

(27)

x ~o.

where C is a constant such that F( oo) = 1, and e is any number I e I < 1. Then F(x) is a distribution function and for each n,

J~ xn dF(x),

n

= 0, 1, ... ,

are the same values for all choices of e, I e I < 1, because of (26). If we use Eq. (18), Section 7.6,

J~ e-az cos tx x~- 1 dx

=

[F(a)f(a 2 + t 2 )1' 2 ] cos[a arctan(t/a)],

then we have as before, for a > 0, a > 0,

e>

(28)

0, - oo < t < oo,

J~ exp(-axll) cos(txll) x~~~- 1 dx = [F(a)fe(a 2 + t 2 )11 2] cos[a arctan(t/a)] choosing ae = 2n

+ 1, t =

0'

tan len.

J~ exp( -axil) cos(txll) x 2n dx = 0,

n = 0, 1, 2, ....

(29)

Then

J~oo exp( -al x Ill) cos(tl x Ill) xn dx = 0,

n

= 0, 1, 2, ... ,

because this is true for an even n from (27) since the exp( -al xI") cos(tl x Ill) is an even function and is also true for an odd n, since the integrand of the above is an odd function. Hence the distribution function

where C 1 is a constant such that F 1 (oo) = 1 and F 1 (x) has the same moments for all I e I < 1.

408

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

F 1(x) obviously has infinite range in both directions. The following sufficient conditions for the uniqueness of the moment problem were given by Carleman (see for example, Shohat and Tamarkin [1 ]). Theorem 11.1.7. A sufficient condition that the moment problem (19) be determined is that (31) L"" 1/(/l2jP'(2j) = =· j=l

More generally, it is also sufficient that

L 1/r2j =

(32)

oo,

j=l

11.2. Smoothness of a Characteristic Function and the Existence of Moments In general, the term "smoothness" refers to properties of functions such as, continuity, Lipschitz continuity (continuity with Lipschitz conditions), differentiability, analyticity, etc., although sometimes the term is used for specific properties (see the end of this section). In the preceding section, we mentioned the relationship between differentiability of the moment generating function and the existence of the moments. We now study in detail a similar relationship for characteristic functions. We shall use the following notations. Let f( x) be an arbitrary function over (- oo, oo ). Define its first difference by

iJ/'j(x)

=

iJhj(x)

=

f(x +h)- f(x- h)

and the higher difference by

We then easily find that

.:J,/'j(x) for n

= 1, 2, ....

=

,t (-1Y( ~

)f[x

+

(n- 2k)h].

(1)

11.2.

409

SMOOTHNESS OF A CHARACTERISTIC FUNCTION

In particular, (2) If f(x) is n-times differentiable at x, then

as h--+ 0.

(3)

Now let f(t) be the characteristic function of a distribution function F(x). Then, using (2),

L1~d(0) =

I: [L1~k'!ixt] 1 _0

dF(x)

= (2i) 2k

I:

(sin xh) 2k dF(x),

and

L1~J(0)/(2h) 2k = i2k I~oo (sin xhfh)2k dF(x),

(4)

where k is a nonnegative integer. From this relation we have the following result due to Lukacs [3], p. 27.

Theorem 11.2.1. If the characteristic function f(t) of a distribution function F(x) satisfies M

= lim inf/

L1~d(0)/(2h) 2k

/

< oo,

(5)

h-+o

then the 2kth moment of F(x) exists as do all the moments a8 of order s < 2k. Moreover, the derivatives j<• 1(t) exist for all t and for s = 1, 2, ... , 2k, and

(6) Proof.

For any numbers a< 0, b > 0,

I~oo

(sin xhfh) 2k dF(x)

~I:

(sin xhfh) 2k dF(x).

J!

If h--+ 0, then the right-hand integral converges to x2k dF(x) by the bounded convergence theorem. Hence, from (4 ), (5) implies

I:

x2k dF(x)

~ M.

Since a < 0, b > 0 are arbitrary, J:O x2k dF(x) ~ M. Hence from Theorem 11.1.1 (iv ), for s = 0, 1, 2, ... , 2k, the moment a, exists.

410

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Now

[j(t +h)- f(t)]/h = J~oo [(ei
lim[j(t +h)- f(t)]/h = h-+o

Joo-oo lim[(ei
(7)

that is, f'(t) exists and

f'(t) = i J~oo xeitx dF(x). In the same way,

This procedure can continue until we get

This shows (6) and completes the proof of Theorem 11.2.1. Note that the proof of the above theorem shows that the condition (5) implies the existence of the derivative off( t) of order 2k, which is continuous. Also note, however, that this statement is not true for an odd integer in place of 2k, as will be discussed later (Theorem 11.2.4 ). Theorem 11.2.1 gives us the following theorem as an immediate consequence.

Theorem 11.2.2. Suppose that the characteristic function f(t) of a distribution function F(x) be differentiable n times at t = 0. (i) If n is an even integer, then the sth moment of F(x) exists for s = 1, 2, ... , n and is ' (8) a = i-'J<'l(O). (ii) If n is an odd integer, then the sth moment of F(x) exists for s = 1, 2, ... , n- 1, and (8) holds for s = 1, 2, ... , n- 1.

11.2.

SMOOTHNESS OF A CHARACTERISTIC FUNCTION

411

In either case, j<•l(t) exists for every t J<•,(t) = i•

J:

x•eitz dF(x)

(9)

for each set of values of s. Case (i) is in fact the same as the preceding theorem. Case (ii) is also the special case of Theorem 11.2.1. If n is odd then n - 1 is even and the theorem applies. The converse of the theorem is true in the following sense.

Theorem 11.2.3. If the kth moment of the distribution function exists, k ~ 1, then the characteristic function f(t) is k-times differentiable and j
However, if k is an odd number, the result is not so complete. In fact even if J
L cosjtf(P logj),

f(t) = C- 1

(10)

j=2

where C · :l:~ 2 (p log j)-1 • The corresponding distribution function F(x) is given by the step function with saltus (2C)- 1 (j2 logj)- 1 at x = ±j, j = 2, 3, .... This example was given by Zygmund [3]. The series 00

L cos jt flog j

C- 1

j=2

is a Fourier series of an integrable function as we saw in Eq. (17), Section 10.5. Hence, from Theorem 4.1.5, the series 00

C- 1 L -sinjt (j logj) i=2

(11)

412

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

is uniformly convergent. (11) is a termwise differentiated series of (10) and consequently f(t) is differentiable at every t. However, the first absolute moment

I

oo

-co

I xl

co

dF(x)

= C- 1 ~2(jlogj)- 1 = oo. )=

Now we shall discuss the differentiability of odd orders of a characteristic function. We begin with the following lemmas which are frequently used in this matter.

Lemma 11.2.1.

Let G(x) be nondecreasing for x the following five statements are equivalent.

I: dG(x) = o(1) R- I: xk+ dG(x) = o(1) R I: xk- dG(x) = o(1)

(a)

Rk

(b)

1

(c)

1

1

~

0. If k

as

R-+-oo,

as

R-+-oo,

as

R-+-oo,

(d) R I~ xk- 1 sin 2 (xfR) dG(x)

= o(1)

as

R-+-oo,

Rk I~ I sin(x/R) lk+I dG(x)

= o(1)

as

R-+-oo.

(e)

Proof.

(a)

~

> 0, then

(b).

(a) means that G(oo) is finite and Rk[G(oo)- G(R)] H(x) = G(oo)- G(x). Then from (a),

~

0. Write

as R--+- oo. Now R- 1

I:

xk+ 1 dG(x)

= -R- 1

I:

= RkH(R)

xk+ 1 dH(x)

+ [(k + 1)/R]

(12)

I:

xkH(x) dx,

which is, by (12), R- 1

I:

xk+I dG(x) = o(1)

+ o( R-

1

I:

dx) = o(1 ).

11.2.

413

SMOOTHNESS OF A CHARACTERISTIC FUNCTION

(b)=> (a). Rk

I2RR dG(x) ~ R- I2RR xk+ 1

1

(13)

dG(x).

J:

From (b), for any e > 0, R- 1 xk+l dG(x) < e for R ~ R 0 for some R 0 • Hence for any integer n ~ 1, we have from (13) with 2n- 1R for R, (14)

for R

~

R 0 • Then Rk

I"" dG(x) = Rk r I

2 "R

n-1

R

<

dG(x)

2"-'

00

2e

L 2-
R

for

~

R0 •

n-1

This shows (b) => (a). (a) => (c). From (12), for any e > 0, RkH(R)

< e

for some R 1 > 0. Then R

I:

xk- 1 dG(x) = -R

=

I:

RkH(R)

xk- 1 dH(x)

+ (k-

< e + (k - 1 )Re

1)R

I:

I:

xk- 2H(x) dx

x- 2 dx

= e + (k - 1)e

for R ~ R 1 • This proves (a)=> (c). (c) => (a). If k ~ 1, this is obvious from R

If 0

<

k

I:

xk- 1 dG(x) ~ Rk

I:

dG(x).

< 1, then, since R

I2R xkR

1

dG(x) ~ 2k- 1Rk

I2R dG(x) R

and the left-hand side converges to zero, the right-hand side converges to zero and (14) holds again. From this, (a) follows as before.

414

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

(b)=> (d). R f~ .xk- 1 sin 2 (x/R) dG(x)

say. Since sin(x/R) x ~o. (2/n) 2R- 1

~

s:

s: + s:

=R

R

(2/n)(xfR) for 0 _xk+l dG(x)

and / 2

~R

~

x

~

~ / 1 ~ R- 1

s:

= 11 + 12 ,

R, sin(x/R)

s:

~

.xk+ 1 dG(x)

xfR for

(15)

(16)

.xk- 1 dG(x).

From (b) and the right-hand inequality of (15), / 1 --+- 0. On the other hand, since (b)=> (a)=> (c), / 2 -.0 from (16). This shows (b)=> (d). (d)=> (b). If (d) holds, then / 1 = o(1) since / 1 is not greater than the integral in (d). Then from the left-hand inequality of (15), (b) follows. (e)=> (b). As in (15), (2/n)k+lR- 1

J:

_xk+l dG(x)

~ Rk

~

J:

R- 1

s:

[sin(x/R)]k+l dG(x) _xk+l dG(x).

(17)

From (e) and the left-hand inequality, we get (b). (b) => (e). Rk J~ I sin(x/R)

Jk+ 1 dG(x) ~

Rk

J:

+ Rk

[sin(x/R)]k+l dG(x)

s:

dG(x).

The first term on the right-hand side converges to zero from (17) and (b). The second term also converges to zero from (a), which is equivalent to (b) as already shown.

Lemma 11.2.2. If k > 0 and G(x) is nondecreasing, the five statements in Lemma 11.2.1, with 0(1) instead of o(1) in each, are equivalent. The proof is carried out in exactly the same way as that for Lemma 11.2.1. We are now in position to show the following theorem given by Pitman [1, 2].

11.2.

415

SMOOTHNESS OF A CHARACTERISTIC FUNCTION

Theorem 11.2.4. Let k be an odd positive integer and let f(t) be the characteristic function of a distribution function F(x). The necessary and sufficient conditions for the existence of the kth derivative j
+ 1- F(x)] =

0,

(18)

z-+oo

and that lim JR x'< dF( x) R-+oo -R

(19)

exist. When these conditions are satisfied, j
(20)

Proof. Suppose that j
pk-11(t) = ik-l

J:

x'<-leitz dF(x)

J:

.x'<-•[(eitz _ 1)/t] dF(x)

from Theorem 11.2.2 (9). Now

{[jlk-11(t) _ j
= -ik-l J~oo x'<- 1 [sin 2 (tx/2)/(t/2)] dF(x)

+ ik J~oo x'<- (sin tx/t) dF(x). 1

(21)

Write M(x) = 1 - F( -x). Then the right-hand side of (21), is

-ik-l J~ xk- 1 [sin 2 (tx/2)/(t/2)] d[F(x)

+ M(x)]

+ik[J~ x'<- 1 (sin t:cft) dF(x)- J:'' x'< dF(x)]

J:'' x'< dM(x)]

-ik[f~ x'<- 1 (sin txft) dM(x) -ik Jltt x'< dM(x) -lit

=

-I.

+ /2- I a - /4,

say. Since k- 1 is even, / 1 is real while / 2

(22) -

/ 3 -

/ 4

is purely imaginary.

416

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

The existence of the limit of (22) when t - 0 [the existence ofj'kl(O)] implies the existence of the limits of / 1 and / 2 - / 3 - / 4 • / 1 is the kth derivative of the real part of f(t), which is cos xt dF(x). Since this is an even function of t, its odd order derivative is zero at t = 0. Hence lim,._. 0 / 1 = 0 and

f:'oo

lim t-+oo

Joo kk- [sin (txf2)/(t/2)] d[F(x) + M(x)] = 1

2

0.

(23)

0

+

Since F(x) M(x) is nondecreasing, (23) implies, by Lemma 11.2.1 [(d)=> (a)] (with G = F M, R = 1/t),

+ lim Rk[F(oo) + M(oo)- F(R)- M(R)] = R-+oo

0,

which is no more than (18). Now,

i-kJ2 = J~ xk- 1 (sin txft) dF(x) -

J:lt xk dF(x)

=

J

=

o( t J:' xk+~ dF(x)) + o( t- J: xk- dF(x)) ,

111

0

[(sin txftx)- 1]xk dF(x) 1

+ Joo

~

1

xk- 1 (sin txft) dF(x) 1

(24)

both terms of which are o(1) as t - 0+ because of (23) [deleting M(x)] and Lemma 11.2.1 [(d) => (b)]. Also we have / 3 - 0 as t - 0. Therefore, we have obtained the fact that j'kl(O) = lim /4 t-+o

exists. This gives us (19) and also (20). Conversely suppose that (18) and (19) hold. From (18)

xk[1 - F(x)] - 0,

xkF(-x)

-o

as

x - oo.

Then by Lemma 11.2.1 [(a)=> (d)], (23) results. That is, / 1 -0. Also, from Lemma 11.2.1 [(a)=> (b) and (c)] and (24), we have

1.-o,

/ 3

-0.

From (19) the limit of / 4 exists. Altogether, J'kl(O) must exist and (20) is obvious. This completes the proof.

11.2.

417

SMOOTHNESS OF A CHARACTERISTIC FUNCTION

We now give the following theorem of Zygmund [3, 4], which is substantially equivalent to Theorem 11.2.4 (Pitman).

Theorem 11.2.5. Let k be a positive odd integer and let f(t) be the characteristic function of a distribution function F(x). If as

t-o,

(25)

then lim

L1~t!f(O)f(2t)k

(26)

t-->-0

exists if and only if lim R-+oo

IR

xk dF(x)

(27)

-R

exists.

Proof.

From (4), with k

+ 1 in place of 2k,

t1~+lf(O) = 2k+I( -1 )
2k+I( -1 )t 2 I~ (sin xt)k+l d[F(x)

+ M(x)J,

(28)

where M(x) = 1 - F( -x) as before. Since k + 1 is even, the condition (25) is equivalent to the condition (18) in Theorem 11.2.4, by Lemma 11.2.1 [(a)<=> (e)]. Hence, by Theorem 11.2.4, with (25), (27) implies the existence of J(O) and consequently (26) exists. If (26) exists, then using (4 ), with k in place of 2k,

L1~tlf(O)f(2t)k = jkt-k I~"" (sin xt)k dF(x) = ikt-k

I~ (sin xt)k d[F(x)

+ F( -x)]

say, and I

I"" ~ t-k I""

1 2 1 ~ t-k

111

111

I

d[F(x)

d[F(x)

+ F( -x)] I

+ M(x)],

(29)

418

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

where M(x) is, as before, 1 - F( -x). This inequality is easily seen from

I F(x + !5) + F(-x~

F(x

!5)- F(x)- F(-x) I F(x) - M(x)

+ !5) + M(x + !5) -

for

!5 > 0.

Since condition (25) is equivalent to (18), (a)<=> (e), Lemma 11.2.1, which is the same as saying that (29) --+- 0. Thus / 2 = o(1 ).

11

-

ik

flit xk d[F(x) + F( -x)] 0

= ik

f

lit

0

[(sinxtfxt)k- 1]xk d[F(x)

+ F(-x)]

o{t f:" xk+t d[F(x) + M(x)]},

=

(30)

where (sin xtfxt)k = [1 + O(xt) 2 ]k = 1 + O(xt) for I xt I ~ 1, is used. (30) is o(1 ), by Lemma 11.2.1 [(a) => (b)] [(a) corresponds to (18)]. Since

f lit xk d[F(x) + F(-x)] = flit 0

-1/1

xk dF(x),

we reach the conclusion. REMARK. As we saw from Theorems 11.2.4 and 11.2.5, for an odd integer k > 0, under condition (18) or (25), the existence of j
f(h)

+ f( -h) -

2f(O)

=

o(h)

as h--+- 0.

(31)

Generally, a continuous function satisfying (31) is said to be smooth at t = 0. If in place of (31), a continuous function f(t) satisfies,

f(to +h)+ f(to- h)- 2j(t0 ) = o(h)

as

h

-o,

(32)

then f(t) is said to be smooth at t0 • From the above remark, we see the following result: If f(t) is smooth at t0 and has a right derivative D +f(t0 ) (or a left derivative) at t0 , then the left derivative (or the right derivative) must exist at t0 and then the derivative should exist at t0 • This means that the graph of a smooth function f(t) can never have an angular point, that is, a point where the right and the left derivatives exist but are not equal to each other. Also it is easily seen that if f(t) is differentiable at t0 , then it is smooth there.

11.3.

419

MORE ABOUT SMOOTHNESS AND EXISTENCE OF MOMENTS

11.3. More about Smoothness of Characteristic Functions and Existence of Moments In the preceding section, we have shown that the differentiability of a characteristic function and the existence of the moments of the corresponding distribution function are equivalent when the order concerned is an even number, while when the order is an odd number, the differentiability at zero of a characteristic function relates to the symmetric improper moment. Now we will be interested in the Lipschitz continuity of characteristic functions and the existence of some sort of moments of distribution functions or the behavior of distributions at infinity. We begin with the following simple theorem.

Theorem 11.3.1. If a distribution function F(x) satisfies

for some r

1 - F(x) = O(x-r)

for large

x,

(1)

F( -x) = O(x-r)

for large

x,

(2)

> 0, then the absolute moment Ps exists for every 0

~

s

< r.

Note that s and r are not necessarily integers. Proof.

For a positive integer m, J

zm+l zm

I X I' dF(x)

from (1 ). Since r > s,

J: I

Jzm+l foo dF(x) dF(x) ~ 2<m+I)s zm zm

~

2<m+I)S

~

2<m+I)s 0(2-m') = 0(2-m(r-sl)

L

2-m
xI' dF(x) <

In a similar way we may show

<

oo

oo

and hence for large R.

J=! I x 1• dF(x)

for large R

> 0.

We shall show the following theorems due to Binmore and Stratton [1].

Theorem 11.3.2. Let 0
A necessary and sufficient condition that

1 - Ref(t)

=

o(t"')

as

t-+ 0+

(3)

420

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

is that

1 - F(x)

= o(x-"')

F( -x)

o(x-"'),

=

(4)

for x--+ oo. (ii) A necessary and sufficient condition that 1 - Ref(t)

= O(t"')

as

t--+

0+

(5)

is that 1 - F(x)

=

O(x-"'),

F( -x) = O(x-"')

(6)

for x--+ oo. Condition (5) is the Lipschitz condition of order a of Re f(t).

Proof.

Conditions (4) and (5) are represented by lim sup

Q.. (x) = I,

z-+oo

with I= 0 and 0
for x > 0. In order to show that (4) and (6) are sufficient in (i) and (ii), it is sufficient to prove that lim sup t-"'[1 - Ref(t)]

~

C..l

(8)

1-+o+

with some constant C.. independent of l. Now note that, for any v > 0, sin 2 v ~

because, for 2v 2

f 02v

~ 2 J:v (sin 2 ufu) du,

n/2,

(sin 2 ufu) du ;;;; (4/n)

and, for 2v

(9)

f2v 0

sin u du

= (8/n) sin2 v

;;:;;; 2 sin 2 v,

> n/2, 2

f2v (sin ufu) du ;;;; .2 f"/2 (sin ufu) du, 0

2

which is, from the above, with v 2

f

2V

0

0

2

= n/4,

(sin 2 ufu) du ;;:;;; 2 sin 2 (n/4)

= 1 ;;:;;; sin 2 v.

11.3.

MORE ABOUT SMOOTHNESS AND EXISTENCE OF MOMENTS

421

Using (9), we have 1 - Ref(t) = 2

J:

sin 2 (xtf2) dF(x)

= -2 J~ sin 2 (xtf2) d[1 - F(x) ~

-2

f oo d[1 0

(note that 1 - F(x) + F( -x) = 1 - Ref(t)

~

-2

f~

= -2 J~

+ F( -x)] fxt (sin ufu) du F(x) + F( -.• J]

-

0

2

f v ;;:x dF(v) is nonincreasing) and hence 1 1

u:t (sin ufu) du] d[x-"Q,.(x)] 2

[f: (sin

2

ufu) du] d[(tfy)"Q,.(yft)],

which is, using integration by parts, equal to

Hence lim sup t-"[1 - Ref(t)] ~ 2/ t-+O+

J"" (sin yfyl+") dy. 2

0

The last integral is finite since 0 < a < 2. Hence we get lim sup t-"[1 - Re f(t)]

~

C,.l

t-+o+

J:

with C,. = 2 (sin 2 yfyl+") dy. For the proofs of the necessity of (4) and (6), it is sufficient to show that lim sup Q,.(x) ~ C,.'m, C,.' constant, where m

=

lim sup1_. 0 + t-"[1 - Re f(t)]. But this is immediate from

f

lul;;:x

dF(u) ~ 7x

f

l/X

0

[1 - Ref(t)] dt,

which is the inequality given in Theorem 3.7.3 (16). This completes the proof. If a

~

1, 0 < a < 2, then condition (3) and the condition that

1 - f(t)

= o(l

t I"),

t---+0,

(10)

422

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

are equivalent to each other, and condition (5) and

1 - J(t) = o(l t I"),

(11)

t--+-0,

are equivalent to each other. It is obvious that (10) implies (3). Suppose that (3) is true. Then, from Theorem 3.7.3 (16),

f

lzi>R

Now

dF(x) = o(R-")

J:

Imf(t) =

f

=

R-oo.

as

(12)

sin xt dF(x)

lzl
sin xt dF(x)

= t Jlzl-;; 111 _ 1 x dF(x)

+ o[f

lzl>lll-1

dF(x)]

+ o(l t I"),

by (12). For t > 0,

s:-

1

x dF(x)

= -

s:-

1

x d[1 - F(x)]

= -t- 1 [1 _ F(t- 1 )]

+ 5:-

1

[1 - F(x)] dx,

which is, from ( 12) (noting a -=1= 1),

s:-

1

x dF(x) =

o(t•-

1)

+ o(J:-

1

x-• dx)

= o(t"- 1 ) + o(t"- 1 ) = o(t"- 1 ). For f~1 _ 1 x dF(x), t > 0, a similar argument applies. We thus obtain Imf(t)

= o(l t I")

as

t--+-

0.

With this together with (3), we have (10). The equivalence of (5) and (11) is obtained in the same way. Hence we have the following theorem.

Theorem 11.3.3. Let 0 < a < 2, a -=1= 1. Let f(t) be the characteristic function of a distribution function F(x). Then

11.3. (i)

423

MORE ABOUT SMOOTHNESS AND EXISTENCE OF MOMENTS

1 - f(t)

=

o(l t

1~)

as

t-o

if and only if (4) holds. (ii) f(t) satisfies the Lipschitz condition of order a: 1 - J(t)

= O(l t 1~)

if and only if (6) holds. This theorem, for 0
1.

< 1, was given by Boas [3].

= 1

The case a= 2 becomes meaningless because, if f(t)

+ o(t2), then F(x) reduces to the unit distribution from Corollary 3.7.1.

REMARK 2. In Theorem 11.3.3, the case a = 1 ceases to be true. This is easily seen from Theorem 11.2.4 (k = 1).

We also remark that since 1 - Ref(t) = J~oo (1 -cos xt) dF(x) and the second order difference

LIJ 1 ~(0) = f(t) - 2f(O)

+ f( -t) =

-2 f~"" (1 - cos xt) dF(x)

are the same up to a constant factor -2, Theorem 11.3.2 is also given in termsofj(t)- 2/(0) + f(-t). Now we shall be interested in the existence of moments of the form f:X, $(x) dF(x) for some $(x) ~ 0. However, we are not going to consider the general function $(x), but a very special kind of function such as I x lk (log+ I xI )a, although the generalizations to $(x) are not hard when suitable conditions are imposed. We prove the following theorem.

Theorem 11.3.4. Let f(t) be the characteristic function of a distribution function F(x). Let k be a nonnegative integer. (i)

If

a> 0,

(13)

then _

1- Ref(t)-

a2 t 2

.... ll4£-

-if- 4f +

... + (-1)

k 1 rt2k t2k

+ (2k)!

+0

(

t2k

)

llogl tlla (14)

as t--+ 0, where a. is the vth moment of F(x).

424

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

(ii) If (14) is true for a > 0, then

J~oo x2k(log+j x i)b dF(x) < oo forb< a- 1.

Proof.

(i)

Since 2k is even, from (13 ), a. , 0

~

~

v

2k, exist.

Now 1 - Re f(t) =

~

{[f(O) - f(t)]

1

2

= -

l

tf'(O)

+ [f(O) -

f( -t)]}

t2

+ 2f /"(0) + ... +

t2k-l

(2k - 1)I j'2k-ll(O)

+~ J'2kl(()t)] (2k)! -

~

+

(;:)!

[ -tf'(O)

/"(0)- ... -

( 2t~\)! j'2k-ll(O)

j'2kl(()'t)]

t2

=

+ ~~

t2k-2

t4

2f a 2 - 4f a4 + .. ·+

(-1)k (2k- 2)1

a2k-2

+ (-1 )k+l _!_2 ~ [Joo-oo eiOxx2k dF(x) (2k)! + where 0 < (),

J:

I()' I <

eiO'xx2k dF(x)],

(15)

1. Therefore,

1- Ref(t)-

a t2•

k

L (-1)•+1 -(2v).1 •=l 2-"-

= (-1 )k+l _!_ ~ 2 (2k)!

[Joo-oo (eiOxt -

1)x2k dF(x)

+ J~oo (eiO'xt- 1)x2k dF(x)] =

o[ t 2k J~oo Isin( ~ Oxt) Ix2k dF(x)] + o[ t 2k J~oo Isin(~ ()'xt) Ix 2 dF(x)]. k

(16)

11.3.

MORE ABOUT SMOOTHNESS AND EXISTENCE OF MOMENTS

Now if I xt

I<

425

1, then

Isin(~21xt) I ~ I xt I = ~ llogl

I t lllogl t II a I x-1 lllogl x-1 I Ia

llogl x II a llogl t I Ia

I Ia llogl t I Ia

and if I xt

I ~ 1,

X

then

I~- 1 -~ llogl llogl x I Ia = Isin(_!__oxt) t- I Ia 2

llogl x I Ia . llogl t I Ia

1

Therefore,

t 2k J~oo sin(

~ Oxt) Ix2k dF( x)

I

=

o(t2k foo

x 2k(log+l

-oo

= 0

Xi)a

dF(x)

1 ) llogl t I Ia

( lloglt2kt I Ia ) .

The same is true for the second member of (16). This proves (14). (ii) We first note that

f

u 0

du 2 .+ 1

Juz>+I 0

u2v -2v-! -

du 2 •

• • •

Ju1cos x dx 0

u2•-2

(2v- 2)!

Using this, we have fork

foo

(1- cos tx) dF(.Y:)-

-oo

= ( -1 )k+ 1

f oo

-oo

dF(x)

+ .. ·+ ~

u2

(-1)• -2-!

+

(-1)•+1(1- cosu).

1, a t2•

k

L (-1)•+1-(2v)! •=

2- • -

1

ftx 0

du 2k+l

fuzk+I 0

du 2k

• • •

Ju1cos v dv. 0

From (14), (a> 0), this integral is 0(1 t l2k/llogltlla).

(17)

426

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

The integrand of dF(x) in the right-hand side of (17) can be written as

fix du2k+l f"zk+l

g(tx) =

du 2k

0

0

• • •

f"• (1 0

- cos u2 ) du 2

This is always nonnegative. For, if tx > 0, it is obvious and if tx < 0, it is still obvious since it is the integral of a nonnegative function taken an even number of times. Hence

f

lzl
g(tx) dF(x) = 0(1 t l2k/llogl t

I Ia).

(18)

Now, for I tx I < 1, g(tx)

~

fix du2k+1 f"•k+l [(tx) k+2f(2k + 2)!.

l(2/n) 2

= (2/n) 2

0

0

2

For t > 0, from this and (18),

f:-· x2k+ dF(x) = 0(1ft2llog t Ia). 2

When k = 0, this order relation is still seen to hold if g(u) is taken for 1- cos u. Writing G(x) =

f:

u 2k+ 2 dF(u),

the above means that G(x) = O(x2 llog x 1-a) for large x and for k Now for A> 0,

f:

~

0.

x 2k I log+ x lb dF(x)

=

J:

x- 2 (log x)b dG(x)

= A- 2 (log A)bG(A)- e- 2 G(e)

+2 - b

J: J:

x- 3 (log x)b G(x) dx x- 3 (log x)b-l G(x) dx,

(19)

11.3.

MORE ABOUT SMOOTHNESS AND EXISTENCE OF MOMENTS

427

which is, from (19), equal to O[(log A)-la-b)]

+ o(JAo Since a - b

+ 0(1) + o(f.A

dx ) x(log x)a-b

dx ) x(log x )a-b+l ·

> 1, the integrals converge as A

--+

oo and thus we have

J~ x 2k I log+ x lb dF(x) < oo. For J~oo, we have the same conclusion in just the same way. This completes the proof. The following theorem also holds. We have only to take the expansion up to the term of the (2k 1)th derivative of f(t) in (15) and proceed in a similar way.

+

Theorem 11.3.5.

Let f(t) be the characteristic function of a distribution function and let k be a nonnegative integer. (i)

If

J~oo I X 12k+1(log+l X I)a dF(x) < oo,

a> 0,

(20)

then

as

t--+

(ii)

t4

a t a =if4! 2

1- Ref(t)

+ .. ·+ (-1)k+1

a t2k

(;)!

+0

(

I t l2k+1 ) llogl tlla (21)

0.

If (21) holds for a > 0, then (22)

forb< a- 1. The interesting cases will be when k = 0, 1 in Theorem 11.3.4 and k = 0 in Theorem 11.3.5, that is, the following special cases. (a)

J:

(log+l xI )a dF(x) < oo,

a> 0,

(23)

implies 1 - Re f(t) = 0[1/(logl t !)a]

as

t--+

0.

(24)

428

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Conversely, (24), a > 0, implies

J: for b

(lob+ I xI )b dF(x) < oo

(25)


implies as

t---+

0.

(27)

Conversely, (27), a > 0, implies

J: for b

x 2(log+l xI )b dF(x) < oo

(28)


J: I

(c)

x l(log+l x

II" dF(x) <

(29)

oo

implies

1- Ref(t) = O[l t 1/llogl t I Ia)

as

t---+

0.

(30)

Conversely, (30), a > 0, implies

J: I for b

xI (log+l x

i)b dF(x) < oo

(31)


Some results were obtained in the foregoing St;ction on the relationship between the existence of absolute moments and the corresponding distribution functions. Here we give a few results on the representation of the absolute moments in terms of characteristic functions. The absolute moments are the ordinary moments if the order is an even integer and, roughly speaking, they are represented by the derivatives of characteristic functions at the origin.

11.4.

429

ABSOLUTE MOMENTS

Let f(t) be the characteristic function of a distribution function F(x).

Theorem 11.4.1. In order that sufficient that

f:'oo I xI dF(x) exist, it is necessary and

J~oo {[1 - Rej(t)]ft2} dt

(1)

J~oo I X I dF(x) = J~oo {[1 - Rej(t)]/t 2 } dt.

(2)

exist. In this case

Proof.

The proof is quite simple if we note that n- 1 J~oo [(1 - cos xt)ft 2 ] dt =

I X I·

(3)

This is obvious since

is either 1 or -1 according as x Then

tA I xI dF(x)

= n- 1 =

n- 1

~

0 or x < 0, respectively.

2]

tA dF(x) J~oo

[(1 - cos xt)ft dt

J""

(1 - cos xt) dF(x).

-oo

t- 2 dt

JA-A

The inner integral is nonnegative and converges to 1 - Re f(t) nondecreasingly and the theorem is obvious.

Theorem 11.4.2. If J:'oo I xI dF(x) < oo, then

J: I

xI dF(x) = -n- 1

J::

[Ref'(t)/t] dt.

(4)

(See Renyi [1], pp. 309, 310.) Proof.

We use the identity lim n- 1 A-+-oo B-+oo

JB (sin xtft) dx = A

sgn x

(5)

430

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

(the limit is boundedly convergent), which is obvious. Then

J~oo I XI dF(x)

J:

J~oo XdF(x) A~r:!oo :n:- 1 B-+oo

=

(sin xtft) dt.

Since the inner integral converges boundedly, it becomes, from (5), lim

fB

:n;- 1

A~-oo

t- 1 dt

foo -~

A

x sin xt dF(x)

=

:n:- 1

f-+oo

(Ref'(t)ft) dt.

~-~

B-+oo

In the same way we have the following theorem.

Theorem 11.4.3. A necessary and sufficient condition for the existence of f:'oo I x IP dF(x), 0 < {3 < 2, is that

J:

{[1 - Ref(t)]/1 t II+P} dt

exist and in that case we have

oo f -oo I x IP dF(x)

=

{3(1 - {3) 1 F(2- {3) sin[(1- f3H:n:]

Joo

1 - Rej(t) d ti+P

o

t.

(6)

(The coefficient of the integral on the right-hand side is interpreted to be 2/:n: when {3 = 1.) The proof is carried out in the same way as in the proofs of the foregoing theorems, if the following formula is used:

Proof.

f

oo

1 - cos xt ti+P

o

!!...]

_ F(2 - {3) . [ _ dt - {3(1 - {3) sm (1 {3) 2

P

Ix I ,

0 < {3 < 2. (7)

The proof of (7) is obtained from Section 7.6, Eq. (23); that is,

f. . 0

oo

F(a)

:n:a 2 ,

t«- 1 cos xt dt = - - cos -

I X I«

O
Note that the left-hand integral is boundedly convergent as is seen from the proof of Section 7.6, Eq. (23). Therefore, we may integrate both sides twice with respect to x and J-+oo turns out to be the ordinary Lebesgue integral since the resulting integral is absolutely convergent. Thus

f

oo

o

1 - cos xt d t 3 -«

t

=

F(a) 1 :n:a (1 - a)(2 - a) xac- 2 cos 2'

This is true for x < 0 if I x

1«-2

X> 0.

is substituted for xac- 2 since the left-

11.5.

BOUNDEDNESS OF SPECTRA OF DISTRIBUTION FUNCTIONS

431

hand side is an even function of x. Writing a= 2- {3, we have (7) for 1 < {3 < 2. For 0 < {3 ~ 1, (7) still holds for x > 0, by analytic continuation, since both sides are easily seen to be analytic in 0 < Re {3 < 2. Now we consider the case {3 ~ 2. Define, for a positive integer p, P

Lp(x)

If

J:X, x 2P dF(x) < J~oo

x2lc

= 1- cosx + ,'&1(-1)k (2k)! · oo, then

Lp(xt) dF(x)

1-

=

oo p f -oo cos xt dF(x) + A'&l p

= 1 - Ref(t) If 2p < {3 < 2p

+ lc~t

(

-1 )kt21c

(2k)!

foo -oo

x 21c dF(x)

t2k ( 2k)!

j<2k,(O).

+ 2, then

f0oo [L p (xt)fti+P] dt = foo0

[L (u)ful+P] du I p

X

Ill.

Writing

we have the following theorem.

Theorem 11.4.4. Let 2p < {3 < 2p

J:X, I x IP dF(x) < (2/Cp)

=

+ 2, p being a positive integer. If

oo, then

J~ (1fti+P) [ 1- Ref(t)

J:1

+lett [t2kf(2k)!]J<2k,(O)] dt

x Ill dF(x).

(8)

11.5. Boundedness of the Spectra of Distribution Functions The following theorem is the restatement of Theorem 8.2.2.

Theorem 11.5.1. Let f(t) be the characteristic function of a distribution function F(x).

432 (i)

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

If f(t) is the entire characteristic function and

f(z) =

A >0,

O(eAizl),

(1)

in the complex plane, then

F(x) = {

~

for x >A, for x <-A,

(2)

that is, the spectrum of F(x) is contained in [-A, A]. (ii) Conversely, the characteristic function f(t) of F(x) with (2), is entire and (1) holds. Proof. Here we give a direct and simpler proof of this theorem (see P6lya [1 ]). The proof of (ii) is immediate since

f(z) =

JA eizz dF(x), -A

which coincides withf(t) on the real axis, is obviously regular for every z; that is, is entire and

If(z) I ~ elziA

tA

dF(x) ~

eAizl.

The proof of (i) is carried out in the following way. Suppose that (1) holds for some A> 0. For some constant C,

I f(z) I ~ CeAizl. Let z

(3)

= iy and take the logarithm of both sides. We then see that loglf(iy) I ~log C +A I y

Let y

I·

(4)

> 0. Then using Theorem 9.5.1, log! f(iy) I = log ~

log

J:

J

zl

e-vz

dF(x)

e-vz dF(x)

(5)

Zl

for any x 1 < x 2 • Choose x 2 so that F(x 2 ) > 0 and x 1 so that F(x 2 ) - F(x 1 ) > 0, which is possible since, if x 1 < 0 and I x 1 I is large enough, F(x1 ) can be made as small as we want and log! f(iy)

I ~ log[ e-11z2 =

J::

dF(x)]

-yx2 + log[F(x 2 )

-

F(x 1 )].

11.5.

BOUNDEDNESS OF SPECTRA OF DISTRIBUTION FUNCTIONS

Divide both sides by y and take lim sup11 __..

00 •

433

Then we get

lim sup logJ f(iy) J/y ~ -x2 v-->-oo

since log[F(x 2 ) - F(x 1 )] is finite. On the other hand, from (4) lim sup logJ f(iy) IIY ~A. y-->-oo

Hence A ~ -x2 or x 2 ~ -A. In other words, F(x) In a similar way, loglf(-iy) I ~log C +A I y

= 0 for x <-A.

I·

(6)

Let y > 0. Again we have logJ f( -iy) I = log

f

oo -oo

evz dF(x) ~log

Choose x 2 and x 1 so that F(x 1 ) < 1, F(x 2 )

-

J'"• evz dF(x).

F(x 1 )

Dividing both sides by y and taking lim sup11 __.. before,

00 ,

zt

> 0. Then, as before,

we have, using (6) as

In other words, if F(x 1 ) < 1, then x 1 ~A; that is, F(x) = 1 for x >A. Hence the spectrum of F(x) is contained in [-A, A]. This proves the theorem. The boundedness of the spectrum is also discussed in terms of the Dirichlet integral Daf of the characteristic function defined in the following theorems.

Theorem 11.5.2. If f(t) is the characteristic function of a distribution function F(x), then Daf(t)

= lim n-1 JT [sin a(t- u)f(t- u)]f(u) du, T-->-oo

a> 0,

(7)

-T

exists for every x and Daf(t)

= J:a

eizt

dF(x),

provided that F(x) is standardized: F(x) = l[F(x

(8)

+ 0) + F(x- 0)].

434

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Proof.

n- 1

rT

[sin a(t- u)f(t- u)]j(u) du

n- 1

=

= n- 1 = n-1

rT Ioo Ioo

[sin a(t- u)f(t- u) du

I~oo eiu dF(x)

IT

.

-oo

dF(x) -T [sin a(t- u)f(t- u)] e'z" du

-oo

e' 1z dF(x)

.

IT--l . -T-I (sin avfv) e'zv dv.

(9)

Since n- 1

as T---+

=

n- 1

T-1 I-T-I (sin avfv) cos xv dv

I xi> a, I xi= a, I xi< a,

0, { ---+ t, 1,

boundedly for x, from Lemma 2.6.4 and

T-t (sin avfv) sin xv dv = n- I-T+I (sin avfv) sin xv dv I-T-t -T-1 1

(noting that the integrand is an odd function of v), which converges to zero as T---+ = boundedly for x again by Lemma 2.6.4, (9) converges to

I

ll--0

eilz

-a-HI

dF(x)

+l I

eilz

{a}

dF(x)

+l I

eilz

{-a}

dF(x),

where {a} is a set of a single point a. This is

I::-HI eilz dF(x) + l[F(a + 0) +

l[F( -a

F(a - 0)] eita

+ 0) -

F( -a - 0)] e-ila,

(10)

which is precisely J~ e'"z dF(u). A simple consequence of Theorem 11.5.2 is:

Theorem 11.5.3. For the spectrum of a distribution function to be bounded, it is necessary and sufficient that the characteristic function f(t) satisfy (11) -CXl < t < ex>, Daf(t) = J(t), for some a > 0.

11.5.

BOUNDEDNESS OF SPECTRA OF DISTRIBUTION FUNCTIONS

435

Corollary 11.5.1. If Daf(t) = 1, -oo < t < oo, holds for every a> 0, then F(x) must be the unit distribution function. Conversely, for the unit distribution function F(x), Daf(t) = 1 for every a > 0. This is immediate. We now give the following theorem.

Theorem 11.5.4. Let O(t) be a nonnegative, nondecreasing function fort> 0. If

f~ [O(t)ft 2 ] dt

=

oo

(12)

and the characteristic functionf(t) of a distribution function F(x) satisfies

f(t) = O(e-OWII) for large It

1. then the spectrum of F(x)

(13)

is (-oo, oo).

The condition ( 12) is best possible in the sense of the following theorem.

Theorem 11.5.5. If O(t) is a nonnegative, nondecreasing function for t > 0 such that

f~ [O(t)/t 2 ] dt < oo,

(14)

then there is a distribution function with bounded spectrum, the characteristic function f(t) of which satisfies (13) for large I t I· Proof of Theorem 11.5.4.

q;(t)

Consider =

f(t)(sin at fat),

a> 0,

which obviously belongs to £2(- oo, oo ). The Fourier transform in £2(-oo, oo) is, by Lemma 8.2.1, equal to [(2n) 112 f2a][F(x +a) - F(x- a)]. If F(x) is constant in some interval (a, p), then F(x +a)- F(x- a) is zero in (a a, p - a), where a is supposed to be so small that a< HP- a). Hf(t) satisfies (13), then q;(t) also satisfies (13) for a fixed a. Then, by Theorem 8.3.2, F(x +a)- F(x- a) must be zero throughout (- oo, oo ). This is a contradiction because, if so, then q;(t) = 0 identically and f(t) = 0 except at t such that sin at = 0, t -=F 0. This is not true since f(t) is a characteristic function. This proves the theorem.

+

XI.

436

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

The proof of Theorem 11.5.5 will be given later as the proof of Theorem 13.3.5. Suppose that a distribution function F(x) is 0 for x 0}, so that F(x) = 0 for x 0 for x >a. In a similar way, supposing that F(x) = 1 for x > B for some B, we define b by (16) b = sup{x: F(x) < 1}. In terms of the corresponding random variable X, X ~ a almost surely in the case (15) and X~ b almost surely in the case (16). Let us denote by Sp the spectrum of a distribution function F(x). Then a = inf SF , b = sup SF are determined by the characteristic function when it is an analytic characteristic function, that is

f(z)

=

J:

eizz

dF(x)

converges and is analytic in some strip containing the real axis in its interior and on the real axis it coincides with the characteristic function f(t) (Section 9.6).

Theorem 11.5.6. Let f(t) be an analytic characteristic function. Let SF be the spectrum of the corresponding distribution function F(x). VVe then see that

inf SF= -lim sup[logf(iy)/y],

(17)

1/-'>00

sup SF= lim sup[logf( -iy)fy]. y-'>oo Proof.

Let a

=

(18)

inf SF be finite. Then for any e > 0,

F(a +e)- F(a- 0) > 0. j(iy)

= Joo e-zv dF(x) -oo

=

e-4.V

J:

=

e-"v dF(u

(19)

Joo e-zv dF(x) a-il

+ a).

(20)

11.6.

437

INTEGRABLE CHARACTERISTIC FUNCTIONS

Obviously

J: rvu dF(u +

a}

~

1, and, therefore,

logf(iy)/y for y > 0. On the other hand, from (20),

logf(iy)fy

=

-a+ y- 1 log

J:

~

(21)

-a

ruv dF(u +a)

~-a+ y- 1 log J~ e-"Y dF(u +a)

~ -a+ y-tlogl e-•v J~ dF(u +a) I = -a- e +

y- 1 log[F(a + e)- F(a- 0)].

Using (19), lim sup[logf(iy)/y]

~

(22)

- a - e.

y-+oo

This with (21) shows ( 17) since e is arbitrary. If a = - oo, (22) still holds with any positive constant in place of - a - e. This shows (17). The proof of ( 18) is carried in the similar way.

11.6. Integrable Characteristic Functions Let f(t) be the characteristic function of a distribution function F(x). We study the relationship between the integrability of f(t) and the properties of F(x). If /(t) e £1(- oo, oo), then the distribution function is absolutely continuous and

F(x)

=

J:oo p(u) du,

p(u)

~

0,

(1)

where p( u) is the probability density and belongs to Lr(- oo, oo) for every 1 ~ r < oo and p(u) e (C). (See Section 2.7.) A similar theorem is true for /( t) e L 2 ( - oo, oo ).

Theorem 11.6.1. If the characteristic functionf(t) e L2( -oo, oo), then the distribution function F(x) is absolutely continuous and the density function belongs to L2(- oo, oo) and hence to Lr(- oo, oo) for any 1 ~ r ~ 2.

438

Proof.

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Let/(t) be the Fourier transform ofj(t) in £2( -oo, oo) and write /(t)

= l.i.m.(Zn)- 112 IT f(u) e-itu du. T->-oo

-T

Then

I 0z /(t) dt

= Iz dt l.i.m.(Zn)-112 IT J(u) e-itu d:,;_ T->-oo

0

-T

= (Zn)-11 2 lim I T f(u) du Iz e-'.1" dt T->-oo -T

0

(by the l.i.m. rule, see Section 6.2) and

I oz /(t) dt

= (Zn)- 1' 2 lim IT [(e-iuz- 1)/-iu]f(u) du. T->-oo -T

Therefore, from the inversion formula (Theorem 4.4.1 ),

F(x) - F(O)

= (2n) 1' 2

I:

/(t) dt

if F(x) is standardized. However, this formula shows that F(x) is continuous and actually F(x) is absolutely continuous. Writing p(u) = (2n)- 11'f(u), F(x) - F(O) = p(u) du and this implies that p(u) ~ 0, p(u) E £1(-oo, oo), since ;(u) E £2(-oo, oo), p(u) E £2(-oo, oo). This proves the theorem.

J;

As a matter of fact, it is true that iff( x) E LP(- oo, oo) for some p, 1 ~ p ~ 2, then F(x) is absolutely continuous and the probability density p(u) E L'(-oo, oo) for any r such that 1 ~ r ~ pf(p- 1). We omit the proof of this fact because the LP theory (1 ~ p ~ 2) of the Fourier transform, which we have not presented, is needed. Theorem 11.6.1 ceases to be true, if f(t) E LP(- oo, oo ), p > 2. If p > 2, then there is a characteristic function f( t) E LP(- oo, oo) such that the corresponding distribution function is singular. This will be shown later in Theorem 13.4.2. Next we consider the case in which the characteristic function enjoys stronger integrability conditions.

Theorem 11.6.2. If the characteristic function f(t) satisfies

I: I

t I'P If(t) I dt < oo,

(2)

where p is an integer equal to or greater than 0, then the corresponding

11.7.

439

ANALYTICITY OF DISTRIBUTION FUNCTIONS

distribution function has a bounded continuous (p F
+ 1)th

derivative

(3)

Proof.

This is immediate from the inversion formula

F(x)- F(O) = (2n)- 1 J~oo [(e-itx- 1)/-it]f(t) dt.

(4)

The integral is absolutely convergent since f(t) E £1(- oo, oo) and the case where p = 0 was given before Theorem 11.6.2. For a general p,

1 Joo

2n

-oo

I dxP+I dP+l

(

1 Joo -oo I t

= 2n

e-itx - 1 ) -it

IIJ(t) I dt

Ip I/( t) I dt <

oo,

which shows that differentiation under the integral sign 1s permitted and (3) is obtained. In particular we have a corollary.

Corollary 11.6.1.

If a characteristic function f(t) satisfies j(t) = 0(\ t

~-P-•)

for an integer p ~ 1 and e > 0, then F(x) has a bounded continuous pth derivative with F
11.7. Analyticity of Distribution Functions We here discuss the analyticity of a distribution function, which will be regarded as an extreme case of Theorem 11.6.2.

Theorem 11.7.1. Let f(t) be the characteristic function of a distribution function F(x). If, for some c > 0, f(t) = O(e-•ltl)

as

It I- oo,

(1)

then F(x) is analytic -oo < x < oo. Furthermore, there exists a func-

XI.

440

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

tion F(z), z = x + iy which is analytic in the strip I y real axis it coincides with F(x).

I<

c and on the

Proof. Much of this proof is also based on the inversion formula. Consider

F 11 (z) = (2n)- 1

J:

[(ritz- 1)/-it]f(t) dt.

(2)

Since the right-hand side is differentiable, it is an entire function. Now suppose I y I ~ c'
IJ:

[(ritz- 1)/-it]J(t) dt

I~ A- 1 =

J:

(M/A)

(1

J:

+ etlvl) lf(t) I dt (1

+ etc') ret dt,

where M is a constant such that I f(t) I ~ Me-et (t > 0). The last integral converges to zero as B-oo, A- oo. Hence F,(z) converges as n- oo uniformly in - oo < x < oo, I y I ~ c'. ylence by the Weierstrass theorem (see the proof of Theorem 7.3.2), the limit F(z) ot F,(z) is analytic in the strip I y I < c. On the real axis, F(z) is no more than the given distribution function F(x) by (2). Now a question comes up: What will happen if (1) is replaced by the weaker condition

J(t) = O(e-cltl")

as

It I__. oo,

(3)

0 < a < 1 ? We will then find that F(x) is not always analytic on the real axis. More precisely, we have the following theorem due to Boas and Smithies [1 ].

Theorem 11.7.2. Let e(t) be an even function which has a continuous second derivative for t ~ t 0 and is such that e(t) decreasingly converges to zero, te(t) increasingly diverges to oo, and e- 1• 10 is convex for t ~ t 0 • Then there exists a distribution function F(x) which is not analytic at t = 0 and the characteristic function f(t) of which satisfies j(t) Proof.

=

O(e-ltl•etl)

as

It 1- oo.

Define t ~ t0 , 0 ~ t ~ t0 ,

(4)

11.7.

441

ANALYTICITY OF DISTRIBUTION FUNCTIONS

where the constants A and B are chosen so that L(x) is continuous, convex, and non increasing over (0, oo ). This is obviously possible. Then T(x) = n- 1 lim T-->oo

r

(sin xyfy)L(y) dy,

-oo

<X<

oc,

(5)

0

exists (Theorem 2.1.3 and Lemma 2.6.2 or see Theorem 4.3.2). Integration by parts gives us T(x)

= =

n- 1

IT (sin xyfy) dy- IT L'(y) dy I" (sin xufu) du] . IT [-L'(y)] dy I:r" (sm. vfv) dv. hm lim [L(t)

7'-->oo

n- 1

7'-->oo

0

0

0

0

0

Since L(x) is convex and has a second derivative L"(x) > O, and -L'(y) is nonincreasing and nonnegative, and moreover L'(y)- 0 as y - 0 since L(y)- 0 as y - oo. For each x ~ 0, J~" (sin vfv) dv = 0(1); therefore,

I: (sin vfv) dv I

I~ I L'(y) I dy I

= o{I~ [-L'(y)] dy} = O[A- L(T)] = 0(1).

(6)

Hence T(x) = -n- 1 I 0oo L'(y) dy

IXIJ (sin vfv) dv,

(7)

0

where the outer integral is absolutely convergent. Now the integral

I~ L'(y) dy = x- 1

a[J: (sin vfv) dv] ax

I~ L'(y) sin xy dy,

-:f::-0,

X

is absolutely convergent [as in (6)]. Therefore T'(x) exists and is given by T'(x) = (nx)- 1 I~ L'(x) sin xy dy,

X

-:f::. 0.

= n-•x- 2 I~ L"(y)(l - cos xy) dy ~ 0,

X

(8)

Again integration by parts shows that T'(x) = -n- 1x- 2L'(y)(l - cos xy)

+ n-•x-

2

loo y=O

I~ L"(y)(l - cos :t"y) dy -:f::. 0.

(9)

#2

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Therefore, T(x) is nondecreasing for -oo < x < oo. Now define

F(x) = [T(x)fA]

+ t·

Then F(x) is also nondecreasing over ( -oo, oo). Let x---+ oo in (7). Since J~" (sin vfv) dv---+ (n/2) boundedly (Lemma 2.6.3), T(oo) =-!I~ L'(y) dy = lL(O) = !A.

Hence F( oo) = 1. In a similar way T(-oo) = -Af2. Hence F(-oo) that F(x) is a distribution function. The characteristic function of F(x) is

= 0. This shows

j(t) = I~oo eitz dF(x) = A- 1 I~oo T'(x)eitz dx, which is, from (8), -(An)- 1 I~oo (eitzfx) dx I~ L'(y) sin xy dy = -2(An)- 1

I~ (cos txfx) dx I~ L'(y) sin xy dy

= -2(An)- 1 lim IR (cos txfx) dx Ioo L'(y) sin xy dy

R-+oo

0

0

= -2(An)-1 lim Ioo L'(y) dy IR (cos tx sin xyfx) dx. R-+oo

0

0

The inner integral converges boundedly to 0, for y, when I t I > y, and to 1 when I t I < y (by Lemma 2.6.4 ). Therefore the last term is -2(An)- 1 Ioo L'(y) dy = 2(An)- 1L(I t I). Ill

Hence for our f(t), (4) is satisfied. Now we show that our F(x) is not analytic at x = 0. Consider the Laplace transform

IJ?(z) = n- 1 I~ e-z"L(u) du,

Z =X+ iy.

(10)

Since L(u)---+ 0 as u---+ oo, the Laplace integral exists for Re z > 0.

11.7.

443

ANALYTICITY OF DISTRIBUTION FUNCTIONS

For Re z

=

=

x < 0, y

0,

IRr~:u£(u) du ~ IR e-xu-ue

to

to

as R-oo, since e(u)- 0, -x > 0. Hence the Laplace transform «p(z) has the convergence abscissa x = 0. Since L(y) ~ 0, «p(z) is singular at z = 0 by Theorem 8.4.1. From this fact we can show that the origin is the singular point of F(x). Suppose that this were false and F(x) were analytic at x = 0. Then -r(x) and consequently -r'(x) would be analytic at x = 0, that is, there would be a function x(z) analytic in I z I < TJ for some TJ > 0, and -r'(x) = x(x) On -TJ <X < TJ· Now because of the second mean value theorem, for real y,

I:' e-iv"L(u) du

= L(R)

I~ cosyu du + iL(R) I~ sinyu du

= o(l)

(R<~ 1 .~ 2
as R'- oo, R-oo. Therefore the integral J;oo e-ivu L(u) du exists. Again, using the second mean value theorem, we see that

I:oo e-iV"L(u) du - I:oo e-
I:

e-ivu(l - e-x")L(u) du

+ O[L(R)],

X> 0,

where R is an arbitrary positive number and 0 is uniform for x > 0, -oo < y < .oo. Hence letting x - 0+, the first term of the last member converges to zero uniformly for -oo < y < oo. Thus, «p(x + iy) converges to «p(iy)

=

:n;- 1

I

-+00

0

e-iv"L(u) du

uniformly for y. This shows that «p(z) is continuous in 0 Rez ~ 0 (for any 0 < TJ' < TJ). Also we see that -r'(x) in (8) can be written -r'(x) =

I

-+00

:n;- 1

0

cos xu L(u) du.

(11) ~

I z I ~ TJ

1,

(12)

444

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Now consider

tp(z)

=
This is analytic in 0 ~ lzl ~ 'f/ 1 , Re z > 0, and continuous in 0 ~ lzl ~ 'f/ 1 , Rez ~ 0. From (11) and (12), tp(iy) = ( -i/n) J;""' sin yu L(u) du is purely imaginary. Define tp 1 (z) = VJ( -z) in I z I ~ 'f/ Re z ~ 0. Then tp 1 (z) is analytic in I z I ~ 'fJ', Re z < 0 and I y I ~ 'fJ', x = 0, tp 1 (iy) = tp(iy). Hence tp 1 and 'Pare analytic continuations of each other and tp(z) becomes analytic in I z I ~ "'· In particular, tp(z) is analytic at z = 0. Since x( -iz) is analytic at z = 0,
,

A more precise result than that of Theorem 11.6.2 was shown by Boas and Smithies [1 ]. Let 'fJ(t) be a function approaching 0 at = more slowly than e-cltl. They showed that if E is any closed set on the real axis, then there is a distribution function whose characteristic function has the same order of approaching 0 as 'fJ(t) and E is the set of singular points of the distribution function. The set E may be the whole real axis or the empty set. We state their results (without proof) in the following theorems.

Theorem 11.7.3. Let e(t) be a function for t > 0 such that e(t) decreases to zero as t-+ = and te(t) increases to =as t-+ =· Then there exists a distribution function F(x) such that its characteristic function f(t) satisfies • lim sup I f(t) I exp -21tl e(ltl) > 0,

(13)

ltl->-oo

lim sup I f(t) I exp(1 - e) ltl e(itl) < =

(14)

ltl->-oo

for every e > 0, and F(x) is analytic on ( -=, =).

Theorem 11.7.4. Let E be any closed set on the real axis. If e(t) satisfies the conditions in Theorem 11.7 .3, then there exists a distribution function F(x) such that its characteristic function f(t) satisfies (13) and (14), and E is the set of the singular points of F(x).

11.8.

445

MEAN CONCENTRATION FUNCTION

11.8. Mean Concentration Function of a Distribution Function Let F(x) be a distribution function. Levy ([3], p. 44) introduced the concentration function Q(h) =

max [F(x -oo<x
+ h + 0)- F(x- h- 0)],

(1)

for h > 0. Obviously Q(h) is a nondecreasing function. This relates to the smoothness properties of a distribution function. For instance, limh-+o+ Q(h) = 0 means that the distribution function is continuous. We investigate the relationship between the behavior of Q(h) at h = 0 and that of the characteristic function f(t) at ±oo. For handling the Fourier analysis, the following function C(h), which is an average of Q(h) in a certain sense, will be more convenient:

C(h) = (2h)- 1

J:

[F(x +h)- F(x- h))2 dx

(2)

for h > 0. This is called the mean concentration function of F(x) (Kawata [4, 5]) or of the corresponding random variable with distribution F(x). Actually the integral (2) exists, because

J:

[F(x +h)- F(x- h))2 dx

~

J:

[F(x +h)- F(x- h)] dx

= .~i~rA [F(x +h)- F(x- h)] dx = lim

.A-+oo

[J

A+h A-h

f-A+h

F(u) du -

-.A-h

F(u) du

]

= 2h.

(3)

This also implies

0

~

C(h)

~

(4)

1.

The relation between C(h) and Q(h) is given by

!Q2(h/2) We have

C(h) ~ Q(h)(2h)- 1

J:

~

C(h)

~

(5)

Q(h).

[F(x +h)- F(x- h)] dx

= Q(h)

446

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

from (3), while for any

~

and any e > 0 such that e < h/2,

+ +

h) - F(x- h)] 2 dx C(h) ~ (2h)-1 Ie+
+

from which it follows that as e - 0+, C(h) ~ l[F(~

+ (h/2) + 0)- F(~- (h/2)- 0))2.

Taking the maximum, C(h) ~ lQ 2(h/2). From (5), it follows that C(h)- 0 as h - 0+ if and only if Q(h)- 0 ash- 0+. Therefore, we have C(h)- 0 ash- 0+ if and only if F(x) is continuous on - oo < x < oo. Let f(t) be the characteristic function of F(x) and let h > 0. Then (sinhtft)f(t) e L 2(-oo, oo). We see from Lemma 8.2.1 that the Fourier transform of this in L 2(-oo, oo) is (n/2) 112 [F(x +h)- F(x- h)]. Hence by the Parseval relation, we have, noting that I f( t) 12 is an even function, C(h) = n-1 I : (sin 2 htfht 2 ) I f(t) 12 dt =

Zn-1 I~ (sin 2 htfht2 ) I f(t) 12 dt.

(6)

Here we remark that if X is a random variable with characteristic function f(t) and X' is a random variable independent of X and having the same characteristic functionf(t) (consequently, having the same distribution), then the characteristic function of X- X' is given by I f(t) 12, since -X' has the characteristic function f( -t) = f(t). The distribution function of -X' is 1 - F( -x + 0) and hence, from Theorem 3.10.1, the distribution function corresponding to lf(t) 12 or X- X' is (7) G(x) = F(x) • [1 - F( -x + 0)]. From Theorem 9.3.1, C(h) can be written C(h) = (2h)- 1

I:

[G(u) - G( -u)] du,

(8)

from which it is obvious that C(h) is a nondecreasing function of h > 0. We summarize what we have obtained in the following theorem.

11.8.

447

MEAN CONCENTRATION FUNCTION

Theorem 11.8.1. The mean concentration function C(h) of a distribution function F(x), whose characteristic function isf(t), has the properties: (i) C(h) is expressed by (6) and (8), (ii) 0 ~ C(h) ~ 1, (iii) C(h) is a nondecreasing function of h > 0, and (iv) C(h) ---+ 0 as h---+ 0 if and only if F(x) is continuous for -oo

<X<

oo.

We shall show the following theorem.

Theorem 11.8.2. For the mean concentration function C(h) for a characteristic function f(t), lim C(h) =lim T- 1 h-+o+

T-+oo

r

I

f(t)

12

(9)

dt

0

and (10)

lim C(h) = 1. h-+oo

Proof. (10) is obvious from C(h) -+I /(0) 12 by the property of the Fejer integral. (9) is a particular case of Wiener's formula (Theorem 5.5.1) with K(x) = sin 2 xfnx 2 • A direct proof is possible based on the assumption that J(t) is the characteristic function and C(h) has (2) as its defining form. C(h) = (2h)- 1 =

J~oo

J~oo dF(u)

[F(x +h) - F(x- h)] dx (2h)- 1

J:::

J:::

dF(u)

F[(x +h)- F(x- h)] dx.

(11)

The function Zh- 1

f

u+h

u-h

[F(x + h) - F(x - h)] dx

is uniformly bounded by 1 and converges for each u to F(u + 0) - F( u - 0) as h ---+ 0+ since (2h)- 1

!Hh F(x +h) dx = f u-h

(2h)- 1

(Zh)- 1

!Hh F(x- h) dx = f u-h

(2h)- 1

and

ash---+ 0.

fu+2h u

F(v) dv---+ F(u + 0)

fuu- h F(v) dv---+ F(u- 0) 2

448

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Therefore, by the bounded convergence theorem, lim C(h) = h-+0+

J'"'

+ 0)- F(u- 0)] dF(u).

[F(u

-oo

The right-hand side is obviously L. Pn 2 , {Pn} being the sequence of the saltus of F(x), which is by Section 3.8,

IT

lim (2T)- 1 T-+oo

I

f(t)

12

dt =lim T- 1

-T

T-+oo

IT

I

f(t)

12

dt.

0

This proves (9). (9) gives us a relation between the behavior of f(t) at infinity and that of C(h) at 0. Another result in this category is the following theorem. ~a< 1. Letf(t) and C(h) be the characteristic function and the corresponding mean concentration function, respectively. Then the condition

Theorem 11.8.3. Let 0

T- 1

I:

I

f(t)

12

for

dt = O(T-«)

T--+ oo

(12)

is equivalent to C(h) = O(h«)

as h--+ 0+.

(13)

We give a lemma.

Lemma 11.8.1. Let 0 < a such that T- 1

then

I: I

~

1. If ~t(x) is a locally integrable function

q;(x) I dx = O(T-"),

I; [l~t(x) lfx

2]

dx = O(T-cx-t),

T--+ oo,

T--+ oo.

The case a= 0 was shown in Lemma 5.5.1. The proof is similar:

I; [I q;(x) l/x

2]

dx

= x- 2

I: I

q;(u) I dul;

+ 2 I; x-

3

dx

I: I

q;(u) I du

(using integration by parts)

~

T- 2

I: I tp(u) I du + 2 I; 0(1fx«+2) dx

= 0(1/Tcx+~).

11.8.

449

MEAN CONCENTRATION FUNCTION

Proof of Theorem 11.8.3.

C(h)

Suppose that (12) holds. From (6 ),

(2/n) J~ (sin 2 htfht 2 ) I f(t)

=

= (2/n)

f

lih

0

+ (2/n) JCX>

1/h

12

> 0)

(h

dt

= 11 + l2,

(14)

say. Using Lemma 11.8.1, 12 ~ (2/n) JCX> [lf(t) l2 /ht 2 ] dt 1/h

11 ~ (2/n)

f

llh

0

=

(sin 2 htfht 2 ) lf(t)

O(h"+Ifh) 12

=

dt ~ (2hfn)

O(h"),

JIM lf(t)

12

0

dt

=

O(h").

These with (14) show (13). Conversely suppose that (13) holds. Then we have

J:

I

f(t)

12

dt = 4x 2

~ n 2x 2

J:

J:

I

(tf2x)[J(t)ft]

12

sin 2 (tf2x) lf(t)ft 12 dt

~ in 2x J~ [sin 2 (tf2x)/(t2/2x)] I f(t) ~

dt

12

dt

!n3x C(1f2x).

(15)

Now log 2

JT lf(t) 12 dt = JT lf(t) 12 dt Jt 0

0

~ J~ x- 1 dx

x-1 dx

1/2

J: lf(t)

12

dt,

(16)

which is, from (15), equal to or less than

!na f~ C(2x)- 1 dx =

o(f~ x-" dx) =

O(T1 -").

(17)

Hence (12) holds. This completes the proof. We now show that C(h) cannot be of order lower than h" (a> 1) when h is small. Actually, we may show that there is an h0 such that, for 0 ~ h < h0 , (18) C(h) ~ Kh, where K = K(h 0 ) is a constant.

450

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

This is shown in the following way. C(h) ;;;:;; (2/n) where

~

J:

(sin 2 htfht 2 )

I/(~) 12 dt,

is chosen so that If(t) \2 > l for 0

~

t < ~ < 1. Then

if h < n/2. This shows (18). For the case a = 1 in Theorem 11.8.3, we have the following result.

Theorem 11.8.4. If T- 1

s:

I

/(t)

12

dt

= O(T- 1 )

as

T-oo,

(19)

then C(h) = O(h)

as

h-0+.

(20)

C(h) = O(h)

as h-0+,

(21)

= O(T-1 log T).

(22)

If

then T- 1 (

I

f(t)

12

dt

Proof. The proof of the first part is carried out in just the same way as in that of Theorem 11.8.3. No change is necessary. For the second part, the proof is also similar. Suppose (21). We have (15). In place of (16), we have, forT> 2,

log 2

s:

lf(t)

12

dt

~

r s:

~n

x- 1 dx

r

lf(t)

C(1/2x) dx

=

12

dt

o(r

x- 1 dx) = O(log T),

which shows (22). We conclude this section with the following remarks. REMARK

1.

If f(t) = 0(\ t 1-112)

(23)

11.8.

451

MEAN CONCENTRATION FUNCTION

for large I t

I, then as h---+- 0+;

C(h) = O(h log h-1) that is,

as h -

J: o+,

[F(x

+ h) -

(24)

F(x))2 dx = O(h2 log h- 1 ).

(25)

F(x)] = O[(h log h-1 )11 2].

(26)

and from (5), max[F(x z

+ h) -

If (23) is satisfied, T- 1 J~ I f(t) 12 dt = O(T- 1 log T). The proof is carried out in a way similar to the proof of Theorem 11.8.3. From / 1 in (14), we have: /1

=

O(h)

+ (2hfn) f

1tli I

1

f(t)

12

=

dt

O(h log h- 1).

REMARK 2. A distribution function is a unit distribution if and only if C(h) is identically 1 for h > 0.

The "if part" is obvious. If C(h) = 1, then (2/n) J~ (sin 2 htfht2 ) I f(t)

12

dt = 1,

from which, (2/n) J~ [1 -

I

f(t)

12 ]

(sin 2 htfht2 ) dt = 0.

This implies that I f(t) I = 1 since I f(t) Corollary 3.6.2, our result follows. REMARK

12 ;;=:;;;

1 always holds. Then from

3. The condition (13) that

C(h) = O(h"')

as h

-o+

is written

J~oo I F(x +h)- F(x) 12 dx = O(hl+"'),

(27)

that is, the distribution function F(x) satisfies the integrated Lipschitz condition of order 1 +a, or F(x) e Lip(1 +a; 2). Theorem 11.8.3 says that, if 0 ;;=:;;;a < 1, then F(x) satisfies the integrated Lipschitz condition

452

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

of order 1 + a, if and only if

T- 1

I:

I

/(t)

12

dt =

O(T-~)

r-

as

00.

We also remark that (27) implies max[F(x z

REMARK

4.

+ h) -

F(x)]

h > 0.

= O(h~ 12 ),

(28)

For the Fourier-Stieltjes transforms,

f(t)

=

J~oo

eizt

dF(x),

F(x) EM,

we may consider the mean concentration function C(h) with the same definition, that is, C(h) = n- 1 J~oo (sin 2 htfht 2 ) I f(t) = (2h)- 1

12

dt

J~oo I F(x +h)- F(x- h) 12 dx,

and have similar results. For instance, Theorem 11.8.1 holds, replacing (ii) by 0 ~ C(h) ~ F(oo)- F(-oo). Theorem 11.8.2 also holds with limA-+o+ C(h) = F( oo) - F(- oo ).

11.9. Some Properties of Analytic Characteristic Functions Let f(t) be the characteristic function of a distribution function F(x). We have shown the following basic results (Theorem 9.5.1) on analytic characteristic functions in Section 9.5. If /(t) is analytic at t = 0, that is, there is an analytic function f(z) in I z I < «5 in the complex plane, which is identical with/(t) on I t I < «5 on the real axis, then there is a function /(z) analytic in some strip -7} 2 < lm z < 7} 1 , 7} 1 , 7J 2 > 0 [the analytic continuation of f(z) in I z I < «5]./(z) is identical with the givenf(t) on -oo < t < oo and/(z) is given by J:'oo eizz dF(x) in -7} 2 < lm z < rJ 1 •

The largest strip -7} 2 < Im z < 7} 1 was called the strip of analyticity ofj(t). We agree to call a characteristic functionf(t) orf(z) with the above property an analytic characteristic function.

11.9.

453

ANALYTIC CHARACTERISTIC FUNCTIONS

In this section we shall give further basic properties of an analytic characteristic function.

Theorem 11.9.1. Let f(z) be the analytic characteristic function in a strip -1] 2 < Im z < 1] 1 , 1] 1 , 1] 2 > 0. Then max_oo
I. f(z) I along .

= 0 for any -1] 2 < T < 1] 1 , that is, the maximum of any horizontal line in the strip is attained on the imag-

mary ax1s.

The proof is quite simple from

I/(z) I = IJ~oo eizx dF(x) L=l+iT ~ f~oo

e-TX

dF(x) =/(iT).

(1)

(1) implies that ifj(i-r) = 0 for some -r, thenf(z) = 0 for every t +iT, which reducesf(z) to 0 in the strip. This is a contradiction sincef(O) = 1. Hence we obtain the following theorem.

Theorem 11.9.2. An analytic characteristic function has no zeros on the segment of the imaginary axis inside its strip of analyticity. To determine whether the distribution function has an analytic characteristic function, the following theorem is as directly useful as Theorem 9.5.2 is, but is a consequence of Theorem 9.5.2.

Theorem 11.9.3. The characteristic function f(t) of a distribution function F(x) is an analytic characteristic function if and only if there is a positive constant a such that, for large x > 0,

1 - F(x)

+ F( -x) =

o(e-rx)

(2)

for all positive r < a. In this case, the strip of analyticity contains the strip I Im z I < a.

Proof. I Im z

J:

Necessity. From Theorem 9.5.2, if f(z) is analytic in then, for any 0 < r
I
J~ e'" dF(u) ~ e'x[1 - F(x)],

454

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

which must converge to zero. Similarly we may show that F( -x)e•z-+ 0 as x-+ oo. Sufficiency.

From (2) we have F( -x)

1 - F(x) = o(e-•z),

= o(e-'z)

as

x-+ oo.

(3)

For any 0 < r' < r,

f

k-o

f

Jk+l er'z dF(x) ~ er'Ck+ll Joo dF(x) k k-o k

= e•'

00

L

e-Cr-r'lk e•k[1 - F(k)]

k=O 00

~

Ce•'

L

e-(r-r')k

[from (3)],

fr-o

which is obviously convergent, where C is a constant. This shows that J;;' e•'z dF(x) < oo. In a similar way, we have J~oo e•'lzl dF(x) < oo. Then from Theorem 9.5.2, f(z) is analytic at least in I lm z I < a. This shows the sufficiency as well as the final statement. Note that in the proof of sufficiency we have used 1 - F(x) = O(e-•z), F( -x) = O(e-•z), for any 0 < r < a, but this is equivalent to (3) for any 0 < r
j(i-r) = where -rJ 2 < like,

T

J: e-•z dF(x),

(4)

< 1J 1 • f(i-r) can be differentiated as many times as we

and J:O -r 2e-Tz dF(x) ~ 0. If d2j(i-r)/d-r 2 = 0, then, obviously, F(x) is a unit distribution, that is f(z) = 1. Hence f(i-r) is a strictly convex Junction of T in -rJ 2 < T < 1J 1 unless f(t) = 1. Now we show the following theorem due to Dugue [2].

11.9.

455

ANALYTIC CHARACTERISTIC FUNCTIONS

Theorem 11.9.4. For an analytic characteristic function f(z), - 'f/ 2 < Im z < 'f/ 1 , 'f/ 1 , 'f/ 2 > 0, the function log f(i-r) is a convex function of Tin -'f/2 < T < 'f/1· J>roof.

~rite

logf(i-r)

= g(-r),

(5)

Since /(iT) has no zero on -TJ 2 < T < 'f/ 1 , from Theorem 11.8.2, logf(iT) is defined uniquely if any branch of analyticity is taken. In other words, in some domain D which contains -TJ 2 < T < 'f/ 1 , logf(i-r) can be thought of as a single valued function. In D,

f(z)

=

eg<-izl

=

exp[A(t, -r)

+ iB(t, -r)],

z = t +iT,

where A(t, -r) = Reg(-iz), B(t, -r) = Reg(-iz), and lf(z) I= eA
[8A(t, -r)f8t]lt-o = 0,

(6)

From the Cauchy-Riemann equation (or the Laplace equation),

Therefore from (6),

That is, A(O, -r) = logf(iT) is a convex function of T. Suppose that tp(z) is analytic in a strip -a < Im z < {3, a > 0, {3 > 0. If (7) 1 tp(t + iT) 1 ~ tp(iT) is satisfied, then tp(x) is called a ridge Junction (Linnik [1]). Then any analytic characteristic function is a ridge function and this property will play an important role in the theory of analytic characteristic functions, particularly in the theory of infinitely divisible characteristic functions, although we are not attempting to go into details in this book. ~e remark that a ridge function is not necessarily an analytic characteristic function. (For such examples, see Linnik [1 ], p. 56.) For more properties of analytic characteristic functions the reader may refer to Lukacs [1, 3].

456

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

11.1 0. Characteristic Functions Analytic in the Half-Plane Suppose that the distribution function F(x) is zero for x < 0. Then its characteristic function f(t) becomes

f(t) = J~

eitx

dF(x).

If we consider the integral g(z)

= J~ eizx dF(x),

z

=

t

+iT,

then this is analytic in the upper half-planeT > 0 andf(t) is the boundary function of g(z) on the real axis, that is,J(t) = g(t), -oo < t < oo. We are now concerned with the converse of this. First we show the following theorem which was given by Marcinkiewicz [1 ].

Theorem 11.10.1. In order that the characteristic function f(t) of a distribution function F(x) be the boundary function, on I t I < a, a > 0, of a function which is analytic in 0 < T < R, I t I < a, it is necessary and sufficient that

for all 0

Proof.

~

r

roo

e-rx

dF(x)

<

(1)

oo

< R.

Sufficiency.

Suppose that ( 1) holds. Define

z = t

+iT.

This is analytic in T < R. This is seen in the following way. Writing iz = s, L(s) = - J;:" e-•x dF( -x) = f;:" e-•x d[1 - F( -x)] is the Laplace-Stieltjes transform and from (1) it converges on 0 > Res> -R. Hence L(s) converges in the right half-plane Res> -R. Therefore, transforming from s to z, f 1 (z) is analytic in T < R. As we saw in Section 9.5,

is analytic in T > 0. Therefore, 'P(z) = f 1 (z) + f 2(z) is analytic in 0 < T < R. It is obvious that 'P(z) is continuous on - oo < t < oo and

11.1 0.

FUNCTIONS ANALYTIC IN THE HALF-PLANE

457

f(t) = r(t). Thus we have shown that if (1) holds, thenf(t) is the boundary function of a function q;(z) analytic in 0 < -r < R, -oo < t < oo. Necessity. Suppose that r(z) is analytic in D: 0 < -r < R, It I
J:

Then f 2 (z) = eizx dF(x) is analytic in T > 0. Hence q;(z) - / 2 (z) is analytic in D. On the other hand, if we consider / 1 (z) = f~oo eizx dF(x), then it is analytic in -r < 0. On I t I < a, q;(t) - / 2 (t) = / 1 (t) and botli q;(z) - / 2 (z) and / 1 (z) are continuous. Hence they are analytic continuations of each other (see Section 1.14 ). In particular, in I t I < a, I -r I < R, / 1 ( z) is analytic. Since cf1 (t), c = F- 1 (0) [if F(O) = 0, there is nothing to show], is thought of as a characteristic function, from Theorem 9.5.1 (or the fact stated in the beginning of Section 11.9), cf1 (t) is analytic in the strip I T I < R. Hence, from Theorem 9.5.2, f~oo erlxl dF(x) < oo for all 0 < r < R. This completes the proof. REMARK.

Condition ( 1) is equivalent to

F(-x} for all 0


=

O(e-rlxl)

as

x-oo

(2)

R.

This is easily seen as in the proof of Theorem 11.9.3.

Theorem 11.10.2. The characteristic function of a distribution function F(x) is the boundary function of an analytic function in the upper half-plane if and only if lim -log F(-x)fx

x-+oo

= oo.

(3)

Proof. From Theorem 11.10.1 and the above remark, the necessary and sufficient condition is that (2) hold for all 0 ;£ r < oo. It is equivalent to the condition that

log F( -x)fx ;£ (log Afx) - r

XI.

458

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

for x ~ X,, for some constants X, and A, which is, in turn, equivalent to (3 ). If an analytic function in the upper half-plane is required to be bounded, (3) turns out to be the condition that F( -x) = 0 for x < 0, or we have the following result.

Theorem 11.10.3. The characteristic function of a distribution function F(x) is the boundary function of a function analytic and bounded in the upper half-plane if and only if F(x) = 0 for x < 0.

J:

Proof. The "if" part is obvious, since the analytic function eizx dF(x), z = t + i-c, is analytic and bounded by 1 in "r > 0 because J: e-1x dF(x) ~ dF(x) ~ 1. We shall prove the "only if" part. As in the proof of Theorem 11.10.1, we write f(t) = / 1 (t) + / 2 (t), where f(t) is the given characteristic function and / 2 (t) = J: e>xt dF(x), f 1 (t) = J~oo eixt dF(x). Then, if q;(z) is the function analytic and bounded in 'f > 0 whose boundary function is f(t), q;(z)- / 2 (z) is analytic and bounded for 'f > 0 andf1(z) is analytic and bounded for 'f < 0. On the real axis, they are identical. Hence by analytical continuation there is an entire function VJ(z) which is rp(z)-/2 (z) in 'f ~ 0, and isf1(z) in 'f ~ 0. Both rp(z)- / 2 (z) and/1(z) are bounded in their respective half-planes; VJ(z) is bounded and analytic in the whole planc;. Therefore, by a well-known theorem of Liouville in complex function theory, it must be a constant in the whole plane. Hence / 1(z) is constant in 'f ~ 0. In particular, J~oo eixt dF(x) = constant, which implies that F(x) = constant for x ~ 0, by the uniqueness property of the Fourier-Stieltjes transform (Theorem 3.5.3). The constant must be zero, because F(- oo) = 0. This completes the proof.

J:

We now show that, in this theorem, the condition that the analytic function be bounded in the upper half-plane is equivalent to the condition that the analytic function be represented by the Poisson integral of the boundary function. More precisely we have:

Theorem 11.10.4. If a characteristic function is the boundary function of a function f(z), analytic and bounded in the upper half-plane, then f(z) is represented by the Poisson integral ofj(t) in the upper half-plane: f(z) =

for

'f

> 0.

3t- 1

J:J(u){"r/[(t- u) 2

+

"r 2 ]}

du,

Z=t+i'f

(4)

11.10.

459

FUNCTIONS ANALYTIC IN THE HALF-PLANE

Conversely, if an analytic function f( z) in -r > 0 is the Poisson integral (4) of a characteristic functionf(t), thenf(z) is bounded in T > 0 andf(t) is the boundary function of f(z). This is, in view of Theorem 11.10.3, equivalent to:

Theorem 11.10.5. If the distribution function F(x) is zero for x ~ 0, then the characteristic function f(t) of F(x) is the boundary function of an analytic function in the upper half-plane, which is represented by the Poisson integral of f(t). Conversely, if the characteristic function f(t) is the boundary function of an analytic function in the upper half-plane, which is represented by the Poisson integral of f(t), then F(x) = 0 for x < 0. It is sufficient to prove Theorem 11.10.5.

The converse part of the theorem is simple to show. If f(z), analytic in the upper half-plane, is written by (4 ), then

Proof.

lf(z) I~ n- 1 J~oo ~ n- 1

r:

lf(u)

I {-r/[(t- u) 2 + -r2]} du

{-r/[(t- u) 2 + -r 2]} du = 1,

since I f(t) I ~ 1. Hence f(z) becomes bounded in T > 0 and the conclusion follows from Theorem 11.10.3. Before proving the first part of the theorem, we note the following relation, which will be useful on some other occasions too: n-1

J:

{-r/[(t _ u)2

+ -r2]} eiuz du =

eitz-tlzl,

T

> 0.

(5)

This was substantially obtained in Section 4.8, Eq. (16). But for convenience we give a direct proof:

(2n)-1i2 f~oo

eitx e-•lxl e-iux

du

= (2n)-1i2 [f~ ei
eW-u)z e•X

dx]

= (Zn)- 112 {-[i(t- u)- -r]- 1 - [-i(t- u)- -r]- 1} =

(2/n)1 12 {-r/[(t- u)2

+ -r 2]}.

Hence by the inversion of Fourier transforms, we have formula '(5).

XI.

460

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

The proof of the first part of the theorem is also simple. Compute the Poisson integral of f(t): P(z)

= n- 1 = n- 1

J: {

-rf(u)f[(t - u) 2

J: {

-r/[(t - u) 2

[F(x) = 0 for x = n- 1 J~ dF(x)

~

J:

+ -r

+ -r

2 )}

2 ]}

du

du

J~ eiux dF(x)

0, from the assumption]

{-r/[(t- u) 2

+ -r

2 ]} eiux

du,

(6)

which is, by (5)

J~ eitz e-•z dF(x) = J~ eizz dF(x). Hence P(z) becomes analytic in -r > 0, and from (6) and the property of the Poisson integral (Theorem 4.8.4 ), P(z) converges to f(t) as -r - 0+. This completes the proof.

11.11. Entire Characteristic Functions I We now study the entire characteristic function that is the characteristic function analytic in the entire complex plane. First we give the condition which is necessary and sufficient for a characteristic function to be an entire characteristic function.

Theorem 11.11.1. The characteristic function of a distribution function F(x) is entire if and only if lim -log[1- F(x)]fx =lim -log F( -x)fx z-+oo

= oo.

(1)

z-+oo

Proof. This is immediate from Theorem 11.9.3. If the characteristic function is entire, then 1 - F(x) = O(e-rz), F( -x) = O(e-rz) for all r > 0. [This is equivalent to Section 11.9, Eq. (2), as was remarked after the proof of Theorem 11.9.3.] Then, for any r > 0,

-log[1 - F(x)]fx > r

and

-log F( -x)fx > r

if x > Xr for some Xr > 0, which is the same as (1 ). The converse argument also applies.

11.11.

ENTIRE CHARACTERISTIC FUNCTIONS

l

461

Before proceeding, we give some definitions on general entire functions. Let IP(z) be an entire function. For every r > 0, IIP(z) 1. I z I ~ r, has a maximum on the circle I z I = r, by the maximum modulus principle. This maximum is denoted by M(r, IP) =max IIP(z) I·

(2)

!Z!Sr

The order

e of the

entire function IP(z) is defined by

e=

lim sup {[log log M(r, IP)]/log r }.

(3)

r->-oo

If e > 0 and is finite, then the type -r of IP(z) is defined by

-r = lim sup{[log M(r, IP)]/rQ}.

(4)

r->-oo

q1(z) is said to be of maximal, intermediate, or minimal type according as, -r = + oo, 0 < -r < oo, or -r = 0. (3) is the same as saying that, for any e > 0, M(r, IP)

< exp(re+•)

(5)

is true for r ~ r0 for some r0 > 0, and at the same time that for any e < 0 (5) does not hold. (5) is expressed in the following way. For any e > 0, M(r, IP) < exp[(y

+ e)rt>]

(6)

for r ~ r0 for some r0 > 0, and for any e < 0, (6) does not hold. IP(z) is said to be of exponential type; it is either of order 1 and of finite type (that is, not maximal type), or of order less than one. We are going to discuss the relationship between the order or type of an analytic characteristic function and the asymptotic behavior of the corresponding distribution function at ±oo. Let f(z) be an entire characteristic function. Then from Theorem 11.8.1, M(r,J) =max 1/(z) I= max[f(ir),f(-ir)]. lzi:Or

Theorem 11.5.1 can be stated in the following form:

(7)

462

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

An entire characteristic function is of order 1 and of type less than A if and only if the spectrum of the corresponding distribution function is contained in the open interval (-A, A). In order to see this, note that the spectrum is a closed set. This statement implies the following theorem.

Theorem 11.11.2. An entire characteristic function which is not identically 1 is not of order less than 1 (see Levy [4]). We shall give the necessary and sufficient conditions for the characteristic function to be the entire function of specified order or type. In the most precise form, the results were obtained by Ramachandran [2]. These results will be given in the following section, and here we make preparations for them. To begin with, we shall prove the following Lemma.

Lemma 11.11.1. The integral I(z) =

I:

exp(izx - kxH«) dx,

a> 0, k > 0,

(8)

defines an entire function of order 1 + (1/a) and of type (cfk) 11'", where

c = a'"/(1

+ a)l+'".

Proof. Since I(z) is an entire characteristic function up to a constant factor, M(r, I) =

I:

exp(rx - kxl+'") dx,

in view of (7). By a simple calculation, we see that the function rx - kxl+« attains its maximum (cfk)11«rl+lll«l at x = x, = [k(1 + a)]-1l«r1''"· Choose a constant p > (1 + a) 1''" and write M(r, I) =

In, + Ioo 0

pa:,

= I1

+I

2,

say. We then see that 0

< 11 ::S;;; px, exp[(c/k)1'«rl+llt«l] ::S;;;

c1r11« exp[(c/k)1'«rl+llt«l],

where c1 is a constant independent of r.

11.11.

ENTIRE CHARACTERISTIC FUNCTIONS

l

463

For x > px, kxl+«- rx is increasing and is greater than xr[p,./(1 +a) - 1] = c-zrx, where c2 is a positive constant. Hence

foo

0 < /2 <

e-car:t: dx- 0

as r--+ oo.

p~

Therefore,

for larger (c 3 being a constant> c2), from which we have log M(r, /) Since rl+ 111«1flog r

for r

~log

--+ oo,

c3

+ (1/a) log r + (cfk)1 1«rl+ 11«.

(9)

we have, for any e > 0,

log c3

+ (1/a) log r + (cfk) 11«rl+ 111«1 ~ (cfk)1 1«rl+(l/,.1(1 + e),

~ r0 ,

r 0 being some positive number. Thus

log log M(r, /)flog r

~

[(1/a) log(c/k)flog r]

+ [1 + (1/a)]

+ [log(1 + e)flog r]. Letting r

--+ oo,

we have

lim sup[log log M(r, /)flog r]

~

1 + (1/a).

r-+oo

On the other hand, for 0 M(r, /) >

Jz,.

CI-..,1~

~ 1JXr ~

=

< 1J < 1,

exp(rx - kxl+«) dx

exp{r(1- fJ)Xr- k[(1- fJ)XrJI+«}

+ a)- '«r · exp{ [k(1 + a)]-11«rl+ll/«l[ (1 1Jk(1 + a)- 11«r 11,. 1Jk(1

1

1',.

. exp[ (1 - 'YJ)(:

r,.

~

1Jk(1

(~ ~ ~,.]}

1 +a- ~1 - 'YJ)I%-1 rl+(1/tZI].

Since for small fJ, 1 + a - (1 - 1J )«-1 ......., a M(r, /)

'YJ) -

+ (a -

1)'YJ, we have

+ a)-11,.,11«

· exp[(1- rJ)(c/k) 11«rl+11«(1- Ct1J)]

(10)

464

XJ.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

for some constant c4 , and for small rJ ( rJ < c4 1 ). Then we see that log M(r, I)

~log

rJ

+ log c + a-

+ (1 -

5

1

log r

C6?J)(cfk)l'«rl+(ll«l

(11)

for some constant c6 • Since for large r, the last term goes to infinity faster than other terms in the right-hand side of (11 ), we may write

Here c7 (?J) may depend on rJ and be less than 1, and r is supposed to be large enough. Therefore, log log M(r, /)flog r = [log c8 (?J)flog r]

+

(1

+

(1fa)),

where Letting r

~

=, ~

lim inf[log log M(r, /)flog r]

1 + (1fa).

(12)

r-+oo

From (10) and (12), we obtain lim[log log M(r)flog r]

= 1 + (1fa).

r-+oo

Now from (9), we see that lim sup[log M(r, l)fr1+(ll« 1]

~

(cfk)l'«.

(13)

r-+oo

From (11), lim inf[log M(r, l)fr1+(ll« 1]

~

(1- c6rJ)(cfk) 11«.

r-+oo

Since rJ is arbitrarily small, this shows that lim inf[log M(r, l)fr 1 +(11« 1]

~

(cfk)ll«,

r-+oo

(13) and (14) give us lim[log M(r, I)fr1+(11« 1] r-+oo

This completes the proof of the lemma.

= (cfk)ll«,

(14)

11.11.

I

ENTIRE CHARACTERISTIC FUNCTIONS

465

We need another lemma:

Lemma 11.11.2. If a distribution function F(x) satisfies, for x ;;;; R, 1- F(x)

+ F(-x) ~ exp(-kxl+"),

k > 0,

(15)

for some R > 0, then F(x) has the entire characteristic function f(t), which is either of order 1 (1/a) and of the type less than or equal to -r, or of order less than 1 (1/a), where T = (cfk) 11", c = a"/(1 a)I+".

+ +

Proof.

For A

s;

+

> R and r > 0,

s;

erz dF(x) = =

e'Z d[1 - F(x)]

-e'Z[l - F(x)]

I; + s; T

e'z[l - F(x)] dx

= e'R[1 - F(R)] - e'A[1 - F(A)]

+

T

s;

e'z[l - F(x)] dx.

Using (15), which implies 1 - F(x) letting A ---+ oo, we have

~

exp( -kxl+") for large x, and

J; erz dF(x) = e'R[1- F(R)] + r J;

s;

e'z[1 - F(x)] dx

~ e'R[1 - F(R)]

+

~ e'R[1 - F(R)]

+ r J~ exp(rx- kxi+") dx.

T

exp(rx - kxl+") dx (16)

Since

roo e'z dF(x) ~ e'RF(R), we have, putting this together with (16),

J~oo e'z dF(x) ~erR+ r J~ exp(rx- kx 1'") dx. In a similar way, we also have

f~oo e-'z dF(x) = in which the relation F(-x)

r: ~

e-'z dF(x)

+

J:

(17)

e'x dF(x),

exp( -kx1+"), x ;;;; R, which is implied

466

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

by (15), is used for the first integral and second. Then

I: e-rz dF(x) ~ r I~ exp(rx -

J~

erz dF(x)

kx1 +«) dx

~erR

+ erR.

for the

(18)

Now from (7), (17), and (18),

M(r,f)

=

max[f(ir),f( -ir)]

~ r I~ exp(rx - kxl+<~) dx + erR. Since log log erRflog r - 1 as r-oo for a fixed R, and from Lemma 11.11.1, log log[r f;' exp(rx- kxl+cz) dx]flog r -1 + (1/a), lim SUPr-+co log log M(r,f)flog r ~ 1 + (1/a). Moreover, again by Lemma 11.11.1, log r f;' exp(rx- kxl+cz) dxfrl+(l/czl- T and log erRfrl+cz- 0, we have the conclusion of the lemma.

11.12. Entire Characteristic Functions II We are going to show the theorems which determine the order and the type of a given characteristic function in terms of the distribution function. Theorems 11.12.3-11.12.6 were obtained by Ramachandran [2]. We begin with the entire function of order one. The following theorem is no more than a restatement of Theorems 11.5.1 and 11.5.6, except for the last clause of the converse part.

Theorem 11.12.1. The characteristic function of a distribution function F(x), which has a bounded spectrum and is not a unit distribution U(x- c) for any c, is an entire function of order one and of exponential type. Conversely, every entire characteristic function f(z) of order one and of exponential type corresponds to a distribution function F(x) with bounded spectrum SF, satisfying inf SF= lim sup[logf(ir)/y],

(1)

r-+co

sup SF= lim sup[logf( -ir)fr],

(2)

r-+co

and the type is given by max(sup Sp, -inf Sp). It is sufficient to show the final statement of the converse part, which is quite easy to do:

11.12.

ENTIRE CHARACTERISTIC FUNCTIONS

II

467

From the definition, the type Tis given by T = lim sup,-+oo log M(r,J)fr. Since M(r,f) = max[J(ir),f( -ir)], we see from (1) and (2) that T ~ max(sup Sp, -inf Sp).

(3)

On the other hand, writing a= -inf Sp, b = sup SF, J(ir)

=

f( -ir)

=

J:

e-•z dF(x)

~ e-ar,

J:

e•z dF(x) ~ ebr.

~

max[e-ar, eb'],

Hence M(r,J) from which we see that T ~ max(b, -a).

(4)

(3) and (4) give us T = max(b, -a). From the final statement of the theorem we immediately have:

Theorem 11.12.2. There is no entire characteristic function of order one and of minimal type. This is so because if such a characteristic function f(z) does exist, then a = b = 0 must hold, where a, b are - inf SF and sup SF , respectively, of the corresponding distribution function F(x). Hence F(x) reduces to the unit distribution function U(x) andf(z) must be identically 1 which is no more than of order 1 and of minimal type. For a characteristic function of order 1 and of maximal type, we have Theorem 11.12.3 which will be stated after the following remarks. Throughout this section, for a given distribution function F(x), we shall invariably use the notations: T(x) = 1- F(x)

g(x)

=

+ F(-x),

(Sa)

(Sb)

{log log[1/T(x)]}flog x.

We also mention the following simple relation for an entire characteristic function, i.e., M(r,f)

~

!T(x)e'z

for all

x

~

0.

(6)

468

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

This is shown as follows: J(ir)

=

J: J:

e-'u dF(u) ~

~ erx j( -ir)

=

dF(u)

roo

e-'u dF(u)

e'x F( -x),

=

J~oo eru dF(u) ~ J~ eru dF(u)

~ e'x J~ dF(u)

=

e'x[l - F(x)].

Hence M(r,J) = max[J(ir),J(-ir) ~ Hf(ir)

+ f( -ir)]

~

ie'xT(x).

Theorem 11.12.3. A necessary and sufficient condition for a distribution function F(x) to have an entire characteristic function of order one and of maximal type is that T(x) > 0

for every x > 0

(7)

and limg(x) = ;r-+oo

oo.

(8)

Proof. Suppose that the entire characteristic function f(z) of F(x) is of order one and of maximal type. Then (7) must hold, because otherwise, the spectrum of F(x) becomes bounded and, from Theorem 11.12.1, f(z) must be of finite type. g(x) is now well-defined. We shall show the necessity of (8). From Section 11.11, (5), and (7) of this section, for any e > 0, there is an R such that for all r ~ R,

for all x > 0. Therefore,

T(x) Choosing x

~ 2R•

~

2 exp( -rx

+ r +~). 1

and taking r = (!x) 11•, we have

T(x)

~

2 exp[- ( !x)I+II!el].

Thus loglog[1/T(x)]

~

log[-log2

+ (!x) +~] 1

~

log[(1-17)(lx)1+ 111•l]

11.12.

ENTIRE CHARACTERISTIC FUNCTIONS

II

469

for some 1 > 1J > 0 and for large x, which is equal to log(1- rJ) + [1 + (1/e)] log!+ [1 + (1/e)] log x. Therefore, lim inf g(x) > 1 + (1/e). x-+oo

Since e is arbitrary, this gives us the necessity of (8). We next show the sufficiency of the condition. Condition (7) shows that F(x) does not have a bounded spectrum and, therefore, in view of Theorem 11.12.1, f(z) is not of order one and of finite type. From (8), for any e > 0, there is an R such that for x ~ R,

g(x)

~

1 + (1/e),

which implies T(x)

~

exp( -xi+Il/al).

Then from Lemma 11.11.2,/(z) is of order equal to or less than 1 + e. Since e is arbitrarily small, the order of f(z) is not greater than one. In view of Theorem 11.11.2, the order off( z) must be exactly one because f(z) is not identically one by (7). As we saw above, since f(z) cannot be of finite type, it must be of maximal type. This completes the proof. Next we discuss the entire characteristic function of order greater than one.

Theorem 11.12.4. A necessary and sufficient condition for a distribution function F(x) to have an entire characteristic function of order 1 + (1/a), a> 0, is that T(x)

> 0

for all

x

>0

(9)

and lim inf g(x)

= 1 +a.

(10)

x-+oo

Proof. The necessity of (9) is again obvious from Theorem 11.12.1 as in the proof of Theorem 11.12.3. Then g(x) is well-defined. The proof of the necessity of (10) runs almost the same way as in the proof of the foregoing theorem.

470

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

For any e > 0, there is an R such that for all r

~

R

for all x > 0, ~

T(x)

Choosing x

~

2 exp( -rx + rl+lllcxl+•).

2R 111czl+• and taking r = (lx)cztll+czal, we have ~

T(x)

where e' = a 2 e/(1

+ ae).

2 exp[ -( !x)l+cz-•'],

From this, just as before, we have

lim infg(x)

~

1 +a- e'.

z-+oo

Since e, and hence e', is arbitrarily small, lim inf g(x)

~

1 +a.

(11)

z-+oo

Now we shall show that in (11) the inequality sign does not occur. Suppose it does. We then find a y > a and an R (depending on y) such that for x ~ R, g(x) ~ 1 + y, which implies that T(x)

~

exp( -x1 +Y).

From Lemma 11.11.2, the order of f(z) is not greater than 1 + (1/y), which is strictly less than 1 + (1/a ). This is a contradiction. The proof of sufficiency is also similar to that of the foregoing theorem. As before (9) shows that f(z) is not of order one and of finite type. From (10), for any e > 0, we find an R such that for x ~ R,

g(x) This implies that, for x

~

~

1 + a - e.

R,

T(x)

~

exp( -xl+cz-•).

By Lemma 11.11.2, the order ofj(x) is not greater than 1 + [1/(a- e)]. Since e is arbitrary, the order of f(x) is less than or equal to 1 + (1/a). Actually, the inequality sign does not occur. Because if it did, then there would be a y >a and an R such that for r ~ R, M(r,f) ~ exp(rl+
~

2 exp[ -rx + rl+
11.12.

ENTIRE CHARACTERISTIC FUNCTIONS

II

471

from which, in the same way we obtained (11 ), we have liminfg(x)

~

z-+oo

1 + y > 1 +a.

This is a contradiction. We have now finished the proof of sufficiency. We now determine the type of f(z) of order greater than one. For this we introduce the function h,.(x) = {log[T(x)]- 1 }/xl+«.

(12)

Theorem 11.12.5. A necessary and sufficient condition for a distribution function F(x) to have an entire characteristic function of order 1 + (1/a), a > 0, and of finite type T > 0 is that T(x)

>0

for every x > 0

(13)

and (14)

lim inf h,.(x) =efT", z-+oo

where c = a"/(1

+ a)l+« (as before).

Proof. The proof is carried out in a way more or less similar to the previous theorems. Therefore, we do not have to give a detailed proof and thus only a sketch of the proof will be given here. (13) is necessary because it is the counterpart of (9) in the previous theorem. Fore> 0, we have from our assumption, M(r,f)

~

exp[(T

+ e)rl+Ut«>]

for larger. From this, for x > 0 and for larger, we obtain T(x)

~

2 exp[-rx

+ (T + e)rl+UI«>].

Letting r =ax«, and letting be any positive number, -log T(x)

~

-log 2 + xl+«[a- (T

+ e)al+UI«>],

which yields lim inf h,.(x) z-+oo

~a-

(T

+ e)al+.

472

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

Let a= {a/[(• + e)(1

+a)]}~.

Then the above turns out to be

lim inf h,.(x)

~

cf(•

lim inf h"(x)

~

cfrx.

+ e)«

and (15)

x ...oo

Here the inequality sign never occurs. If it does, then there is a k > cfr", such that h"(x) > k for large x and we obtain

M(r,f)

~

exp[(•'

+ e)el+!I/a:)],

where •' = (cfk} 11" < •· This inequality means that the type of f(z) is less than or equal to •' < T, which contradicts the assumption. For the sufficiency, for any k < cfr~, we obtain, for any e > 0,

M(r,f)

~

exp[(•'

+ e)rl+(l/a:)],

(16)

where •' = (cfk)IIa:. From (14), we have first of all that lim infx_.oog(x) = 1 +a; that is, f(z) is of order 1 + (1/a) (by Theorem 11.12.4). From (16), we now see that the type of f(z) is equal to or less than •'; that is, the type of f(z) is equal to or less than •· Finally, we can prove that the inequality never occurs. In this way we can prove the theorem. A theorem concerning an entire characteristic function of order greater than one and of maximal or minimal type will now be given, for which a proof is not supposed to be necessary. (i) A necessary and sufficient condition for a distribution function F(x) to have an entire characteristic function of order 1 + (1/a), a > 0, and of minimal type is that

Theorem 11.12.6.

T(x) > 0

for every

x > 0,

lim inf g(x) = 1 +a,

(17)

(18)

x ...oo

and lim h"(x) =

+=·

(19)

,1; ...00

(ii) A necessary and sufficient condition for F(x) to have an entire characteristic function of order 1 + (1fa}, a > 0, and of maximal type

473

NOTES

is, that (17) and (18) hold and lim inf h"(x)

= 0.

(20)

:z:-+oo

By the theorems we have obtained here, we see that there is an entire characteristic function of any order e and of any type -r, for e < 1, 0 ~ T ~ oo, and for Q = 1, 0 < T ~ oo. Moreover, there is an entire characteristic function of infinite order, for instance the characteristic function of the distribution for which the probability density is given by C Jx J-lzl, C being a normalizing constant. Also, the characteristic function of the Poisson distribution exp(eiz - 1) is of infinite order. REMARK.

NOTES 1. Concerning Theorems 11.2.4 and 11.2.5, another way of handling them was given by Feller [1], Vol. II, p. 528.

2. When the distribution function F(x) is symmetric, detailed studies have been made of results similar to those given in Theorem 11.3.4 by Cramer-Leadbetter when k = 0, 1. (See Cramer and Leadbetter [1], Section 9.3.) For this case better results are obtained in Theorem 11.3.4 (ii), that is, the same conclusion holds with b < a so that in Section 1 1.3, Eqs. (25) and (28) hold for b < a. 3.

In connection with Theorems 11.3.4 and 11.3.5, Lamperti [1] gave the result that if 1 - Rf(t) = la1 t 2 + 0(1 t I"H), e > 0, t -+ 0, then f~oo I x I'*"' dF(x) < oo for e' < E. This is shown in a way similar to the proof of Theorem 11.3.4. Further generalizations will be possible in the direction of Theorem 11.3.4.

4. The converse of Theorem 11.6.1 is obviously true. Then naturally the problem arises: When does the characteristic function belong to L' (- oo, oo) ? This actually is the problem of the integrability of Fourier transforms, which is the analog of the absolute convergence of Fourier series. (See Hille and Tamarkin [1].) 5.

In Theorem 11.7.2, the condition that e(t) be twice differentiable can be removed. (See Boas and Smithies [1].)

fr

In connection with Theorem 11.8.3, Makabe [1] has shown that T-t If(t) I" dt O(T-") (as T -~ oo) implies that F(x) e Lip(a/2) and is implied by F(x) e Lip a, where f(t) is the characteristic function of a distribution function F(:<).

6.

=

7.

A more precise inequality relation than (1) in Section 11.9 was obtained by Lukacs [4]: If f(z) is an analytic characteristic function with the strip of analyticity -TJ• < lm z < Tit• then lf(t +iT) I;:;;; 1/(iT) I;:;;; lf(t +iT) I + {2/(iT)[/(iT)- Ref(t + iT)]}111

for

-n. <

T

<Tit·

474 8.

XI.

DISTRIBUTIONS AND THEIR CHARACTERISTIC FUNCTIONS

We mention the following inequality relation for an analytic characteristic function

/(•) Re[/(iT) - f(t

+ iT)]

~

4- 1 Re[/(iT) - f(2t

+ iT)]

when iT is contained in the strip of analyticity of /(16). (See Lukacs [3], p. 146.) 9.

Let G(x) be a distribution function and consider the integral transform g(s)

=

J:

K(s, x) dG(x),

s being real. Lukacs [3], p. 75, gave the necessary and sufficient condition for g{s) to be the characteristic function of G(x), that is, that K(s, x) should be exp isx. 10. As was stated in Section 11.12, there is an entire characteristic function of any order equal to or greater than 1. A more precise result was obtained by Marcinkiewicz [2]. Theorem. There is no entire characteristic function of finite order p > 2 for which the exponent q of convergence of a-points for some a is less than p. (For terminology, see Titchmarsh [3], p. 273.) As a consequence of this theorem, we have the result: k

> 2,

cannot be a characteristic function. A more general result than this was obtained by Lukacs (see Lukacs [3], p. 146). 11. Ostrovskii showed the more general theorem: Let F(w) and/(•) be entire functions and F(w) ¢ constant. Set !p(%) = F[/(•)]. If !p(%) satisfies the conditions that J!p(X + iy) J ~ M(J y J, !p), M(J y J, 9') = maxl•l-lwl J9'(•) J, then /(•) is either a polynomial of degree not exceeding 2 or /(•) is an entire function of order 1 and not of minimal type (Ostrovskii [1]). 12. Valiron [1] introduced the idea of the precise order of an entire function. Let e(r) be a nonnegative function defined for r > 0 such that e'(r) exists and lim ,....,.,

T(l'(r) log T = 0.

An entire function is said to be of precise order e(r) if lim sup log M(r, r-+oo

f)/rfl''' =

e,

0


oo,

where M(r, /) = maxl•l-r Jf(•) J. Rossberg [1, 2] made an extensive study of the precise order of entire characteristic functions.

XII Convergence of Series of Independent Random Variables

12.1. Convergence of a Sequence of Random Variables We give here the basic notions and results concerning the convergence of a sequence of random variables. Let {Xn(w),

n = 1, 2, ... }

(1)

be a sequence of random variables on the fixed probability space (.Q, SiJ, P), namely a sequence of the measurable functions on (.Q,SiJ, P). In probability theory, the following four modes of convergence of (1) are most important: almost sure convergence, convergence in probability, convergence in distribution, and convergence in the mean LP, or convergence in the norm LP, p > 0. In the terminology of real analysis, almost sure convergence is called almost everywhere convergence, and convergence in probability is called convergence in measure. If there is a measurable set (that is, a set of SiJ), A c .Q with P(A) = 1 475

XI I.

476

SERIES OF INDEPENDENT RANDOM VARIABLES

such that X 11 (m) converges to a random variable X(w) pointwise at every wE 11, then X,(w) is said to converge to X(w) almost surely or almost everywhere. If for any e > 0, P[l X,(w)- X(w) I > e] converges to zero for some random variable X( w ), then X,( lo) is said to con•cerge to X (w) in probability. Let F,(x), n = 1, 2, ... , be the distribution function of the random variable X .. (w). If F,.(x) converges weakly to a distribution function, then X 11 ( w) is said to con'lierge in distribution. Let X .. ( w) E LP( Q), n = 1, 2, ... , p > 0. If there is a random variable X(w) E LP(Q) such that E

I X,(w)-

X(w)

IP =

J I X,(w)- X(w) IP dP---->- 0 D

as n---->- oo, then X,.(w) converges to X(w) in LP or in LP-norm. in probability theory, we are concerned mostly with the case p ;s 1. If nothing is indicated for a function X(w) on Q, we mean it to be a random variable. The notation LP is used for LP(.!J) if no confusion is expected.

Theorem 12.1.1. X .. ( w) converges to some X( w) almost surely if and only if Xm(w)- X,.(w)---->- 0 a.s. as m,n---->- oo; namely, for any e > 0, there is an n0 = n0 ( e, w) and a set A with P(A) = 1, such that

I Xm(w)for n,m

X,.(w) I <

E

(2)

;s n0 •

This is obvious, since almost sure convergence is pointwise convergence on A. As to convergence in probability, we must mention first the following theorem.

Theorem 12.1.2. When X,( o>) converges in probability to a random variable, the limit random variable is uniquely determined except in a set of probability zero.

Proof. Suppose that there were two random variables X(w) and Y(w) such that X .. ( w) would converge to X( w) as well as to Y( w) in probability. We are going to show that P[X(w)-=!= Y(w)] = 0.

(3)

12.1.

477

CONVERGENCE OF RANDOM VARIABLES

If I X(w)- Y(w) I > e (e > 0) for an w, then either I X(w)- X,(w) I > e/2 or I Y(w)- X,.(w) I > ef2 must be true for any n because, if not, I X(w)- Y(w) I ~I X(w)- X,.(w) I + \ X,(w)- Y(w) I ~e. Hence

{I X(w)-

Y(w)

I>

E} c

{I X(w)- X (w) I > e/2} U {I Y(w)- X (w) I > e/2}. 11

11

Therefore, P[l X(w)- Y(w)

I>

e]

+ P[\ Letting n

--+-

oo,

I > e/2] Xn(w) I > e/2].

~ P[l X(w)- X 11 (w)

Y(w)-

(4)

we should have P[l X(w)- Y(w)

I>

e] = 0.

(5)

Now, if X(w)-::/=- Y(w) for an w, then for this w, I X(w) - Y(w) for some integer k and the converse is true. Hence 00

{X(w)-::/=- Y(w)}

= U {I

X(w)- Y(w)

k-1

I>

I> 1/k

1/k}

and consequently, 00

P[X(w)-::/=- Y(w)] =

L P[l

X(w)- Y((•J)

I>

lfk]

=

0

k=l

from (5). This shows the theorem. A useful result is given in the following theorem.

Theorem 12.1.3. If {X,(w), n = 1, 2, ... } converges to a random variable X(w) in probability, then there exists a subsequence {X,t(w), k = 1, 2, ... }, which converges to X(w) almost surely. Suppose that X,(w)--+- X(w) in probability. Using the same reasoning by which we obtained (4 ),

Proof.

{I Xn+•(w)-

Xn(w)

I>

e} c

{I Xn+•(w)- X(w) I > e/2} U {I Xn(w)- X(w) I > e/2}

for every e > 0 and for all positive integers n, v. Therefore, as

n-+-oo

(6)

for every e > 0 and every positive integer v. In other words, for a given

XII.

478 E

SERIES OF INDEPENDENT RANDOM VARIABLES

> 0, there is an n0 ( E) such that, for n

~

n0 ( E), (7)

for v = 1, 2, .... Let us choose E = 1f2k, k being any positive integer. Then n ~ mk,

for

(8)

+

where mk = n0(1f2k). Let n 1 = m 1 , n 2 = max(n 1 1, m 2 ), n 3 = max(n 2 1, m 3 ), •••• Then n 1 < n 2 < · · · and n~.: ---+ oo as k ---+ oo. From (8) with v = 1,

+

Define the set and let Then

P(Bn) <

= = L P(Ak) ~ L 1f2k = k-n

112n-1.

k-n

For a given e > 0, and for m such that 1f2m- 1 < positive integer p,

E,

we have, for every

for w E Bmc· Therefore, the set {I Xn m+:P ( w) - Xn ,. ( w) I > E} is a subset of Bm for p = 1, 2, ... and, consequently, their union is also a subset of Bm. Write

s. ==

= n= u {I xltq+p (w) -

q-1

Then, since

S. c

Xnq( w) I >

p-1

= U {I Xnm+P(w)-

E }.

Xnm(w) I > e},

p=1

P(S.) Letting m

~ P[ ---+

UI XRm+,.(w)- Xnm(w) I > e] ~ P(B

71,)

< 1/2

71' - 1 •

p-1

oo, we have P(S.) = 0.

(9)

12.1.

Now, since

<

479

CONVERGENCE OF RANDOM VARIABLES

1fn,

s. c

for e > e' > 0, for any e with 1/(n + 1)

s. c s..

u 00

1/(n+U;:;ie<1/n

Thus

~

e

Sv
s.

c

u S. u u

s1/(n+l).

00

O<s<1

u 00

S. c

=

n-1 1/(n+U;:;is<1/n

81/(n+U

n-1

and Therefore, from (9),

P( U s.) ~ f P(S ,
0<•

1)

= 0.

n=1

Since, by Theorem 12.1.1,

{Xntc(w)

+ X(w)} = US., •>0

Theorem 12.1.4. The sequence {Xn(w), n = 1, 2, ... } converges to a random variable in probability if and only if (10)

as n, m--+ oo. Proof. The "only if" part has been shown in the proof of the foregoing theorem [Eq. (6)]. The "if" part is shown in the following way. From the foregoing theorem, we can choose a subsequence {Xn (ru), k = 1, 2, ... } of {Xn(w)}, which converges to a random variable X(ru) almost surely. Then observing that

P{l Xn(w)- X(w)

I>

e} ~ P{l Xn(w}- Xntc(w} I > e/2} + P{l Xntc(w)- X(w) I > e/2}

and noting that both probabilities on the right-hand side converge to zero for e > 0, we have the result. REMARK.

It will be obvious that condition (10) is equivalent to (11)

as n --+ oo uniformly for all integers

11

~

0.

XII.

480

SERIES OF INDEPENDENT RANDOM VARIABLES

The inclusion relationships among three methods of convergence: almost sure convergence, convergence in probability, and convergence in distribution, are given in the following theorem.

Theorem 12.1.5. (i) The almost sure convergence of {Xn(w), n = 1, 2, ... } to X( w) implies the convergence in probability of X 11 ( w) to X( w). The converse is not true. (ii) The convergence of X 11 (w) in probability implies the convergence of X 11 (w) in distribution. The converse is not true. We may describe this as a.s. conv.

=>

conv. i.p.

=>

conv, i.d.

a.s. conv.

<;:f=.

conv. i.p.

<;:f=.

conv. i.d.

Proof. (i) The first part. Suppose that Xn( w) converges to X( w) almost surely as n - oo. It is easy to see that

n u •-On {I Xn+v(w)- X(w) I 00

{Xn(w)- X(w)} =

00

~ e}.

I

e>O n-1

Then,

+ X(w)} = u n u {I Xn+v(w)- X(w) I > e} 00

{Xn(ro)

e>O n=l

(since (U n U A)c Therefore,

=

00

v=O

n u nAc).

{Xn(w) + X(w)} ~

n u {I x. +.(w)- X(w) I > e} 00

00

n-1 .. =o

for any e > 0. Since P[X11 (w)

-f X(w)]

= 0,

The left-hand side is

P[

.Q {I Xn+v(w)- X(w) I > e}] = !~ P[ .Q {I Xn+•(w)- X(w) I > e}]

!~~

~lim n-+oo

P{l X 11 (w)- X(w)

I>

e}.

(12)

12.1.

481

CONVERGENCE OF RANDOM VARIABLES

Hence lim P{J X,.(w)- X(w) J > e} = 0. n-+.::o

For the converse part, it is sufficient to give a counterexample. Let us take Q = [0, 1] and let . -» be the class of all Borel sets on it. Let P be the Lebesgue measure. For any positive integer n, choose v so that 2• :::;;: n < 2•+1, v - oo with n. Write n = 2• + k, k = 0, 1, 2, ... , 2•- 1 • Define X 11 (w) by wE [k/2•, (k + 1)/2•], (13) Xn(w) = { elsewhere,

~:

if n = 2• + k. It is easily seen that

P[l X,.(w)

I>

1/2• e] = { 0

for for

0 < e < 1, e ~ 1.

Hence X 11 ( w) - 0 in probability. · But, it can be seen that X 11 ( w) does not converge to any random variable. Let us take any w E (0, 1). We may find an infinite number of intervals of [kf2•, (k + 1 )/2•] which contain (•J. Such a sequence of intervals depends on w and is denoted by {[k 111 /2"', (k,,. + 1)/2"'], m = 1, 2, ... }. Let nm = 2 111 + k,.. Then xllm(w) = 1, but XII(OJ) = 0 for n # n,n• Hence Xn( w) does not converge at w. (ii) The first part. Suppose that X,(w)- X(w) in probability. Let the distribution functions of X 11 (w) and X(w) be F 11 (x) and F(x), respectively. Let x be any point of continuity of F(x). Then, for every e > 0, P[X11 (w)

< x]-

~ P[X11 (w) ~ P[X 11 (w)

P[X(w)

< x

+ e]

< x]- P[X(w) < x + e, < x, X(w) ~ x + e].

X 11 (w)

< x] (14)

If X 11 (w) < x and X(w) ~ x + e, then X 11 (w)- X(w) < -e. Therefore the last probability in ( 14) is not less than P[X11 (w)- X(w)

< -e]

~ P[l X 11 (w)- X(w)

I>

e],

which converges to zero by assumption. Hence lim sup P[Xn(w) < x] ~ P[X(w) < x n-+oo

+ e].

(15)

XII.

482

SERIES OF INDEPENDENT RANDOM VARIABLES

Since x is a continuity point of F(x), we have, letting e ---+ 0, lim sup P[X11 (w) < x]

~

F(x).

(16)

n-+oo

Quite similarly we can show that lim inf P[Xn(w) > x]

~

F(x)

(17)

n-+oo

at a continuity point x of F(x). (16) and (17) give us lim F11 (x)

= F(x).

n-+oo

The second part of (ii) is apparent from the fact that the random variable is not uniquely determined by its distribution function except in the case in which F(x) is a unit distribution U(x - a). This completes the proof. In spite of Theorem 12.1.5 (ii), we have the following result.

Theorem 12.1.6. If X 11 (w) converges in distribution as n---+ oo and the limit distribution function is a unit distribution U(x- a) for a constant a, then X 11 (w) converges to a in probability. Proof. We may, without loss of generality, suppose that a= 0. Let F11 (x) be the distribution function of X 11 (w).

<0, X> 0.

X

(18)

We have P[l Xn(w)

I>

e] ~ P[Xn(w) > e]

= 1- F11 (e)

+ P[X (w) < 11

-e]

+ F (-e)---+ 0 11

by (18). We now consider LP convergence.

Theorem 12.1.7. If {X11 (w), n = 1, 2, ... } is a sequence ofrandom variables which belong to LP(Q), or equivalently are such that E I X 11 IP < oo,

12.1.

483

CONVERGENCE OF RANDOM VARIABLES

p > 0, and Xn(w) converges to X(w) in LP, X(w)

E

LP, then Xn(w)

converges to X( w) in probability.

Proof.

Write

{I Xn(w)-

X(w) I > e} = s. for any e > 0. We have

J I Xn(w)- X(w) IP dP;;;; J D

80

;;;; eP

I Xn(w)-

X(w) IP dP

fs. dP = eP(S.).

[This is no more than the special case of the Markov inequality (Corollary 11.1.1 ). ] From this, our conclusion is immediate. In general, the converse is not true because X 11 ( w) must be presupposed to have finite E I Xn(w) IP. Moreover, even if El Xn IP < oo, the converse is still not true. We can have a sequence {Xn(w)} such that Xn(w)--+ 0 almost surely even though X 11 (w) does not converge to zero in LP. This is seen from the obvious example: D = [0, 1] ; .91 is the class of Borel sets in [0, 1], Pis the Lebesgue measure, and

0 < w < 1/n, elsewhere. X 11 ( w) converges to zero almost everywhere, but

However, we have the following theorem.

Theorem 12.1.8. Let p > 0. If X 11 ( w) E LP and for some random variable Y(w) E LP, I X 11 (w) I ~ Y(w) almost surely, then the convergence of X,(w) in probability to X(w) implies that X(w) E LP and X 11 (w) converges to X(w) in LP. We shall use for the proof, the fact that if {S 11 } is a sequence of measurable sets such that P(S11 )--+ 0 as n--+ oo, then for any X(w) E £1, fs,. X(w) dP--+ 0 as n--+ oo.

Proof. Suppose that X,(w)--+ X(w) in probability. We shall first show that X(w) E LP. Write Es = {I X(w) ~ N} for any positive integer N and sn = {I Xn(w)- X(w) I > e}.

XII.

484 Suppose first p

s

SERIES OF INDEPENDENT RANDOM VARIABLES

1. Using Theorem 11.1.1 (ii ), for any e > 0,

[feN 1 x<w> 1p dPrp;;;:; [feN 1 x<w>- x"<w> 1p dPrp +[feN 1 xn(w) 1p dPfp

; ;:; [J

ENnSn

I X(w)-

X,(w)

IP dP]l!p

+ [J E.vnSn I X(w)- Xn(w) IP dP]l!p c

(19) where n is any positive integer. Since, on E.v, using Theorem 11.1.1 (ii ),

[J

E.vnSn

I X(OJ)-

; ;:; [J

E.vnSn

X,(w)

= N[P(Sn)]l!p

we have,

IP dP]l!p

I X(OJ) IP dP]llp + [J

;;;:; N[fsn dPrp

I X( OJ) I ;;;:; N,

E.vnSn

I Xn(w) IP dP]!Ip

+ [fsn 1 Y(w) 1p dPrp

+ [fsn I Y(OJ) lp dP rp

and

J

c

E.vnSn

I X( OJ)-

X,( OJ)

IP dP;;;:;

J

c

Sn

I X(w)

- X,(OJ)

IP dP;;;:;

t:P.

So from ( 19 ), letting n ----.. oo and using the fact mentioned above and P(S,)---+ 0, we have

[feN I X(OJ) lp dPrp;;;:; e + [fe.v I Y(w) lp dPrp ;;;:; e

+ [E I Y(w) IP]liP.

Since N is arbitrary, this shows that X( OJ) E LP. If 0 < p < 1, we may use the inequality (iii) of Theorem 11.1.1 to obtain the same conclusion.

12.1.

485

CONVERGENCE OF RANDOM VARIABLES

Once this has been shown, it is easy to show the theorem. For, if p

[f

!J

~

1,

I X,.(w)- X(w) IP dP rp

~[Is~ I Xn(~)-

X(w) IP dP rp

+ [fs. l xn(w) 1P dPrp + [fs. l x(w) 1P dPrp ~ e + [fs. l Y(w) IP dPrp +Us.. I X(w) IP dPrp. From P(Sn)---+ 0, this gives us the convergence of X,(w) in LP to X(w).

Theorem 12.1.9. Letp > 0, and X,(w) E LP. Then El Xm- X 11 IP ---+0 if and only if X,(w) converges in LP to some random variable X( w) E LP. We omit the proof because this theorem states the completeness of LP-space, which is already known from real analysis (see, for example, Royden [1 ]).

Theorem 12.1.10. If X 11 (w) E LP converges in LP to X(w) E LP, then there is a subsequence {X,k(w)} of {X,(w)} which converges to X(w) almost surely. This is an immediate consequence of Theorem 12.1.3 and of the uniqueness of the limit of convergence in probability mentioned after Theorem 12.1.1. Finally we remark that if X,---+ X, Y,---+ Y, both a.s., i.p., or in LP, then X,+ Y,---+ X+ Y a.s., i.p., or in LP, respectively. The proofs will be left to the reader. We also remark that if X, ---+ X in distribution and Y 11 ---+ C in probability, where C is a constant, then X 11 + Y, converges to X+ C in distribution. This is shown in the following way. Suppose the distribution function F11 (x) converges to a distribution function F(x) of X. Then P(Xn

+ Yn < x) =

+ Y, <X, I ¥ C I > e) + P(x,. + Y, < x, I Y,- C I ~e)

P(Xn

~ P(l Y,.- C

11 -

I>

for any e > 0

e)+ P(Xn < x- C +e)

XI I.

486

SERIES OF INDEPENDENT RANDOM VARIABLES

because X,+ Y, < x, Y,- C;::;; -e imply X,< x- C +e. Henl.:e lim sup P(X,. + Y, < x);;::;; F(x- C +e)

n-+oo

(20)

if x - C + e is a continuity of F. On the other hand, using the fact that Y,.-C;;::;; e and X, < x-C-e imply X,.+ Y, < x, we have

+ Y,

< x) ;::;; P(Y,- C < e, X,.< x- C- e) = P(X, < x- C- e)- P(Y,.- C;::;; e, X,1 < x- C- e) ;::;; P(X,. < x- C- e)- P(Y,- C;::;; e).

P(X,.

Hence lim inf P(X,.

n-+oo

+ Y,. < x);::;; F(.'C- C- e)

(21)

for a continuity point x - C - e of F. If x - C is a continuity of F, then from (20) and (21), we have, letting e--+ 0 (in such a way that x- C ± e is a continuity of F), P(x, + Y,. < x)--+ F(x- C), which is what we wanted to show. Note that the statement that if X,, Y, converges in distribution to X, Y, respectively, then X,+ Y, converges in distribution to X+ Y, is not true because the distributions of X and Y do not determine the distribution of X + Y.

12.2. The Borel Theorem The result in the title is a basic one in probability theory. The following theorem is called the Borel-Cantelli lemma.

Theorem 12.2.1. Let E 1 , E 2 , ••• be any sequence of measurable sets. L:, 1 P(E,) < oo, then P(lim sup,_,. E,) = 0.

If

Proof.

00

n:=l U~n Ek and P(lim sup E,.) = P( {) UEk) = P( lim 0 Ek) n-+oo k=n n-+oo k-n = lim P( 0 Ek) = lim lim P( UEk) By definition, lim SUPn-+oo E,.

=

n=l

n~oo

k=-n

n-41-oo 1n-+oo

m

;;::;; lim lim n-+oo m-+oo

L

P(Ek)

= 0

k=n

from the assumption. This proves the theorem.

k=n

12.2.

487

THE BOREL THEOREM

The conclusion can be said to be that the probability that En occurs infinitely often is zero. Now we introduce the notion of the independence of a sequence of measurable sets. If, for any finite subset Ei 1 , Ei., ... , Ei,. of {En, n = 1, 2, ... }, (1)

holds, then {En, n = 1, 2, ... } is called a sequence of independent sets. We mention that if {En} is a sequence of independent sets, then so is {Enc}. The proof is carried out without any particular difficulties and so it is left to the reader. We shall now prove that the counterpart of the Borel-Cantelli lemma holds with the independence condition of sets.

Theorem 12.2.2. Let {E,, n = 1, 2, ... } be a sequence of independent measurable sets. If L;"' P(En) = oo, then P(lim supn_.oo En) = 1. Proof.

In order to prove this theorem, we use an elementary inequality:

(2) where ak is any number such that 0 ~ ak ~ 1. This is immediate from the inequality 1 - ak ~ e-"k, taking the products of the both sides. Let ak stand for P(Ek) in (2). It then turns out that

n > m. If :E;"' P(Ek) Hence

=

oo, then the left-hand side converges to zero as n ___. oo. 00

IT [1 k=m

P(Ek)] = 0

for all

(3)

m.

Now

lim limP( m-+oo n-+oo

lJ Ek) =

k-m

lim lim [ 1- P( {) Eke)], n1__,.oo n-+oo

k=m

XII.

488

SERIES OF INDEPENDENT RANDOM VARIABLES

which is, by the independence of E{.c, P(lim sup m-).oo

Em) =

lim lim [ 1 m-+-oo

=lim m-+oo

n-+-oo

TI P(Ekc)] =

1

k-m

!~ [1 -kQ,[1- P(Ek)]]

=

1

by (3 ). This shows the theorem. Theorems 12.2.1 and 12.2.2 are together called the Borel theorem. From these theorems we immediately have:

Corollary 12.2.1. If {En} is a sequence of independent sets, then P(lim supn-+oo En) is either zero or one according as ~ P(En) converges or diverges. We give a simple consequence of the corollary.

Theorem 12.2.3. Let {Xn ( w ), n = 1, 2, ... } be a sequence of independent random variables. Then Xn(w) converges to zero almost surely if and only if

(4) is "convergent for every constant C > 0. Proof. Suppose X,(w) converges to zero almost surely. Write Ell = {I Xn I ~ C}. We then see that {Ell} constitutes a sequence of independent sets. This is obvious from the definition of independence of Xn. There is a set A, with P(A) = 1, such that for every wE A, Xn(w)--+ 0. Hence for a fixed C (but arbitrary), I X,(w) I < C except for a finite number of n for each w E A. That is, lim SUPn-+oo En E Ac, and then P(lim sup En)= 0. Thus from Corollary 12.2.1, L P(En) < oo for every C > 0, which is what we required. Conversely, if (4) converges for a constant C, then P(lim sup E,) = 0 by Corollary 12.2.1. Hence at any wE (lim sup E,y, I Xll(w) I ;;;;; C except for a finite number of n. Since Cis arbitrary, X,(w)--+ 0 almost surely.

12.3. The Zero-One Law Let X(w) be a random variable on (Q,.~. P). The minimal a-field over the class of sets {w : X( w) < x }, - oo < x < oo, is called the

12.3.

489

THE ZERO-ONE LAW

a-field generated by X(w), and is denoted by .~(X). Needless to say, is a subclass of ..9J. The a-field generated by a finite number of random ~·ariables X 1 ( w ), X 2 (w), ... , X,(w), which is denoted by ~~(X 1 , X 2 , ••• , X .. ), is defined to be the minimal a-field containing all sets of the form {w : X 1 ( w) < x 1 , X 2 (w)<x 2 , ••• ,X.. (w)<x,}, for any set {x1 ,x2 , ••• ,x11 } of real numbers. It is obvious that .3.S'(X1 ) is a subclass of .~(X 1 , ••• , X,), or more generally, ;B'(Xi,, ... , Xik) is a subclass of d3'(X 1 , ••• , X .. ) for any subset (i 1 , i 2 , ••• , i,.) of ( 1, 2, ... , n ). Suppose that we are given an infinite sequence of random variables

.~(X)

(1) and consider the sequence of a-fields, .~(X,),

. )J(X",

x,+l),

~(X.. ,

x .. +l, x .. +2 ),

(2)

This becomes a nondecreasing sequence of a-fields. Form the union of a-fields of (2), that is, 00

U·;j'(X,., Xntt•· . . , X,~+k) =lim ~(X.. , ... , Xn+A·)· k=O

k-oo

This is again a a-field and is denoted by .~(X,, Xn+I• .. . ) or supm;;:,·~(Xm)· Furthermore, we consider the sequence of a-fields sup Si.S'(Xm),

sup

~(X,..),

(3)

m;s:;n

This is obviously a nondecreasing sequence of a-fields. The intersection

n 00

n=l

sup ..~(Xm) = lim sup93'(X,.) n__,.oo

m~n

m~n

of all a-fields in (3) is again a a-field and is called the tail a-field of the sequence (1). It is naturally described as lim sup, .... =·~(X.,). Any set which belongs to the tail a-field is called a tail set. Any function which is measurable with respect to the tail a-field is called a tail function. For example, lim infn-+oo w ), lim SUPn-+oo X .. ( w ), or lim, ....oo N- 1 L~= 1 X,(w), ... are tail functions, in so far as they are well-defined. We show for instance that lim infn-+oo X 11 ( w) is a tail function. It is sufficient to show that E = lim inf, ....= {w : X,( w) ;s; x} is a tail set for

x. (

XII.

490

SERIES OF INDEPENDENT RANDOM VARIABLES

every x. Let E,.(x) = {co: X,.(w) ~ x, X,.+l(w) ~ x, ... }, which is a set of <~(X,., X,+ 1 , ••• ). lim,._,. E,. = U;;'= 1 E,.(x) is a tail set. For wE E,.(x), obviously infk;,;n Xk(w) ~ x and hence lim inf,._,.oo X,.(x) ~ x. That is, lim E,.(x) c E. (4) 00

n-+oo

On the other hand, take any wE E. Then lim,._,. [infk;;:n Xk(w)] ~ x, namely, for any e > 0, there is an n0 such that X,.( w) ~ x- e for n ~ n 0 • Therefore, wE E,..(x- e) and consequently wE U;;'= 1 E,.(x- e). Since this is true for all e > 0, we have, for ek ---+ 0, 00

n 00

wE

00

U E,.(x -

ek)

=

k-1 n-1

lim En(x), n->oo

that is,

E c lim En(x). n->oo

This with (4) gives us E = limn-+oo En(x) and thus E is a tail set. We need to introduce a definition. Two classes of sets, C 1 and C 2 , are called independent if any sets A and B, A E C 1 , B E C 2 , are independent of each other (according to the definition given in Section 12.2), namely, P(A n B) = P(A)P(B). We now give a lemma.

Lemma 12.3.1. Let {Xn(w), n = 1, 2, ... } be a sequence of independent random variables. Then B(X1 , ••• , Xn) and ~(Xn+l, X,., ... ) are independent. Proof. Let E,. be any set of the form {X1 (w) < x 1 , •• • , X,(w) < xn} for some set (x 1 , ••• , xn) of real numbers and let Fn,m be any set of the form {Xn+l(w) < Xn+l, ... , Xn+m(w) < Xn+m} for some (xn+l, ... , Xn+m)· Because of the independence of {Xn(w)}, (5) Actually, this equation holds for every En E -~(X, ... , Xn) and for every Fn,m E ~(Xn+I, ... , Xn+m)· This should be quite easy to see, but for the sake of completeness we shall show it. Let En be a fixed set as defined above. For each fixed En, we consider the class Cn,m of sets G for which P(En n G) = P(En)P(G). It is then obvious that Cn,m is a a-field. Since ~(Xn+l, ... , Xn+m) is the minimal a-field over the class of sets of the form (Xn+l < Xn+l, . .. , Xn+m < Xn+m),

12.3.

491

THE ZERQ-ONE LAW

we see that ~(Xn+l•· .. , Xn+m) c Cn.m· This is equivalent to (5) with a fixed En and any set Fn.m belonging to <~(Xn+l, . .. , Xn+m)· In the same way, for each Fn,m E .!:i.S'(Xn+I, ... , Xn+m), (5) holds for any En E ~(XI, ... , Xn)· This completes the lemma. Now we are in a position to prove the following important theorem of Kolmogorov [2], which is called the zero-one law.

Theorem 12.3.1. If {Xn(w), n

= 1, 2, ... } is a sequence of independent random variables, then the probability of any tail set of {Xn} is either zero or one. Moreover, any tail function is constant almost surely.

Proof. Let E be any tail set. Then E belongs to all n = 0, 1, 2, .... Hence, by Lemma 12.3.1,

~(Xn+l,

Xn+ 2 , • •• ),

P(En 11 E) = P(En)P(E)

for every En

E ~(XI, ... ,

Xn)· Since this is true for all n's,

P(F 11 E)

for any set FE

~(XI,

=

P(F)P(E)

X 2 , ••• ). Taking F = E, we have P(E)

=

[P(E))2,

which implies that P(E) is either zero or one. This proves the first part. Let X(w) be a tail function. Then, since {X(w) < x} is a tail set for every x, from the first part, it is either zero or one. In other words, the distribution function F(x) of X( w) is either zero or one for each x. Since, if F(xi) = 0 for some xi, then F(x) = 0 for all x <xi and if F(x 2 ) = 1 for some x 2 , then F(x) = 1 for all x > x 2 , we conclude that for some m F(x) = 0 for x < m and F(x) = 1 for x > m. This completes the proof The following theorem is a simple consequence of the zero-one law.

Theorem 12.3.2. If {Xn(w), n = 1, 2, ... } is a sequence of independent random variables, then (i) Xn(w) either converges almost surely or diverges almost surely; (ii) the series L;;:I Xn(w) either converges almost surely or diverges almost surely.

Proof. We prove only (i). The proof of (ii) will be left to the reader since it is carried out in a similar way.

492

XII.

SERIES OF INDEPENDENT RANDOM VARIABLES

Consider the set E = {lim SUPn-+oo Xn( w) = oo }. Then E is a tail set because it is the set on which lim SUPn-+oo Xn( w) > x for all x. Thus P(E) = 0 or 1. If P(E) = 1, then Xn(w) diverges almost surely. If P(E) = 0, then lim supn-+oo Xn(w) < oo almost surely, and, in this case, there are the two possibilities that, for the set F = {lim infn-+oo Xn( w) = -oo }, P(F) = 0 or 1. If P(F) = 1, then Xn(w) diverges to -oo as n---+ oo almost surely. If P(E) = 0, P(F) = 0, then -oo
Xn(w) < oo

n-+oo

almost surely [(E U F)0 has probability 1]. Now, since lim SUPn-+oo Xn( w) and lim infn-+oo Xn( w) are tail functions, each of them is constant almost surely. Moreover, {lim SUPn-+oo Xn(w) #-lim infn-+oo Xn(w)} is also a tail set. Hence this set is of probability 0 or 1. If the probability is 0, then Xn(w) converges almost surely as n---+ oo, and if it is 1, then Xn(w) diverges almost surely as n---+ oo. Thus, we have the conclusion (i).

12.4. The Equivalence Theorem It is the purpose of this section to show that the three notions of convergence (almost sure convergence, convergence in probability, and convergence in distribution) are identical for the convergence of a series of independent random variables. Once we have this result, we may handle the problem of the almost sure convergence of a series of independent random variables by use of characteristic functions of random variables, since the convergence in distribution is completely in the category of the convergence of characteristic functions based on the Levy continuity theorem (Theorem 9.3.7). We begin with the following obvious theorem.

Theorem 12.4.1. Let {Xn( w ), n = 1, 2, ... } be a sequence of independent random variables with characteristic functions Un(t), n = 1, 2, ... }. In order that the series (1)

converge in distribution, it is necessary and sufficient that m

n A
k-n+l

(2)

12.4.

THE EQUIVALENCE THEOREM

493

converge to 1 as m-oo, n- oo locally uniformly (uniformly in every finite interval of t). This is so because the distribution function of L~-~ Xk converges weakly to a distribution if and only if n~-dk(t), which is the characteristic function of Lt 1 xk, converges locally uniformly, and this is equivalent to the local uniform convergence of (2) from the elementary theorem on infinite products. Concerning the convergence of D;"' ft(t), much more precise results are known and will be given in Section 12.9. Theorem 12.4.1 enables us to show the following corollary.

Corollary 12.4.1. For a series of independent random variables, the notions of convergence in probability and convergence in distribution are equivalent to each other. Proof. Since convergence in probability implies convergence in distribution for a general sequence of random variables, by Theorem 12.1.5 (ii), it is sufficient that convergence in distribution imply convergence in probability. Let j,.(t) be the characteristic function of X,.(w), where the {X,(w)} are supposed to be independent random variables. Suppose that L;"' X,( w) converges in distribution, then, from Theorem 12.4.1, llZ'=n+l/k(t) converges to 1 uniformly locally as m-oo, n- oo. Since llZ'=n+l/k(t) is the characteristic function of LZ'=n+l Xk(w), P(LZ'=n+l Xk(w) < x) converges weakly from the Levy continuity theorem to the unit distribution (since 1 corresponds to the unit distribution) as m-oo, n- oo, which means that P(l S,,.- S,. I > x)- 0 for any x > 0, where S,. denotes L~-o Xk. This proves the corollary.

Now we prove the following equivalence theorem.

Theorem 12.4.2. For a series of independent random variables, almost sure convergence, convergence in probability, and convergence in distribution are equivalent to each other. Proof. Because of Corollary 12.4.1 and Theorem 12.1.5 (i), it is sufficient to show that convergence in probability implies almost sure convergence. Let L:, 1 X,.(w) be a given series, where {X,., n = 1, 2, ... } is a sequence of independent random variables.

XII.

494 Let

Sn(w) =

SERIES OF INDEPENDENT RANDOM VARIABLES

n

L Xk(w),

k=l

Suppose that Sn = Sn(w) converges in probability to a random variable s = S(w). Then for any e > 0, P(l sn- s I >e)---+ oo. From this and Theorem 12.1.3 we can find a subsequence {nko k = 1, 2, ... } of {1, 2, ... } such that Snk converges almost surely to S and

P(l Snk+l- Snk I > e/2) < 1f2k, P(l Snk- S

I>

ef2) < 1J2k.

(3) (4)

Let nk:::;;; n ~ nk+t· If Snk,n > e, Sn,n~:+l > -e/2 for e > 0, then > ef2. Therefore, if we define the set

Sn~:,nk+l

Ik =

1Y [ . n {Snk.i ~ e, s,k,n > e}] n {S,,,k+l > -e/2},

n-n~:+l

•-n~:+l

where

is the set on which for at least one value of n such that nk ~ n :::;;; nk+1, s,k,n > e holds, then "'k+l n lk c: U {Snk,i ~ e} n {S,k•"'k+l > ef2} c: {Sn~:.n~:+l > e/2}. n-n~:+l i-nk+l

n

Thus we have, from (3), (5)

Since

we have

P(Sn,nk+l > -e/2);;;; 1- P(Snk,nk+l ~ -e/4)- P(Snk,n;;;; e/8).

(6)

From the convergence of S, in probability [Section 12.1, Eq. (7)], max

nk:::in::;ink+l

P(l snk,n I > e/8)---+ 0

as k---+ oo.

From this and (3), (6) can be written

P(Sn,nk+l > -e/2) ;;;; 1 - "'ko

(7)

12.4.

495

THE EQUIVALENCE THEOREM

where 'Y/k--+ 0 ask--+ oo. Now the sets

n n

{Sn1,.i

i-n~t+l

< e, Sn",n > e} n {Sn,nk+l > -e/2},

n = nk + 1, ... 'nk+I' are disjoint and the two sets under ni=nk+I are independent. Therefore,

= =

(1- 'YJk) P (

nk+l

U

n=n~t+l

(1 - 'YJk) P(Sn",n

>

nn

i-n~t+l

{Sn",i ~ e,

S,".n > e}

e, for at least one n, nk

Denoting the set {Snk,n > e for at least one n, nk we have

)

+ 1 ~ n ~ nk+I)·

+ 1 ~ n ~ nk+I} =

]k>

Combining this with (5), we have (8) Similarly, we may show that, if ]k' denotes the set {Sn",n < -e, for at least one n, nk + 1 ~ n ~ nk+I }, then (9) Let Ek denote ]k U ]k'• that is, the set on which some n, nk + 1 ~ n ~ nk+I· Then

I Sn",n I ~ e,

for

Take k so large that k > 3, 1 - 'YJk > !. We then have

P(Ek) < 2-(k-2>. Therefore, I;f:, 2 P(Ek) < oo. Then from the Borel-Cantelli lemma (Theorem 12.2.1), P(lim SUPn-+oo Ek) = 0, which is the same as saying that there is a k 0 = k 0 ( w) such that for k > k0 ,

XII.

496

SERIES OF INDEPENDENT RANDOM VARIABLES

almost surely. Namely, S,k·" _.. 0 with nk ~ n ~ nk+1, ask- oo almost surely. Since S"k converges to S almost surely, Sn = S"k•" Snk converges almost surely to S. This completes the proof.

+

12.5. The Three Series Theorem Let (1)

be a series of independent random variables. We are going to study the conditions for the almost sure convergence of (1 ). In order to do this we shall need the notion of truncation of a random variable, which is due to Kolmogorov [1]. Let X(w) be a random variable and let c be a positive number. We define another random variable xc( w) by

Xc(w)

=

for any w on otherwise.

X(c.o) 0,

{

{I X(w) I ~ c},

This random variable Xc( w) is called the truncated random variable of X(w) at c. If F(x) is the distribution function (left-continuous) of X(w) then the distribution function of xc( ())) is given by

P(x)

P(O+)- P(O)

~

= { =

x > c, x < -c,

for for

(2)

1 - F(c) + F(-c + 0).

It is easily seen that

'if' xc

=

J

x dF(x),

lxl
(3)

Lemma 12.5.1. Let {X11 (w), n = 1, 2, ... } and {Y11 (w), n = 1, 2, ... } be two sequences of random variables. If 00

L P[X (w)-# 11

Y 11 (w)] < oo,

(4)

n~l

then converges)

=

P( ~ Y w) 11 (

11-1

converges).

(5)

12.5.

497

THE THREE SERIES THEOREM

Hence if :E~ 1 X,.(w) converges almost surely, then verges almost surely.

:E~ 1

Y,.(w) con-

This is an immediate consequence of Borel-Cantelli lemma (Theorem 12.2.1). In fact, (4) implies that P[lim sup,.__.. {X (w) =F Y 11 (w)}] = 0, which is the same as saying that the probability that X 11 (w) =F Y 11 (w) for an infinite number of n is zero. This proves (5). 00

11

Before giving the main theorem, we mention an inequality in the following lemma.

Lemma 12.5.2. For a real a and t, (6) This is obvious from 1

ei'.t- 1 - a t - (a2t 2f2)

I

=

II:t du

J: I: dv

~I I:t du I: dv

I:

eiw dw

dw

I=

I I

at

13

/6.

Now we present the following theorem of Kolmogorov [1 ], which is called the three series theorem.

Theorem 12.5.1. Let (7)

be a series of independent random variables. The necessary and sufficient condition for (7) to converge almost surely, is that the following three series should converge for any constant c > 0, 00

L P(i X,.(w) I ~c), n-l

(8)

00

L var Xn"(w), ,._,

(9)

and 00

:L wxn•(w),

n-1

(10)

where X,.•(w) is the truncated random variable of X 11 (w) at c. If (8)-(10) converge for some c > 0, then they converge for every c > 0.

Xll.

498

SERIES OF INDEPENDENT RANDOM VARIABLES

Necessity. Suppose that (7) converges almost surely. Then Xn(w)- 0 almost surely as n- oo. Hence from Theorem 12.2.3, it follows that (8) converges for any c > 0. Let 1-'nc = 'i!'Xnc· Then the variance is Proof.

var Xnc =

J:

(x - l-'nc) 2 dFnc(x),

where Fnc(x) is the distribution function of Xnc(w). Letfnc(t) andfnc(t) be the characteristic functions of Fnc(x + 1-'nc) and Fnc(x), respectively. We see that fnc(t) = exp( -il'nct) fnc(t) (11) and hence lfnc(t) I =

fnc(t) = =

J:

eizt

I/nc(t) I· dFnc(x

1 - (t 2 J2)[1

+ 1-'nc) =

+ Ec{t)]

f~oo exp[i(x - 1-'nc)t] dFnc(x)

J~oo (x -!-'nc) 2 dFnc(x),

where e (t)- 0 as t - 0. This follows from Lemma 12.5.2 with a = x- 1-'nc and the fact that I a 3 t 3 l ~ 2clal 2 ltl 3 , {11-'nc I ~ c, I X I ~ c). Hence (12) Let 7J be a positive number so small that I ec(t) I < ! and {t2 /4) var Xnc<1 for ltl < "'· This choice of 7J is possible since var Xnc is bounded by f:'oo X 2 dFnc(x) ~ c2 • Hence lfnc(t) 1 ~ 1 -{t2 /4) var xnc ~ exp[-(t2 /4) var xnc]. Now

0 /tc(t) I= Ik=n+l 0 fnc(t) I ~ exp(- k-n+l f var X,/). Ik-n+l If :L;"' var xnc = oo, then the right-hand side converges to zero as m - 00 for I t I < "'· This contradicts the fact that TIZ'=n+lfkc(t)- 1 for every t as m-oo, n- oo (Theorem 12.4.1). Hence (9) converges. Now we shall show that f];"' fnc(t) converges for every t. Since

fnc(t) =

J:

= 1

exp[i(x - ,u,/}t] dFnc(x)

+ k=2 I (iktkjk!) Joo

-co

(x -1-'nc)k dFnc(x),

12.5.

499

THE THREE SERIES THEOREM

it suffices to show that

is convergent. Actually, this is absolutely convergent, since

I I

n=l

k=2

(It lkfk!) f'' I X - !1-nc lk dF,.c(x) -oo

00

~I n=1

00

I [It 1"'+2/(m + 2)!J(2c)"' var xnc m=O 00

=I var xnc 1t 12 I [(2c 1t 1)"'/m!J n=1 m-o 00

~

It 12e2cltl L var Xn <

oo,

n=t

using the convergence of (9) already shown. Therefore, from (13)

exp(it L~= 1 !1-kc) must converge for every t. Taking t = 1, we see then that there is a sequence {Pn} of integers such that :L~- 1 !1-kc + 2pnn converges as n --+ oo. Hence as

n--+ oo.

(14)

Now !1-nc--+ 0, because Xn(w)--+ 0 a.s. and then Xnc(w)--+ 0 boundedly, which gives us wxnc( w) - 0. Therefore, from (14), Pn must be a constant integer p for n greater than some integer n0 • This gives us the convergence of :L;"' !1-nc· Sufficiency. Suppose that (8)-(10) converge for some c > 0. In view of Lemma 12.5.1 and the equivalence theorem (Theorem 12.4.2), it suffices to show that :L xnc(w) converges in distribution, that is, Dfn"(t) converges locally uniformly. From (12), (15) fnc(t) = 1 + O(t2 var Xnc). Hence for I t I ~ T, T being an arbitrarily fixed pos1t1ve number, n:=tfn"(t) converges uniformly in I t I ~ T. From the convergence of (10), using (13), we see that D;"' fnc(t) converges uniformly in I t I ~ T. The last statement of the theorem is included in the proof of sufficiency. Thus the theorem is proved. An immediate consequence of Theorem 12.5.1 is the following:

500

XII.

SERIES OF INDEPENDENT RANDOM VARIABLES

Theorem 12.5.2. Suppose that {Xn, n = 1, 2, ... } is a sequence of independent random variables. (i) If L:,t var Xn < oo, then L;;'= 1 (Xn- EXn) converges almost surely. (ii) Suppose that I Xn I ~ K, n = 1, 2, ... , K being a constant independent of n. Then L~ Xn converges almost surely if and only if the following two series converge: 00

(16)

L;WXn n-1

and 00

L var Xn. n-t

(17)

Proof. The proof of (i) is, as a matter of fact, substantially included in the proof of the foregoing theorem We have (18)

where fn(t) is the characteristic function of Xn- W Xn. We obtain (18) directly from e>d = 1 iat O(a 2t 2) as we did (12). The convergence of L var Xn immediately gives us the conclusion (using the equivalence theorem). (ii) is a particular case of the foregoing theorem. It suffices to take K fore.

+

+

The three series theorem (Theorem 12.5.1) can be stated in a different way. Let {Xn , n = 1, 2, ... } be a given sequence of independent random variables. Define the variable Yn by Yn = min(1,

that is, Yn(w) Then

= 1 if I Xn(w) I ~ 1, and

I Xn 1), Yn(w)

(19)

=

Xn(w) if I Yn(w)

I ~1. (20)

Theorem 12.5.3. A series of independent random variables (21) converges almost surely if and only if the following two series converge: (22)

12.6.

SUFFICIENT CONDITIONS FOR CONVERGENCE

501

and 00

L WYn

2

(w).

(23)

n=1

Proof. Suppose first that (21) converges almost surely. Then from the three series theorem, the series (8)-(10) converge with c = 1. Now

Thus (22) converges by the convergence of (8) and (10) with c = 1. We mention that the absolute convergence of L g'Xn 1 follows because of the nonnegativeness of g>yn· In a similar way

I ~ 1] + g'(Xn1 ) 2 = P[l Xn(w) I ~ 1] + var Xn 1 + (WXn1 ) 2•

g>yn 2 = P[l Xn(w)

(24)

Hence from the convergence of (8)-(10) with c = 1 and the convergence of L (WXn 1) 2 which follows from the absolute convergence of L g'Xn\ L g'Yn 2 converges. Suppose next that (22) and (23) converge. From the convergence of (22) and (23), 1: var Y,. converges. Hence, noting that I Yn I ~ 1, we see from Theorem 12.5.2 (ii) that 1: Yn converges almost surely. From (24),

from which we see that vergence of (23 ). Therefore,

1: P[l X,.(w) I ~ 1] < oo, in view of the con-

00

L P[X (w) =/=11

Yn(w)] < oo.

n-1

Hence from Lemma 12.5.1, pletes the proof.

L Xn converges almost surely. This com-

12.6. Sufficient Conditions for the Convergence of a Series From Theorem 12.5.1 (three series theorem), we derive several sufficient conditions for the convergence of a series (1)

XII.

502

SERIES OF INDEPENDENT RANDOM VARIABLES

where {X,(w)} is a sequence of independent random variables. First we give the following theorem.

Theorem 12.6.1. Let {X,(w), n = 1, 2, ... } be a sequence of independent random variables. If for some p, 1 < p ;;:;; 2, the series 00

I wxn

(2)

n-1

and 00

L WI Xnlp

(3)

1!=1

are both convergent, then the series (1) converges almost surely. This was shown by Wintner [2, 3] and also by Marcinkiewicz and Zygmund [1] with 'if Xn = 0. The case p = 1 is also true, which is included in the subsequent theorem (Theorem 12.6.2) and the convergence of (2) is superfluous. Proof. Let c be any positive number. Let F 11 (x) be the distribution function of X n • Then

Therefore, the convergence of (3) implies 00

L w I xn• 12 <

(4)

oo.

11=1

Then 00

00

n=l

n=1

L var X,.";;:;; L WI Xn" 1

2

< oo.

Hence the series given in Section 12.5, Eq. (9) converges. P(l X 11

I ~ c) =

J

lxl:;:;c

dF11 (x) ;;:;; c-P

J

lxl;;:;c

I x IP dF (x) 11

;;:;; c-P

WI

X 11 IP.

Hence the convergence of (3) implies the convergence of the series of Section 12.5, Eq. (8). Finally,

12.6.

503

SUFFICIENT CONDITIONS FOR CONVERGENCE

where

00

~

L c-lp-ll WI XIP <

oo.

p=l

This shows, from the convergence of (2), the convergence of (10) in Section 12.5. Therefore, from Theorem 12.5.1, we have the conclusion. Another criterion is given in the next theorem.

Theorem 12.6.2. Let {Xn, n = 1, 2, ... } be a sequence of independent random variables. If, for some 0 < p ~ 1, (5) then the series (1) converges almost surely. This was given by Wintner [2]. The case p = 1 is obvious from the Fubini theorem on integration. In fact, if (5) is true with p = 1, N

lim W

L I Xn(w) I =

n-1

n~oo

N

lim n-+oo

L

WI Xn(w) I < oo.

n-1

Hence from the monotone convergence theorem (Fubini) on integration, L::'= 1 I Xn(w) I < oo a.s. This conclusion says more than required, namely, we have obtained the result that the series (1) converges absolutely almost surely. This conclusion is also true for 0 < p ~ l. As a matter of fact, we can show the sharper conclusion, under condition (5 ), that ( 1) converges absolutely almost surely. This will be shown later (Theorem 12.11.2). Here we prove the above weaker result in a very simple way. Proof.

Since, for 0 < p

~

1,

I eitz- 1 I = 21 sin(xt/2) I ~ 21 sin(xt/2) IP ~ 21 xtf2IP ~ 21 -P I xt IP, we have, letting fn(t) be the characteristic function of Xn,

I fn(t)

- 1I=

IJ~oo

(eizl -

1) dFn(x)

I

~ 2P-ll tIP f~oo I X IP dFn(x) = 2P-ll tIP E I Xn IP.

XII.

504

SERIES OF INDEPENDENT RANDOM VARIABLES

:L::

Hence 1 11 - fn(t) I converges locally uniformly. From this, the locally uniform convergence of n::dn(t) follows, and the equivalence theorem gives the conclusion. We remark that this theorem ceases to be true for p > 1. In other words, there is a series ( 1) which does not converge in distribution (or almost surely) and for which (5) is satisfied for p > 1. Define 0, x ~o. Fn(x) = t, 0 < x ~ 2/n, 1, 2/n < x,

j

for n = 1, 2, .... Then

Hence for p > 1, L WI Xn IP < oo. Since the characteristic function is

fn(t)

= ![1

+ e'(2tfn)] = eittn cos(tfn),

we have N

llfn(t) = n-1

N

N

n-l

n-1

n eit/n n cos(tfn).

(6)

The second factor on the right-hand side of (6) converges as N-+ oo since 00

L [1 n-1

00

cos(tfn)] = 2

L sin (t/2n) ~ 2

n-1

00

!t2

L (1/n

2)

< oo,

n-1

but n:= 1 eittn = exp(it L~= 1 (1/n)) does not converge. Hence ll!'=dn(t) does not converge.

12.7. Convergence Criteria and the Typical Function The three series theorem can be stated in a different form, using the function (1)

where F(x) is a given distribution function and h is a positive constant. This function was effectively used by several authors. In particular,

12.7.

CONVERGENCE CRITERIA AND THE TYPICAL FUNCTION

505

systematic use of it was made by Gnedenko and Kolmogorov [1] and Kunisawa [1, 2] for the limit theorems in probability theory. We call lPF(h) the typical function of F(x). If f(t) is the characteristic function of F(x), then (1) can be written in the form lPF(h)

= h J~ e-ht Ref(t) dt.

(2)

This was observed by Kunisawa. (See Note 4.) Actually, lPF(h)

= h J~ e-ht dt J~oo cos tx dF(x) = =

J:

[h 2 f(h 2

+x

2 )]

J:

dF(x) J~ he-ht cos tx dt

dF(x).

We state the simple properties of lPF(h) in the following lemma.

Lemma 12.7.1. The typical function lPF(h) of a distribution function F(x) is a nondecreasing function of h > 0 and lim (/JF(h) = 1, h-+oo

lim (/)(h) = 0.

(3)

h-+o+

+

Since the function h2 /(h 2 x 2 ) of h is nondecreasing, the first statement is obvious. (3) is also immediate. The following theorem is due to Kunisawa [1, 2].

Theorem 12.7.1. Let (4) be a series of independent random variables Xn( w) with distribution function Fn(x). For the almost sure convergence of (4), it is necessary and sufficient that 00

L [1 -

(/JF/, (h)] <

00

(5)

k-1

and (6)

converges for some h > 0, where (/JF/, (h) is the typical function of F(x 1-'i'),

+

XII.

506

SERIES OF INDEPENDENT RANDOM VARIABLES

If (5) holds and (6) converges for some h > 0, then they hold for every h > 0. Proof. We shall prove this theorem from Theorem 12.5.1. Suppose first that (5) holds and (6) converges for some h convergence of (6) implies that 1-tnh-+- 0 as n-+- oo. Take n0 so large that for n ~ n0 ,

> 0. The

I ~-tn I ~ hf2.

(7)

Now

We have

=

since [,ukh - (h/2), /-tkh

1

SP(I Xk(w)

+ (h/2)]

I~ h)

for k

~

c [ -h, h] from (7). Hence, from (5),

00

L P[l Xk(w) I ~ h] <

oo,

k-l

which is (8) of Section 12.5 (with h for c). Then

Ioo

-oo

x2

x2

+ h2 dFk(x + /-tkh) =

n0

Ioo -oo

~

1 5h2

~

5h1z

(x _ l-tkh)2

h2

+ (x- /-tkh)2 dFk(x)

I I

lz-p~l:ii2h

lzl:i!h

(x-

(x - /-tkh) 2 dFk(x)

~-tl)2 dFk(x)

12.8.

507

RADEMACHER AND STEINHAUS FUNCTIONS

where Xkh( w) is the truncated random variable of Xk( w ). Hence from (5), :E var Xkh(w) < oo which is (9) of Section 12.5. The convergence of (6) is nothing but (10) of Section 12.5 with h for c. Therefore, from Theorem 12.5.1, the series (4) converges almost surely. The converse is shown similarly. Suppose that (4) converges almost surely.

foo

x2

x2

-oo

+ h2 dF(x + ~kh) =

Joo -oo

(x _ ~kh)2 (x- ~kh)2 + h2 dFk(x)

say. Then 0 ~ 11 ~ h- 2

for k

~

J

lz-,..~l
(x - ~kh) 2 dFk(x)

n0 , and

Now from Theorem 12.5.1, (8)-(10) there converge for every c > 0. Hence :E var X'f' < oo, :E P[l Xk(w) I ~ h/2] < oo. Therefore from (8), :E [1 - (Pk'(h)] < oo. Condition (6) is again nothing but (10) of Section 12.5. The last statement is obvious. This completes the proof.

12.8. Rademacher and Steinhaus Functions Before continuing the study of the convergence of a series of independent random variables, we give some basic properties of a sequence of specific functions called Rademacher functions. The investigation of the convergence problem of a series of Rademacher functions made by Khinchin and Kolmogorov [1] motivated Kolmogorov to the further development of the theory of series of independent random variables. Define the function rn(t), n = 1, 2, ... , over [0, 1] by

rn(t) = { ~~n(sin 2 nt),

t =J: kf2n, t = k/2n,

0

~

t

~

1,

(1)

XII.

508

where k

SERIES OF INDEPENDENT RANDOM VARIABLES

= 0, 1, ... , 2". The function r,(t) is called a Rademacher function.

r 1(t)

r 2(t)

=

=

~)'

1

m

( 0,

-1

m

(

0

at

1 0, 2' 1,

1

tn

(o.

-1

m

( ;2 ' i2 ) ' ( i2 '

0

at

0,

~'

1)'

i2 ). ( i2 · i2·)· 1

2

1) '

3

22' 22' 22' 1'

The function r ,(t) can also be defined in the following way. Let 0 ~ t ~ 1, and form the dyadic expansion oft. If t is not dyadic rational, then t can be expanded uniquely into an infinite series

where e, = e,(t) = 0 or 1 for n = 1, 2, .... If t is dyadic rational, t can be represented in two ways:

t

= .e 1e2

•••

e,_,O 11 ··· = .e 1 e2

•• •

e,_,1 00 ···.

For t for which e,(t) is uniquely determined, and t -=1:- 0 or 1, (2)

For t for which e,(t) is expressed two ways and t

= 0 or 1, (3)

The following properties are easy to show: (a)

J:

(b)

{r, (t ),

r, 2 (t) dt n

= =

(4)

1, 1, 2, ... } is a normalized orthogonal system:

f r,(t)r,(t) dt l 0

=

{

0,, 1

m-=j:.n, m=n,

(5)

12.8.

509

RADEMACHER AND STEINHAUS FUNCTIONS

and

J:

r11 (t) dt

= 0.

(6)

Let m > n. Then r 11 (t) is constant in 2 11 intervals of equal length, which a~e denoted by 1 1 , / 2 , ••• , 1 2n. Each h (k = 1, ... , 211 ) is partitioned into 2m-n subintervals of equal length on each of which r,.,(t) is constant, and take the values 1 and -1 alternately. Hence f 1k rm ( t) dt = 0, and

f

rlll(t)rn(t) dt

=

(-1)k+ 1

lk

Therefore,

f

rlll(t) dt

=

0,

k

=

1, ... , 211 •

lk

J:

r"'(t)r,.(t) dt

J Lk

=

1

rm(t)r 11 (t) dt

=

0.

This shows (b) for m -::/=- n. The !=ase m = n, which is nothing but (4 ), is obvious. (6) is obvious too. The Rademacher system is not complete. Since the graph of r11 (t) is symmetric with respect to the point (!. 0),

J:J(t}r 11 (t} dt

0

=

for any

n > 0,

for any integrable functionf(t) such thatf(t) = q;(t- !), where q;(u) is any even function over [- !, !]. It is of interest to notice that for each t which is not dyadic rational, {r11 (t), n = 1, 2, ... } gives us a sequence of 1 and -1, in other words, it provides a sequence of signs. To choose a value oft which is not dyadic rational is equivalent to choosing a sequence of signs. So if we consider a senes (7) L a 11 r,.(t),

+

00

n-1

then we have a series of the form Ln- 1 ±a11 • Now we take [0, 1] for a probability space in which the a-field of all measurable sets is considered and the Lebesgue measure is taken to be the probability. Then each r11 (t) is thought of as a random variable. We then have the important fact that {r11 (t), n = 1, 2, ... } is a sequence of independent random variables. Each r11 (t) has a distribution function a 11 (x) and a characteristic function fn(t) given by for for for

- oo < x ~ -1, -1 < x ~ 1, 1 < x,

(8)

510

XII.

SERIES OF INDEPENDENT RANDOM VARIABLES

and

fn(t)

=

COS

t,

(9)

for all n. The proof is simple. Since rn(t) assumes the values 0, ±1, only, it suffices to verify, for independence, that

where m is the Lebesgue measure and ek = 0 or ± 1, k = 1, 2, ... , n. This can be done easily from the definition of rn(t). (8) and (9) are immediate. The series (7) was called the Bernoulli distribution function in Section 3.10. (6) and (4) are written, respectively, var rn(t) = 1.

and

(10)

Then from Theorem 12.5.2 (ii), we have:

Theorem 12.8.1. The series an

real,

(11)

converges almost everywhere if and only if

L an2 <

oo.

(12)

Note here that var anrn(t) = an 2 var an(t) = an2. This theorem was first shown by Khinchin and Kolmogorov [1]. Khinchin [2] has also shown the following theorem.

Theorem 12.8.2. For p > 1, (13) where AP and BP are constants depending only on p. This was generalized to the case in which general independent random variables are considered. This will be shown later on [Theorem 13.5.2]. More detailed properties of the series of Rademacher functions will be given in Section 13.5. Another very important sequence of functions will now be discussed.

12.8.

511

RADEMACHER AND STEINHAUS FUNCTIONS

Let t = . e1e2 e3· · · be the dyadic expansion of t, 0 < t < 1, which is uniquely determined except on a set of measure zero. (The set of dyadic rationals is countable and of measure 0.) For such t we define {On(t), n = 1, 2, ... } by 01(t) = .e1e3e& · · ·, 02(t) = . £2£5£9 ••• ,

(14)

03(t) = . £4£s£13 • • ·,

Using On(t), we define

n = 1, 2, ....

(15)

This was defined by Steinhaus [3] and is called the Steinhaus Junction. The function On(t) has the property that, for any measure setS in [0, 1],

m{t : On(t)

E

S} = mS,

n

= 1, 2, ....

(16)

+

We shall prove this for 01(t). Suppose that S = (pf2q, (p 1)/2q), 0 -;;;, p -;;;, 2q- 1, p, q being integers, p ~ 0, q ~ 1. If O(t) E S, then by definition, the first q digits are determined, that is, O(t) becomes .T1T 2 ••• Tq•• ···,where T1, T 2 ,. •• , Tq (they are ±1) are the same for every t such that O(t) E S. The original expansion oft is (17)

where the number of • (place of some ei) between Tk and For instance, if T 1 = 1, T 2 = 0, then the set of t's,

Tk+l

is k.

consists of two intervals ( 1

1

1 )

2' -z+ 23

and

In other words, it consists of two intervals of length 1/23. In a similar way, we find that the set oft's, which are represented by (17) (where T1,T 2 ,. •• ,Tq are fixed), consists of 2qlq-1112 intervals of length 1f2q
XII.

512

SERIES OF INDEPENDENT RANDOM VARIABLES

altogether there are !q(q - 1) + q places and then the length of each interval is 1f2lqlq-ll+f1.] Therefore the measure of the set oft's of the form (17) is

which is equal to the measure S. Hence we get (16) for sets of the form indicated there. Now once this is shown, it is also true for the union of disjoint such intervals. Since any interval is the union of disjoint such intervals, (16) is true for any interval s because of the continuity property of the measure. The set of S for which (16) holds constitutes a a-field, and we see that (16) continues to hold for any Borel set S. Since O(t) is a measurable (Lebesgue) function, (16) should hold for all measurable sets S. As in the case of the Rademacher function, we may show that {On(t), n = 1, 2, ... } is a sequence of independent random variables on ( [0, 1], !?, m), where .9' is the class of Lebesgue measurable sets. If

0 ~ p ~ 2q-t, 0 ~ r ~ 2•- 1 , then we see that m{0 1(t) E [a 1 , b1 ), 02 (t) E [a 2 , b2 )} and m{0 1 (t) E [a 1 , b1 ) }m{0 2 (t) E [a 2 , b2 }} are 1/2q+a and (1/2q)(1/2"), respectively, and they are equal. The proof of the general case will be left to the reader. Because of relation (16 ), the distribution function of On(t) is given by 0, { F(x) = x, 1,

x ~o.

0

<X~

1,

(18)

1 <X.

that is, the uniform distribution. Now, for the Steinhaus function 8n(t), we may show that it is a normal orthogonal system.

m -=1= n, m=n, because for m -=1= n,

I: em(t)8,.(t) dt

=

I:

(19)

exp{2ni[Om(t)- (Jn(t)]} dt

= I~ exp[2niO,,(t)] dt

I: exp[ -2ni0n(t)] dt

(20)

12.9.

513

PRODUCTS OF CHARACTERISTIC FUNCTIONS

from the independence of {On(t) },

- dt = Jl exp(Znix) dF(x) Jl exp( -Zniy) dF(y), Jl8m(t)8n(t) 0

0

0

where F(x) is given by (18), and

J 8m(t)8n(t) dt = Il exp(2:n:ix) dx Il exp( -may) dy l 0

_

0

0

= 0.

The case m = n is obvious. Here we considered the complex-valued random variables 8n(t), but the validity of (20) can also be obtained in a way similar to the case of real-valued random variables. As a matter of fact, if we write
where S;, ... , Sik is any set of two-dimensional Borel sets. This, of course, implies that each of {


Theorem 12.8.3. For the almost pure convergence of L an8,(t) (an may be complex valued), it is necessary and sufficient that

This also follows from Theorem 12.5.2 (ii). A direct proof may be found in Kaczmarz and Steinhaus [1].

12.9. Convergence of Products of Characteristic Functions Let Un(t),

n = 1, 2, ... }

(1)

be a sequence of characteristic functions. We are going to discuss the

XI I.

514

SERIES OF INDEPENDENT RANDOM VARIABLES

convergence of n

(2)

!p,.(t) = fl!k(t) k-1

as n ---+ oo. If X,. ( w ), n = 1, 2, ... , are independent random variables and J,.(t) are the characteristic functions of X,.(w), n = 1, 2, ... , then qJ,.(t) is the characteristic function of the partial sum

(3) So for the convergence of S,.( w) in distribution or for the almost sure convergence of S,.( w) (they are equivalent because of the equivalence theorem), the locally uniform convergence of qJ,.(t) is necessary and sufficient in view of Levy's continuity theorem (Theorems 9.3.4 and 9.3.7). Here it is convenient to state the following lemmas.

Lemma 12.9.1. If (2) converges to a characteristic function almost everywhere, then it converges locally uniformly and the series I: X,.( w) converges almost surely. Lemma 12.9.2. If I j,.(t) 12

---+

1, then for some m,., j,.(t)eim,.t

---+

1

locally uniformly. First we show Lemma 12.9.2. Let Xn(w) be the random variable corresponding toj,.(t). Then is the characteristicfunction of Xn(w)- X,.'(w), where X,.'(w) is a random variable independent of X,.(w) and with the same distribution function as X,.(w). Take the median m,. of X,.(w), that is, Proof.

I j,.(t) 12

P(l Y,. I~ e)

=

P(Y,. ~e)+ P(Y,. ~-e)

~

P[{X,.- m,.- (X,.'- m,.)

~

e} n {X,.'- m,.

+ P[{X,.- m,. -(X,.'- m,.) ~ -e} ~

P({X,.- m,.

~

e}

n {X,.'-

+ P({X,.- m,. ~ -e} n

n {X,.'-

m,. ~ 0})

{X,.'- m,.

~

~

0}),

0}]

m,. ~ 0}]

12.9.

515

PRODUCTS OF CHARACTERISTIC FUNCTIONS

which is, by the independence of Xn and Xn',

P(l

Yn I ~ e) = lP(Xn- mn ~ e) =

lP(I Xn-

+ lP(Xn- mn ~ e)

mn I ~ e).

Now since I fn(t) 12 ---+ 1, Yn converges to zero in probability (e.g., by Theorem 9.3.5) and hence the left-hand side P(l Yn I ~e) of the above equality converges to zero and hence Xn - mn converges to zero in probability. This shows the result.

Proof of Lemma 12.9.1. lim I~Pn(t) n-+oo

12 =

n

IT Ifk(t) 12 n-+oo k=l lim

exists for every t since n~=l I !k(t) 12 is now nonincreasing. Let IPn(t)---+ f(t) a.e., where f(t) is a characteristic function. Then I~Pn(t) 12 ---+I f(t) 12 • Hence, from the foregoing lemma, IPn(t)eim,.t converges locally uniformly for some mn. Since IPn(t) converges almost everywhere, eimnt converges to one a.e., but this implies that eim,.t converges to one everywhere. This is seen from

(The left-hand side is sin antfant = =

21 (ant)- 1 sin lant cos lant I ~ I cos !ant I [l(1

+ cos ant)]112.)

Then from Theorem 9.3.6, eim,.t converges uniformly to one. Hence IPn(t) converges locally uniformly. Then L Xn converges in distribution, which yields the almost sure convergence of L Xn by the equivalence theorem. Because we shall use it later, we shall state as a lemma what we have shown in the proof of Lemma 12.9.2.

Lemma 12.9.3. If X(w) and X'(w) are independent and

a

is a median

of X( w ), then for any x > 0,

P(l

X- X'

I ~ x)

~

lP(I X- a I ~ x).

We now show the following theorem (Kawata and U dagawa [1 ]).

XII.

516

SERIES OF INDEPENDENT RANDOM VARIABLES

Theorem 12.9.1. If Un(t), n = 1, 2, ... } is a sequence of characteristic functions, then m

lim lim

n

n-+oo m-+oo k-n+l

I!t(t) I

(4)

is either zero almost everywhere or one almost everywhere. The existence of lim lim in (4) is obvious. Write g(t) = limn-+oo limm-+oo I ft(t) I· Obviously g(t) is an even function. From the elementary result concerning infinite products, we see that g(t) is 0 or 1 for each t. Now,

Proof.

nr=n+l

is a characteristic function. Therefore, from Theorem 3.7.1 (iv), we have, for any h > 0 and for any t,

Letting m

---+

oo, n ---+ oo, we have

(5)

If g(t) = 1 except on a finite set, then g(t) = 1 a.e. Next suppose that g(t) = 0 on an infinite set.

Case (i). Suppose that there is a sequence h.(>O) such that h.---+ 0, g(h.) = 0, 'II= 1, 2, .... If there were a set E with mE > 0, such thatg(t) = 1 onE andg(t) = 0 on mEc, then by the Lebesgue density theorem, almost every point of E should be a point of density of E; in other words, taking a point t 0 of density of E (t 0 E E), m{(t0

-

h., t0 +h.) n E}f2h.

as

---+1

v-+ oo.

On the other hand,

m{(t0

-

h., t 0 +h.) n E}f2h.

= ~

(2h.)- 1 {[1

f

lo+hv

lo-hv

g 2 (u) du

+ g2(h.)]f2}112 =

by (5). This contradicts (6). Hence we have g(t)

= 0 a.e.

(1/2)112

(6)

12.9.

517

PRODUCTS OF CHARACTERISTIC FUNCTIONS

Case (ii). Suppose that the origin is not the limiting point of {t : g(t)

= 0}. If {t :g(t) = 0} has no limiting point, then g(t) = 1 a.e. Next suppose that there is an h0 such that

h0 = inf S > 0,

(7)

where S is a set of point t with the property that t > 0, g(t) = 0 and t is not an isolated point of {t : g(t) = 0 }. Then there is an h' such that

.y2 >

h'Jh0 > 1,

g2(t) dt

~

g(h') = 0.

We then see that

1= 2io

rh

h'

1

h0

2h

f

h'

= - --,

-

-h'

h' ( 1 )1/2

-h 2 0

zio

rh

h' g 2 (t> dt ~h0

g2(t) dt

u[l + g 2 (h')]}1' 2

(by (5))

<1.

This is again a contradiction. That is, there is no such h 0 satisfying (7). Altogether we have shown that g(t) = 0 a.e. or g(t) = 1 a.e. From Theorem 12.9.1, we may show the following theorem (Loeve [1], p. 197).

Theorem 12.9.2. Let Un(t)} be a sequence of characteristic functions. Suppose that, for
(8)

a.e.

*

Then either (i)
From the foregoing theorem,

Ig(t) 12 =

m

lim lim n-+oo m-+oo

TI 1/k(t) 12 k-n+1

is 0 a.e. or 1 a.e. In the case g(t) = 0 a.e., 00

00

TI 1/k(t) 12 ~ k-n+1 TI 1/k(t) 12 ~g(t) = k-l Hence
= 0 a.e.

0 a.e.

XII.

518

SERIES OF INDEPENDENT RANDOM VARIABLES

Suppose next that I g(t) 12 = 1 a.e. Since
and we may write 00

I gn(t) 12 = k=n+l TI Ifk(t) 12• Again by Corollary 9.3.1, I gn(t) 12 = J:'oo eixt dGn(x) a.e. for some Gn(x) E M. I
with some distribution function Fn(x). Then, from · I gn(t) 12 , we have

11f'(t) 12

=

f~oo

eixl

dFn(.1C) f~oo

eixl

I


I

(/Jn(t)

12

(9)

dGn(x)

for every t, since the left-hand side and the both of the right-hand side integrals are continuous. Now I g(t) 12 = 1 a.e. and I gn(t) 12 --+ 1 a.e. Then I gn(t) 12 converges to 1 everywhere. This is seen from (5) with I gn(t) 12 for g 2 (t). Hence I gn(O) 12 --+ 1 or Gn(oo)- Gn( -oo)--+ 0. Taking t = 0 in (9),

11f'(O) 12 = [Fn(oo)- Fn(-oo)][Gn(oo)- Gn(-oo)] = [Gn(oo)- Gn(-oo)].

Letting n--+ oo 11f'(O) 12 = 1, which is the same as 1p(O) pletes the proof.

=

1. This com-

We shall now discuss the average of the product of characteristic functions. For a characteristic function f(t), or more generally for a Fourier-Stieltjes transform f(t) of a function belonging to M, lim (2T}-1 T-+oo

JT

-T

f(t) dt

(10)

12.10.

519

UNCONDITIONAL CONVERGENCE

exists and is F(O+)- F(O- ), F(x) being the corresponding distribution function or a function belonging to M. (10) is now denoted by 1[/]. We now show (Kawata and Udagawa [1]) the following theorem.

Theorem 12.9.3. Let Un(t)} be a sequence of characteristic functions. Suppose that n (11) lim f1 ft(t) = tp(t) a.e. n-+oo k=l

Then

lim.L[

n-+oo

TI lA 1

2]

1(11P 12 )·

=

k-1

(12)

(12) means that both sides of (12) exist and are equal to each other.

Proof. Since I~Pn(t) 12 = n~=l I fk(t) verges decreasingly to I~P(t) 12 a.e.,

12 ,

where IPn(t)

=

n~=lft(t), con-

(13) for every n Let

~

1.

be the mean concentration function iff is a characteristic function or a Fourier-Stieltjes transform of a function belonging to M. As in the proof of Theorem 12.9.2., tp(t) can be written as a FourierStieltjes transform of some function belonging to M. From Theorem 11.8.1 (iii), Theorem 11.8.2, and (13), we have

C(h, IPn) ~lim C(h, IPn) h-+o+

Since C(h, IPn)

---+

=

1[11Pn 12 ] ~ 1[11P 12 ].

C(h, tp) for every h > 0, we have, letting n

n-+oo

---+

oo,

n-+oo

Since C(h, tp) converges to L[IIP 12 ] as h---+ 0+, we get the theorem.

12.10. Unconditional Convergence Let us take a series of random variables (1)

520

XII.

SERIES OF INDEPENDENT RANDOM VARIABLES

Let {Pn, n = 1, 2, ... } be a rearrangement of the set of positive integers (1, 2, 3, ... ), that is, every integer n ~ 1 appears once and only once in {Pn}. The series

(2) is called a series obtained by a rearrangement of (1) or briefly a rearrangement of (1). If (2) is convergent for every rearrangement {Pn} of positive integers, then (1) is said to be unconditionally convergent. It is an elementary fact that for a series of numbers, the notions of unconditional convergence and of absolute convergence are equivalent to each other. However, for a series of functions or of random variables, they are not identical. Let {rn(t)} be the sequence of Rademacher functions and (3)

be a Rademacher series. In (3 ), choose {an} so that (4)

(5) an being real. By Theorem 12.8.1, (3) converges almost everywhere. If {Pn} is a rearrangement of positive integers, then L a~ .. < oo. Therefore, again by Theorem 12.8.1, L:=I ap,.rp,.(t) converges almost everywhere. Hence the series (3) is unconditionally convergent almost everywhere. But (3) does not converge absolutely because I rn(t) I = 1 almost everywhere and

Needless to say, this may happen since the exceptional sets oft for the convergence of series may depend on rearrangements of the series and the number of rearrangements is more than countable. We begin with the following theorem (Wintner [3], p. 11 and Kawata and Udagawa [1]).

Theorem 12.10.1. Let (6)

12.10.

521

UNCONDITIONAL CONVERGENCE

be a series of independent random variables which converges almost surely to S(w). If (2) is a rearrangement of (6) and converges almost surely to S 1 (w), and F(x) and F 1(x) are the ·distribution functions of S(w) and S 1 (w), respectively, then

(7) for some constant a.

Proof. Let the characteristic functions of Xn(w), S(w), and S 1 (w), be fn(t), f(t), and g(t), respectively. {p1 , p 2 , ••• } is the rearrangement of the sequence of positive integers that we shall consider. Take {p 1 , ••• , pt}. Then this is contained in {1, 2, ... , m(k)}. Write (8) where hk(t) is some characteristic function. We then have

"" Jj(t) 12 ~ I n Jj(t) 12 ~ I n JPi
Letting k

-+

m(k)

k

J-1

J-1

oo, we have (9)

Similarly, since {1, 2, ... } is a rearrangement of {p 1 , p 2 , ••• }, we have (10) (9) and (10) give us

lf(t) 12 =I c(t) 12 •

(11)

Since j(t) and g(t) are characteristic functions, there is an TJ such that for I t I < TJ, If(t) I > 0 and I g(t) I > 0. Hence letting k -+ oo in (8), hk(t) converges to some function h(t) for I t I < TJ uniformly and

j(t) = g(t)h(t)

for

It I <

TJ

and, from (11), 1 h(t) 1 =

Now write

1

for

It I <

TJ·

(12)

522

XII.

SERIES OF INDEPENDENT RANDOM VARIABLES

Choose a subsequence {Hnk(x)} of {Hk(x)} so that Hnk(x) converges weakly to a nondecreasing function H(x) using the Helly theorem. Then, by Theorem 9.6.1, hnk(t) converges locally uniformly to h•(t) and H(x) is a distribution function whose characteristic function is h•(t). Obviously h(t) = h•(t) in I t I < 'YJ and from (12) I h•(t) I = 1 for I t I < 'YJ· Hence from Corollary 3.6.2, h•(t) = ei.at for some a. Write nk fork in (8) and let k-+ oo. We then have

f(t) = g(t)h•(t)

for

- oo

< t < oo.

That is,

f(t) = g(t)ei.at, which is equivalent to F(x) = F 1(x- a). This proves the theorem. Because of this theorem, a convergent series obtained by a rearrangement of a convergent series of independent random variables has a sum whose distribution function is generally different from that of the sum of the given series. But if all rearrangements are convergent, that is, the given series is unconditionally convergent, then the situation becomes different and actually we have the following theorem (Wintner [3] and Levy [2]).

Theorem 12.10.2. If the series (1) of independent random variables is unconditionally convergent, then all the distribution functions of the series obtained by rearrangement of ( 1) are identical. Proof. Let fn(t) be the characteristic function of Xn( w ). It is sufficient to show that for each t, every sequence n~=dpk(t) converges to the same limit as n -+ oo, where {h, k = 1, 2, ... } is any rearrangement of {k, k = 1, 2, ... }. Suppose that ll~dn(t) :;t=O for a particular t. For such t, ll::"=dp,.(t) :;t=O. This is a consequence of

which is just relation (11) in the proof of the foregoing theorem. Now we use the well-known result on an infinite product, that an unconditionally convergent infinite product n;:o an is absolutely convergent; that is, n;:o (1 - 11 - an 1) is convergent and all values of the products are the same. (See for example, Knopp [1].) In view of this theorem, for the t being considered, ll~dp,.(t) has the same value for all rearrangements {Pn}·

12.10.

523

UNCONDITIONAL CONVERGENCE

If for some t, ll::':dn(t) = 0, and fn.(t) = 0 for some n 0 , then n::'=dp,.(t) = 0 for any rearrangement. Finally, suppose ll::':dn(t) = 0 for some t, but fn(t) =I= 0, n = 1, 2, .. , for the same t. Then for large N, I n~=dn(t) I < E for a given E > 0. Now let {Pn} be an arrangement of positive integers and suppose that {p 1,p2, ... ,pn,lN>} contains {1, 2, ... , N}. Then for the t being considered, I N m!Nl /Pn(t) 1 ~ fn(t) < E.

I IX

Jl

I

Therefore, n:::dp,.(t) = 0. Altogether, we have the required conclusion. We now give a necessary and sufficient condition for the unconditional convergence of (1) which is derived from the three series theorem (Theorem 12.5.1 ). This was given by van Kampen [1 ].

Theorem 12.10.3. A series L Xn( w) of independent random variables is unconditionally almost surely convergent if and only if the following three series converge for some c > 0, 00

L P(l xn I ~ c),

(13)

n-1

00

L var xnc,

(14)

n-1 and 00

L I wxncl.

(15)

n-1

If these three series converge for some c, then they do for every c > 0. Before proving the theorem, we remark that the condition that (14) be convergent, can be replaced by the convergence of (16)

This is seen in the following way. We have 00

00

00

n-1

n=1

n=l

L W(Xnc)2 = L var xnc + L (WXnC)2,

where

XII.

524 since

I g"xnc I ~ c.

SERIES OF INDEPENDENT RANDOM VARIABLES

Therefore, 00

00

n-1

n-1

L W(Xnc) 2 :::;;: L var xnc + c L I g"xnc I· Hence the convergence of (14) and (15) implies the convergence of (16). Conversely, from 00

00

n-1

n-1

L var xnc ~ L

W(Xnc)2,

it is evident that the convergence of (16) implies the convergence of (14). Proof of Theorem 12.10.3. The proof is quite simple because the necessary and sufficient condition for unconditional convergence is the convergence of (8)-(10) of 8ection 12.5 for all rearrangements of :E Xn. But the convergence of (8) and (9) of Section 12.5 is not altered by changes in the order of the terms since they are nonnegative. The series (8) and (9) of Section 12.5 are (13) and (14), respectively. The convergence of series (10) of Section 12.5 is equivalent for all arrangements to the absolute convergence of :E wxnc, which is the convergence of (15). The last statement is immediate. In the same way, from Theorem 12.6.1, we have the following theorem.

:E Xn( w) of independent random variables is unconditionally almost surely convergent if, for some p, 1 < p ~ 2,

Theorem 12.10.4. A series

00

L I g"Xn I < n=1

00

(17)

and

Finally we give an immediate consequence of Theorem 12.5.2.

Theorem 12.10.5. Let (Xn, n = 1, 2, ... } be a sequence of independent random variables with EXn = 0. (i) If :E::': 1 var Xn < oo, then :E::': 1 Xn is unconditionally convergent. (ii) If I Xn I ~ K, K being a constant independent of n, then :E::': 1 var Xn < oo is necessary for the unconditional convergence of ~1Xn.

12.10.

525

UNCONDITIONAL CONVERGENCE

The same reasoning gives us the following theorem from Theorem 12.7.1. Let C/>p.(n) be the typical function of Fn'(x) = Fn(x + 1-'n"), where Fn(x) is the distribution function of Xn(w) and f.tn" = 'i!'Xn"·

Theorem 12.10.6. A series L Xn(w) of independent random variables is unconditionally almost surely convergent if and only if 00

L [1 -

C/>F~(c)]

< oo

(18)

n-1

and (19)

for some c > 0. If (18) and (19) conv--erge for some c > 0, then they do so for any c > 0. The following theorem given by Loeve [1], p. 539, is shown by means of a result on infinite products used in the proof of Theorem 12.10.2. A series L~ 1 Xn(w) of independent random variables converges unconditionally almost surely if and only if L~ 1 I fn(t)- 1 I converges on a set of positive measure.

Theorem 12.10.7.

Proof.

Suppose that 00

L lfn(t)- 11

(20)

n-1

converges on a set of positive measure. Then the product TI~dn(t) is unconditionally convergent on a set of positive measure [since (20) is independent of the arrangement of {fn(t) }]. Here the convergence is in the sense of the infinite product and the limit is not zero. Hence, from Theorem 12.9.2, TI~:dn(t) converges almost everywhere for any arrangement to a characteristic function. This shows that L Xn( w) converges for any arrangement almost everywhere in view of Lemma 12.9.1. Next suppose L Xn( w) converges unconditionally almost surely. Then TI~dn(t) converges unconditionally for every t. Hence from a theorem on infinite products used in the proof of Theorem 12.10.2, L 11 - fn(t) I < oo for every t. The proof of the following theorem due to Karlin [1] will be found in Section 13.8 after Theorem 13.8.2.

XII.

526

SERIES OF INDEPENDENT RANDOM VARIABLES

Theorem 12.10.8. Let {Xn, n = 1, 2, ... } be independent and wxn = 0, Xn(w) E LP, p ~ 1, n = 1, 2, .... If L:,l Xn(w) converges almost surely to a random variableS( w) and S( w) E LP, then I;:,1 Xn( w) is unconditionally convergent.

12.11. Absolute Convergence Here we shall discuss the absolute convergence of a series of independent random variables. We begin with the following theorem, which is a simple consequence of the three series theorem (van Kampen [1]).

Theorem 12.11.1. Let (1)

be a series of independent random variables. This series converges absolutely almost surely if and only if the following two series converge for some c > 0: (2) LP(IXnl
IWI Xnl

(3)

0•

If (2) and (3) converge for some c > 0, then they do so for every c > 0.

Proof. The sufficiency of the convergence of (2) and (3) will be obvious from the three series theorem if

But this is a conse<)uence of the convergence of (3) for varl Xn lc ~

I

izl:iic

~

C

'if' I Xn lc,

x 2 dFn(x) ~ c

I

lzl:;ic

I X I dFn(x)

where Fn(x) is the distribution function of Xn(w). Conversely, if L I Xn(w) I < oo almost surely, then, from the three series theorem, (2) and (3) follow for every c > 0. The last statement is obvious. We shall now show that the conditions in Theorem 12.6.2 are also sufficient for absolute convergence (van Kampen [1 ]).

12.11.

527

ABSOLUTE CONVERGENCE

Theorem 12.11.2. The series (1) of independent random variables converges absolutely almost surely if, for some p, 0 < p ~ 1, (4)

Proof. This is a consequence of Theorem 12.11.1. First, (4) implies the convergence of (2) because

(4) also implies the convergence of (3). For,

;;;; cP-1

J

IX,.(w)l
I Xn I dP

(since 0 < p

~

1).

That is,

We shall now give a theorem of a different type.

Theorem 12.11.3. Let {Xn(w), n = 1, 2, ... } be a sequence of random variables with Xn(w) eLl, and suppose that the origin is a median of Xn(w). If 1 Xn(w) converges in V to S(w), and S(w) is bounded above (or below), then 1 Xn(w) is absolutely convergent almost surely.

:E::

Proof.

:E::

Define

n = 1, 2, 3, ....

(5)

Namely, if Xn(w} = 0 if Xn(w} =I= 0. {Yn(w)} then constitutes a sequence of independent random variables. Yn(w) assumes three values: 0, -1, and 1, and is symmetric since P(Yn = 1) = P(Xn(w) > 0) = P(Xn(w) < 0) = P(Yn = -1) (since 0 is a median of Xn)· Also we see that g' Y n = 0.

XII.

528

SERIES OF INDEPENDENT RANDOM VARIABLES

Suppose that S(w) < K.

(6)

Now since we have

~~~~ WSnYk = ~~~ w( jtt XjYk) = ~~~ jt. W(XiYk)·

WSYk =

If j oj::. k, Xj and Yk are independent and W(XjYd = 'if'Xj WYk = 0 since go Yk = 0. Therefore,

(7)

and we have n

L ;-.qxjYkYI) =

6'SYkYI =lim ll__.,OO

c-f(XkYkYI)

+ ;-qxiYkYt)

j=l

(8)

fork oj::. I. More generally, we have ifSYkYI · · · Y 111

= 0

(9)

for a distinct set (k, I, . .. , m} because II

=

L ef(Xj Yk Yr · · Ym) w I xk I yl . .. Y,, +
=

w

WSYk Y 1 •

• •

Y,, = lim

n---+oo j=l

1

Ill+ go

xk 1 (:fyl ... tify

1

xl

1

eryk ... ery"'

+ ... = o.

From (7)-(9), WS

fl

(1

+ Yj) =

IJ"S( 1

+ jtt Yj + j~ YjYk +);
=

WS+

L ?5'1 k-l

n

(10)

Xkl•

n = 1, 2, ... , where it is noted that W S is finite because Sis finite as a limit in £1 of Sn and S E £1. On the other hand, since 1 + Yk ~ 0, we have, using (6) and the independence of Y 1 , Y 2 , ••• , »

ws TI (1 + j=l

YJ ~ K ?8

n

TI (1 + j=l

n

YJ = K

TI 18(1 + j=l

Yj) = K,

12.11.

529

ABSOLUTE CONVERGENCE

that is, II

ff' S

0

(1

j-1

+ Yi)

~

K.

(11)

(10) and (11) give us n

wk-1 L xk 1

1

ws <

~ K -

oo.

Since this is true for every n, w :Lt:1 I xk I < oo, which implies the almost sure convergence of :Lk: 1I Xk I· When S(w) is bounded below, we may consider -S(w) in place of S( w ), which then reduces to the above case in which S( w) is bounded above. This completes the proof.

Corollary 12.11.1. If the partial sum S 11 (w) = I:~- 1 Xk of independent bounded random variables converges almost surely to a bounded random variable S(w) and the origin is a median of X(w), then :L Xn(w) converges absolutely almost surely. Also, we have the following theorem.

Theorem 12.11.4. Let {X11 (w), n = 1, 2, ... } be a sequence of independent random variables. Suppose that

I Xn I ~an<

a.s.,

oo

a 11

>0, n=1,2, ... ,

(12)

n = 1, 2, ... ,

(13)

WXn= 0,

and 00

L var X < 11

(14)

oo.

n-1

Let S be the sum I:;;'= 1 X 11 ( w) (which converges a.s. because of Theorem 12.5.2). If Sis bounded above (or below), then 00

L (a11 )-1 var X <

oo.

(15)

n = 1, 2, ... ,

(16)

11

n-1

In particular, if

for some c > 0, then

I: I X

11

(w) I < oo a.s.

Proof. Note first that (13) and (14) imply that I:~- 1 Xk(w) converge in £2 to S(w).

530

XI I.

SERIES OF INDEPENDENT RANDOM VARIABLES

Let Yk = Xkfak> k = 1, 2, .... We then see as before, noting that WS = 0, that (17)

On the other hand, using a.s., then

I Yk I ~

1 a.s., we see as before that if S < K

w[ S J] (1 + YJ] ~ K ka W(1 + Yk) (17) and (18) give us (15). The case S case.

= K.

(18)

> -K is reduced to the above

In a similar way we may show the following result.

Theorem 12.11.5. Let {Xn(w), n = 1, 2, ... } be a sequence of independent random variables. Suppose (12) and (13). If there is a subsequence {Snt(w), k = 1, 2, ... } of {Sn(w)}, Sn(w) = L~- 1 Xk(w) such that or fork= 1, 2, ... , where K is a constant independent of k, then (15) holds. In particular, if (16) holds, then :L Xn(w) is absolutely convergent almost surely. In this theorem, we do not have to assume in advance either ( 14) or the convergence in L 2 of the series. Proof. Let us take any n, and choose nk so large that nk ~ n. In place of the expectation of the product on the left-hand side of ( 17), consider

and proceed in just the same way as in the proof of Theorem 12.11.5, which gives us the same conclusion.

12.12. Essential Convergence For a given series of random variables, (1)

12.12.

531

ESSENTIAL CONVERGENCE

if, for some sequence {an} of constants, 00

L

[Xn(w)- an]

(2)

n-1

converges almost surely, then the series (1) is said to converge essentially. Because of the equivalence theorem, the series (1) of independent random variables is convergent essentially if and only

n fn(t) e-ianl N

lim

(3)

N-+oo n-1

exists locally uniformly for some {an}· Let us write (4)

where Xn' is independent of Xn and has the same distribution function as Xn. The variable Xns is sometimes called the symmetrized random variable of Xn.

A basic theorem on essential convergence is the following:

Theorem 12.12.1. The series (1) with independent {Xn} is essentially convergent if and only if L~ 1 Xn 8(w) is convergent almost surely. Proof. This is an immediate consequence of Lemma 12.9.2. Let fn(t) be the characteristic function of Xn(w). Then Xn 8 (w) has the characteristic function I fn(t) 12• Therefore, if L Xn"( w) converges almost surely, 1fn(t) 1 converges as n - oo and limn ....."" n~n 1fk(t) 1 = 1 then (Theorem 12.4.1). Then, from Lemma 12.9.2, nr=nfk(t)eiA,.t---+- 1 for some sequence {A,} of constants, which is equivalent to the result that L Xn is essentially convergent. The converse is obvious, since the con1fn(t) 1 vergence of n:=dn(t)eia,.e implies that of

n:=l

2

2

n:=l

2.

Theorem 12.12.1 can be also stated in the following way:

Theorem 12.12.2. For the essential convergence of (1) with independent {Xn}, it is necessary and sufficient that (5)

as m ---+- oo, n ---+- oo almost everywhere.

XII.

532

SERIES OF INDEPENDENT RANDOM VARIABLES

We have only to notice that the almost everywhere convergence can be replaced by locally uniform convergence, because of Lemma 12.9.1 applied to Til fk(t) 12 • Let Qm.n(h) be the concentration function (Section 11.8) of :Lr=n+l Xk(w), Xn being independent random variables. Levy has shown that Q(h) = lim lim Qm, 11 (h) (6) n-+oo m-+oo

exists for every h > 0, and we have the following important result (Levy [2, 3] and Tucker [1, 2]).

Theorem 12.12.3. For a series (1) of independent random variables, Q(h), defined by (6), is either identically one or identically zero for all h > 0. If Q(h) = 1 identically, then (1) is essentially convergent, and if Q(h) = 0 identically, then, for any sequence {a 11 , n = 1, 2, ... } of constants, :L;'=1(Xn - an) diverges almost surely. A similar theorem holds for the mean concentration function (Section 11.8 anci Kawata [4]) and as a matter of fact, we may show that the above two theorems are equivalent to each other. Let Cm,n(h), h > 0, m > n, be the mean concentration function. Then

cm,n(h) = (2/n)

I~ kft I f.t(t) 12(sin2htfht2 ) dt,

(7)

where fk( t) is the characteristic function of Xk( w ). It is obvious that cm,n(h) does not increase as m increases and does not decrease as n increases. Hence lim lim cm,n(h) = C(h) (8) n-+oo

m-+oo

exists for every h > 0.

Theorem 12.12.4. Let {X 11 ( w ), n

= 1, 2, ... } be a sequence of independent random variables with characteristic function fn(t). Then C(h) defined by (8) is either indentically one or identically zero for every h > 0. If C(h) is identically one, the series :L:: 1 Xn( w) is essentially convergent and if C(h) is identically zero, then, for any sequence {an, n = 1, 2, ... }, :L::1(Xn - an) is divergent almost surely.

12.12.

ESSENTIAL CONVERGENCE

533

Proof. From Theorem 12.9.1, the first statement of the theorem is obvious. Actually, if TIZ':n+l I ft(t) I - 1 almost everywhere, then C(h) == 1. If Tik'=n+l I ft(t) I - 0 almost everywhere, then C(h) == 0. If C(h) = 1, then this is the case when Tik'=n+l I ft(t) I - 1 a. e. and the conclusion follows from Theorem 12.12.2. The case C(h) = 0 corresponds to the case Tit=n+l I fk(t) I -0 a.e. and this gives us Tit=n+ 1 /t(t)eia,.t- 0 a.e. Hence L::'= 1 (X 11 - an) never converges almost surely, by Theorem 12.4.1. This completes the proof. We now show that Theorems 12.12.3 and 12.12.4 are equivalent to each other. From Section 11.8, Eq. (5),

(9) We have shown in the proof of Theorem 12.12.4 that we have two possibilities: (i) that Cm,n(h)-+ 1 as m-oo, n- oo for every h > 0 or (ii) that cm,n(h)- 0 as m-oo, n - 00 for every h > 0. In case (i), from the right-hand side of (9), Qm,n(h)-+ 1 as m- oo, n- oo for every h > 0 (note that Qm,n(h) ~ 1). In case (ii), from the left-hand side of (9), Qm,n(h)- 0 for all h > 0. This shows that there are also only two possibilities: that limn-+oo limm-+oo Qm,n(h) exists and is identically 1 or that limn-+oo limm-+oo Qm,n(h) exists and is 0 for every h > 0. This consideration gives us the equivalence of the above two theorems. The following corollary, which was given by van Kampen [1 ], is substantially implied by the foregoing theorem.

Corollary 12.12.1. Let Sn(w) be the nth partial sum of L:'= 1 X 11 (w) of independent random variables. If there is a subsequence {snk( w)} of partial sums that converges almost surely, then L:'= 1 X 11 ( w) is essentially convergent. Consider

Then this must converge to one for every h > 0. On the other hand, as we have seen, C(h) always exists and therefore C(h) must be one. The corollary then follows from Theorem 12.12.4. We must discuss the LP-convergence of series of independent random variables, but since we need some knowledge of the properties of the finite sums of independent random variables, we shall do this in Section 13.8.

XII.

534

SERIES OF INDEPENDENT RANDOM VARIABLES

NOTES 1. For more about the convergence of a sequence of random variables see, for example. Loeve [1]. 2. Concerning almost sure convergence, we mention the following theorem. X,.(w)--. X(w) a.s. if and only if P(l X,.- X I ;;;;; e for all n ;;;:; m)--. 1 as m--. oo or, equivalently, P(l X,.- X I > e for some n;;;:; m)--. 0 as m--. oo (see Chung [1], p. 62).

3.

A partial generalization of the zero-one law is found in Kolmogorov [2], p. 60.

4. In connection with the proof of necessity in Theorem 12.5.1 (10) (three series theorem), Chow has shown that if {X,.} is a sequence of independent random variables and E sup,.;;: 1 JX,.J < oo, and l:: X,. converges almost surely, then l:::,, EX,. converges (Chow [1]).

5. If, in defining the formula of a typical function tPF(h), Ref(t) is replaced by 1/(t) 1", then we get IJI(h)

=

h J~ e-ltl lf

1· dt =

r:

[h"f<x·

+ h')] df.'(x).

where f.'(x) is F(x) • 1 - F( -x). Using IJI(h) and tP(h), Kunisawa developed the theory of the limit theorems (Kunisawa [1, 2]).

6. Theorem 12.7.1 was proved by Kunisawa [1] without appealing to Theorem 12.5.1; and, actually, he derived Theorem 12.5.1 from Theorem 12.7.1. 7. Direct proofs of Theorems 12.8.1 and 12.8.2 were given by Kaczmarz and Steinhaus [1]. 8. The proof of Theorem 12.11.3 is an analog of the proof of a theorem known as Sidon's theorem: Let {nrc, k = 1, 2, ... } be a sequence of positive integers such that nrc+lfnrc ;;;:; (J > 1, where (J is a constant independent of k. If l:: (arc cos nrcX + bu sin nrcX) is the Fourier series of a bounded function, then l:: < oo (Sidon [1] and Zygmund [5], Vol. 1, p. 247). 9. A direct proof of Theorem 12.11.3 is possible. It was given by Y. S. Chow (not published). Substantially, Theorem 12.11.4 is due to van Kampen [1].

10. The method of the proof of Theorem 12.11.4 enables us to show the following result: Let gm,n be the a-field generated by Xm+ 1,. .. , x .. , n;;;:; m, where {X,.} is a sequence of independent random variables. Let gm be the a-field generated by Xm. Suppose (12) and (13) in Theorem 12.11.4. If there is a subsequence S,.l:(w) of partial sums of l:: X,.(w) such that S,.k(w) is uniformly bounded a.s. on some set B of gm,l• for some fixed m and l (l > m) with P(B) > 0, then (15) of Section 12.11 holds. In particular, if (16) of Section 12.11 holds, then l:: X,.(w) is absolutely convergent almost surely. 11. A product similar to that in the proof of Theorem 12.11.4, where we have a sequence of trigonometric functions with gaps instead of Y 1 , was introduced first by Riesz [2], and is called the Riesz product (see also Keogh [1]).

NOTES

535

12. Concerning essential convergence, van Kampen [1] has shown that if I: Xn is a series of independent random variables and there is a sequence {bn, n = 1, 2, ... } such that {Sn - bn} is uniformly bounded on a set C with P(C) > 0, Sn being the partial sums, then I: X,. converges essentially. He also showed that if, for every e > 0, there exist a constant M = M(C) and the sets Cm with P(Cm) > 1 - e, such that I Sm I ;:;;; M on Cm, m = 1, 2, ... , then I: Xn is essentially convergent.

XIII Properties of Sums of Independent Random Variables; Convergence of Series in the Mean

13.1. Continuity and Discontinuity Properties of the Sum of a Series Let {Xn(c.o), n = 1, 2, ... } be a sequence of independent random variables. Suppose that 00

S(c.o) =

L Xn(w)

(1)

n=l

converges almost surely. We shall study mostly the continuity or discontinuity properties of S( w ). Let P(F) be the point spectrum of a distribution function F(x). If P(F) is empty, then of course F(x) is continuous for all x. Let us begin by recalling that if F 1 (x) and F 2 (x) are two distribution functions, then the point spectrum of F 1 (x) • F 2 (x) is the vector sum P(F1 ) ( +) P(F2 ) of the point spectra of F 1 and F 2 (see Section 3.9); in other words, if 536

13.1.

CONTINUITY AND DISCONTINUITY PROPERTIES

537

+

P(F1 ) = {a 1 , a 2 , ••• }, P(F2 ) = {b 1 , b2 , ••• } then P(F1 • F 2 ) = {ai b1 , i, j = 1, 2, ... }. Furthermore, if F 1(ai 0) - F 1(ai - 0) = Pi, F 2(bi 0) - F(bi - 0) = qi (Pi, qi are the saltus of F 1 and F 2 at ai and bi, respectively), then the saltus of P(F1 • F 2 ) at ck is

+

+

(2) where Li,i,a;+b;-ck indicates the sum regarding all pairs (i,j) such that ai bi = c", ck being a point of P(F1 • F 2 ). This implies that if one of F 1(x) is continuous (for all x), then (F 1 • F 2 ) is continuous. Let the point spectrum P(F) be {x., v = 1, 2, ... } and p. be the saltus at x. of F(x). Then the point spectrum of F(x)•[l - F( -x)] is {x,. - x., fl, v = 1, 2, ... } and the saltus at x,. - x. is p,.p., as is easily seen. In particular, the origin is a point of the point spectrum and the saltus at the origin is given by :L p. 2 • If f(t) is the characteristic function of F(x), then as was seen before (see Section 3.8) or directly from the above,

+

(3) Suppose that we are given ( 1 ), the series of independent random variables that converges almost surely. Let F,(x) be the distribution function of X,(w), n = 1, 2, .... Let {xlnl, x4nl, ... } be the point spectrum of F,(x) and p~nl be the saltus at x;nl, v = 1, 2, .... Define p(nl = max p(nl. (4)

•

•

We then have, with these notations, the following theorem.

Theorem 13.1.1. (i) If one of F,(x) n

=

1, 2, ... , is continuous then

S( w) has a continuous distribution function.

(ii) If every Fn(x) has discontinuities, then the necessary and sufficient condition for the distribution function of S( w) to be continuous, is that 00

IlP(n)

(5)

n-1

diverge to zero. This theorem is due to Levy [2]. (i) is obvious and (ii) of this theorem is equivalent to the theorem now given.

538

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Theorem 13.1.2. Let fn(t) be the characteristic function of Fn(x), n = 1, 2, .... The necessary and sufficient condition for the distribution function of S( w) to be continuous, is that (6) diverge to zero (Kawata and Udagawa [1]). Here .L[f] = limT-+oo (2T)-l r:.TJ(t) dt. The equivalence of the previous two theorems is immediate from the following inequality relations .

.L[Ifn 12] = L [p~nl] 2

•

(p
by

L p~n) ~ p
~

p
~

L [p~nl]2 =

(3) 1

.L(I fn 12) .

•

The proof will be given after the following theorem.

Theorem 13.1.3. If f~c(t), k = 1, 2, ... , n, are characteristic functions of X"' which are independent, then (7)

Proof It is sufficient to show the case n = 2. I f~c(t) 12 is the characteristic function of F~c8 (x) = F~c(x) • [1 - F~c( -x + 0)], which is symmetric, k = 1, 2. Hence the point spectrum of F~c8 is {0, y~" 1 , -y~" 1 , 1-' = 1, 2, ... }, y,. > 0; that is, points of P(F~c8 ) are located symmetrically with respect to the origin. Hence we may write

say, where q~" 1 (!-' =I= 0) is the saltus of Fk8 (x) at y:f'', q~kl is the saltus of Fk8 (x) at the origin, and Gk(x) is some continuous function belonging to M 0 , k = 1, 2, .... Then

where 11 (t)] 2 (t) is the Fourier-Stieltjes transform of the convolution of two functions belonging to M 0 , one of which is continuous. Hence by

13 .1.

539

CONTINUITY AND DISCONTINUITY PROPERTIES

the statement below (2), .L(/tf2) = .L(]tf2) = 0.

=

0. By the same reasoning, .L(/2]

1)

Now

11(t)l2(t)

=

[q~11 + 2 L q~11 cos(y~11 t)] ''*0

. Since .L[cos(ylflt)] = 0, y~ 2 l > 0, we see that

[q~2l

+ 2 L q~2l cos(y!2lt)] . •*0

.L[cos(y~llt) cos(y~ 2 lt)]

=

0 for

y~lel

> 0,

Now we go back to the proof of Theorem 13.1.2, which is almost immediate.

Proof of Theorem 13.1.2. Letf(t) be the characteristic function of S(w). From Theorem 12.9.3 and Theorem 13.1.3,

n

= lim n-+oo

IT .L[I ire 121. le-1

(9)

The continuity of the distribution function of S( w) is equivalent to .L(I/1 2) = 0 and hence the theorem is obtained from (9). We shall show another theorem given by Levy [2].

Theorem 13.1.4. If the distribution function of the sum S( w) of the series (1) of independent random variables has discontinuities, then there is a sequence {an, n = 1, 2, ... } of constants such that 00

L P[Xn(w) ::;t: an] <ex>.

(10)

n-1

Proof. Let f(t),fn(t) be the characteristic functions of S(w), Xn(w), respectively. If the distribution function of S( w) has discontinuities, then .L[I/1 2] > 0. Therefore from Theorem 13.1.3, the infinite product .L[I In 12] ( > 0) converges. Then the symmetrized random variable [Section 12.12, Eq. (4)] X n8 ( w) of X n ( w) has the characteristic func-

n::1

540

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

tion I fn( t) 12 and the saltus of the distribution function F n•(x) of x,.•( w) at the origin is given by L[lfn 12 ] and is written by P(X,.• = 0). Now from the convergence of ll~ L[lfn 12 ], we have }::;'= 1 (1 .A'[ I fn 12 ]) < oo, that is, 00

L P(Xn

8

n=I

=J= 0) <

(11)

00.

Let an be a median of X 11 (w). Then by Lemma 12.9.3, we have for any e > 0,

Letting

E

=

1/k, k = 1, 2, ... , we have

Now

P(Xn8 =J= 0)

=

p( k-I 0 {I Xn

= lim

k-+oo

8

I>

k-I })

=

P( lim k-+oo

{I

Xn

Is >

k-I })

P(l Xn 8 I > k- 1 ).

From (11), 00

00

> ~

L

lim P(l n-I k-+oo 1 2

1

= 2

xns I >

k- 1 )

00

L

lim P(l Xn - a 11 I > k-I) n=I k-+oo 00

L P(Xn -

n-I

an =J= 0),

which shows the theorem. Suppose that there is a sequence {an, n = 1, 2, ... } for which (10) holds. Write P(Xn - an = 0) = Pn, n = 1, 2, ... ; then the distribution of S( w) is either the unit distribution or the distribution which has a continuous component, and a purely discontinuous component (in the Lebesgue decomposition of the distribution), or a continuous distribution function, according to whether Pn 2 = 1, p, or o, where 0 < p < 1. This is immediate from the above proof. REMARK.

n

Let F(x) be a distribution function. By Lebesgue decomposition, we may write (12)

13.1.

CONTINUITY AND DISCONTINUITY PROPERTIES

541

+ +

where a P y = 1, a, p, y ~ 0 and Fd(x), Fa(x), and F8 (x) are distribution functions that correspond to the distributions having no point of increase except at points in the point spectrum, the absolutely continuous distribution function, and the singular distribution function, respectively. If one of a, p, and y is one and the other two are zero, then F(x) is called the pure distribution function. Fd(x) is also called the purely discontinuous distril:ution function. A theorem called the "pure theorem," which is due to Jessen and Wintner [1 ], is useful, particularly in order to discuss the distribution of the sum of a series of random variables.

Theorem 13.1.5. Suppose that (1) is an almost surely convergent series of independent random variables. Suppose that the distribution of each Xn( w ), n = 1, 2, ... , is purely discontinuous. Then the distribution function of S( w) is pure in type, that is, either purely discontinuous, absolutely continuous, or singular. The following proof was adopted from Breiman [1 ]. Proof. Let Fn(x) be the distribution function of Xn(w), n = 1, 2, ... , Pn be the point spectrum of Fn(x), JP,. dFn(x) = 1, and Pn be a countable set. Let P = U::'=x P n. Then P is again countable. Consider the smallest group G of points containing P, namely, the set consisting of all points of the form m 1x 1 mixi, where x 1 , x 2 , ••• , xi E P and m 1 , ••• , mi are any integers (positive or negative). Then G is also a countable set. Let B be a Borel set on (- oo, oo) and let E be a set of w, on which :L::'=x Xn( w) converges. P(E) = 1. Consider B ( +) G, which is the vector sum of Band G on (-oo, oo) (see Section 3.9), that is, the set of points of the form x + y, x E B, y E G. For each y E G, B ( +) {y} is a Borel set and since G is countable U11 B ( +) {y} = B ( +) G is a Borel set. Let

+ ···+

A = { w : S( w)

E

G ( +) B} n E.

(13)

Now we show that A is a tail set. Let w be any element of E. If S( w) E G ( +) B, then for every n ~ 1, :L::'=n+I Xk( w) takes the value of G ( +) B. This is obvious because, in general, if x E G ( +) B, y E G, then x - y E G ( +) B, in view of the definition of G. Let x = S( w ), y = :L~-x Xk( w ). Hence 00

{w:S(w)eG(+)B}nEc{w:

L

k=n+I

Xk(w)eG(+)B}nE.

542

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Conversely, if wEE and Lk=n+lXk(w) E G(+)B, then S(w) = L~- 1 Xk(w) + Lk=n+l Xk(w) E G ( +) B, since L~=l Xk(w) E G. That is, {w:S(w)eG(+)B}nE:::::>{w:

f

Xk(w)eG(+)B}nE.

f

Xk(w)e G(+)B} n E.

k-n+l

Therefore we get {w: S(w) e G(+)B} n E= {w:

k-n+l

Since this is true for every n, A is a tail set. Now then, from Theorem 12.3.1, for each Borel set B,

P(A) = {

~r

(14)

1.

Let us take countable sets for B. Then we have the alternatives: (i) There is a countable set B 0 with P[ S( w) e G ( +) B 0 ] = 1 or (ii) for every countable set B, P[ S( w) e G ( +) B] = 0. In the first case, let D = G ( +) B 0 , then P[ S( w) e D] = 1 and since D is countable, the distribution function of S( w) is purely discontinuous. Suppose that (ii) is the case. Then P[S(w) e G (+) B] = 0 for all countable sets B. Since B e G ( +) B (0 e G), P[ S( w) e B] = 0, or changing notation, P[ S( w) e H] = 0 for all countable sets H. In this case (14) still holds for any Borel set B which is not countable. Now consider all Borel sets that are not countable and have measure zero. Then again for each such set, P(A) = 1 or P(A) = 0, from which we derive that either there exists a noncountable Borel set B 1 of measure zero such that P[S(w) e G (+) B 1] = 1 or P[S(w) e G (+) B] = 0 for all noncountable Borel sets B of measure zero. Suppose that the first case occurs. Then we have P[ S( w) e D 1] = 0 for some noncountable set D 1 of measure zero [setting G ( +) B 1 = D 1] and P[ S( w) e H] = 0 for all countable sets H. In other words, the distribution function F(x) of S(w) has no point spectrum and the increase of F(x) occurs on D 1 • This means that F(x) is singular. Finally, suppose that P[ S( w) e G ( +) B] = 0 for all noncountable Borel sets B of measure zero. Since B e G ( +) B, P[ S( w) e B] = 0 for all Borel sets of measure zero (even if B is countable because of our supposition that P[ S( w) e H] = 0 for all countable sets). Hence we have

13.2.

543

INTEGRABILITY OF THE SUM OF A SERIES

J{s<w>sB} dP = 0, which is the same as JB dF(x) = 0 for all Borel sets of measure zero. In consideration of the completed measure generated by F(x), F(x) becomes absolutely continuous. This completes the proof.

13.2. Integrability of the Sum of a Series Let S(c.o) be the sum of a series of independent random variables. We shall give a result on the integrability of S( w); from the result we shall show that, denoting the distribution function of S(c.o) by F(x), under some conditions, F( -x) + 1 - F(x) has some restraints like those of a normal distribution function as x ---+ oo. We now take, just for convenience, a series of the form (1)

where {an , n = 1, 2, ... } is a sequence of constants and {Xn (w ), n = 1, 2, ... } is supposed to be a sequence of independent random variables. The almost sure convergence of (1) is equivalent, from the three series theorem, to (2)

if WXn = 0 and an 2

=

var Xn is finite for n

=

1, 2, ....

Theorem 13.2.1. Suppose that Xn( w) has a symmetric distribution function and

I Xn(c.o) I ~ C

a.s.

(3)

for n = 1, 2, ... , where C is a constant independent of n. If (4)

then the sum S( w) of ( 1) satisfies

W exp(A.S 2 ) < oo

(5)

for any A. > 0. This theorem was shown by Paley and Zygmund [1] and Wintner [1, 3] for the case in which Xn( w) has a Bernoulli distribution function (Section 3.10), or what is the equivalent, Xn(c.o) is a Rademacher function rn(t). Their method is valid for the theorem in consideration.

5#

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Proof. Since Xn has a symmetric distribution, WXn = 0 and (4) implies (2) because of (3); thus the series (1) converges almost surely. From (3), moments of Xn( w) of all orders exist. Let us denote the moments in general by We calculate the moments of Sn(c.o) = L~- 1 a.XA:(c.o). Letfk(t) be the characteristic function of a.Xk( w ). Then

. dm [fl(t) .. ·/n(t)] am(Sn) = z-m dtm

I t-o

by Theorem 11.2.2, which is

Hence, writing Ch .... ,;n = ml/j1 1 · · · jnl,

am(Sn) =

L

C; ,; ,... ,;a; (a 1X 1 )· ··a; (anXn)·

jt+···+j,.-m

1

a

"

1

"

Now since Xn has a symmetric distribution, the moment of Xn of odd order vanishes. So the above equality becomes

am(Sn) =

L

2kl + ... +ik, -m

c2kl''"'ikna2kl(a1Xl) ... a2kn(anXn)

if m is even. If m is odd, am(Sm) = 0. Now we use, for the multinomial coefficient

ci ..... ,i,.,

(6)

the relation

(7) This is easily shown from

c2kl, ... ,2kn

(2m)! ml

ckl''"'kn

fi j-l

(k;)l (2k;)l '

where (2m)'

.....:..........,.:. 1 -· =

m.

(m

+ 1)· ··(2m) ~ (2m)m,

k;l

1

(2k;)l

:-:;:; _1_

(k;+ 1) .. ·(2k;)-

2kJ

are used. We then see from (6), using (7), that a2m(Sm) ~ mm

L

kt+"·+kn-m

ckl'"''k,.a2kl(a1Xl)·. ·a2kn(anXn)·

13.2.

545

INTEGRABILITY OF THE SUM OF A SERIES

Since by (3),

a 2 m(Sn) ~ mm

L

k 1 +···+k,.-m

= mmC2m(al2 ~

Ck 1•···•k,.(a 1 C)2k1

• • •

(anC)2k,.

+ ... + an2)m

Ammm,

(8)

where 00

A= C 2

L an <

(9)

oo.

2

n=l

From Fatou's lemma,

m= 1,2, ....

(10)

n-+oo

We also have m = 0, 1, 2, ... ,

(11)

since a 2 m+l(S) exists from (10) and S has a symmetric distribution. Now from (9), for any it > 0,

by the well-known Stirling formula:

m! = (2n) 112mm+!e-m[1

+ o(1)].

If it is a positive number such that

0
(13)

then the series on the right-hand side of ( 11) is convergent and hence 00

L (1/m!)a m(S)itm < 2

oo,

m=O

that is,

We now remove the condition given in (13). Let us take any it > 0. Choose N so large that iteA.v < 1, where A.v = C 2 :L::'=.v+I an 2 , and con-

546

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

sider the series :L:=N+l anXn(w) = S(w)- SN(w). For this series, (13) holds with AN in place of A. Hence

W exp -A.(S- SN) < oo. Since I SN I ~ C L~=l I an I a.s., exp A.S.v is bounded, and we have W exp - A.S < oo. This completes the proof. From Theorem 13.2.1, it is easy to show the following theorem.

Theorem 13.2.2. Let {Xn(w)} be a sequence of independent random variables satisfying the conditions in Theorem 13.2.1. Let F(x) be the distribution function of S( w) = 1 Xn( w ). Then for any A. > 0,

:L::

1 - F(x) = O(exp -A.x2)

F(x) Proof.

O(exp -A.x2)

=

as

x - oo,

as

x - -oo.

(14)

Using Theorem 13.2.1,

J:

oo > Wexp A.S 2 =

exp A.x2 dF(x)

~ f~ exp A.u 2 dF(u) ~

exp A.x2 [1 -

r:

+ exp A.u dF(u) F(x)] + exp A.x 2 F( -x) 2

for any x, which shows (14 ). The integrability of the sum of a convergent series of independent random variables, without the boundedness condition imposed in (3 ), will be discussed in Section 13.8.

13.3. Magnitude of the Characteristic Functions of the Sums of Series In the foregoing section, we showed that at infinity, some restraints are imposed on the distribution function of the sum of a series of independent variables. In this section, we shall show that the magnitude of the characteristic function at infinity is somewhat restricted as well. We consider again a series of independent random variables 00

S(w)

=

L anXn(w)

n-1

(1)

13.3.

547

MAGNITUDE OF THE CHARACTERISTIC FUNCTIONS

and assume that (2) (3)

where an 2 = var X n • The series ( 1) is then almost surely convergent and S( w) is defined. First we show the following theorem.

Theorem 13.3.1. Suppose that (2) and (3) are satisfied. If f(t) is the characteristic function of S(w), as defined by (1), then

-oo 0, where Tn

(4)

= Tn(c) is defined by (5)

c being an arbitrarily chosen positive constant, and Fn 8 (x) is the symmetrized distribution function Fn(x) * [1- Fn(-x 0)] of Xn(w).

+

This theorem and the subsequent results were substantially given by Wintner [3], who actually dealt with the Bernoulli convolution. If Xn(w) is uniformly bounded by C, then by choosing c to be C and noticing that Tn 2 = 2an2 [which is obvious from the fact that Fn8 (x) is the distribution function of Xn(w)- Xn'(w), where Xn'(w) is independent of and identically distributed with Xn( w )], we have the following theorem immediately from Theorem 13.3.1.

Theorem 13.3.2. If (2) and (3) are satisfied and

I Xn(w) I ~ C

(6)

a.s.,

C being a constant independent of n, then for the characteristic function f(t) of S( w ), we have

f(t)=O(exp[-at 2

L

la,.t!::>0- 1

for some positive constant a.

an 2 an 2 ]).

-00

< t < oo,

(7)

548

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

It is sufficient to show Theorem 13.3.1.

Proof of Theorem 13.3.1. Xn(w). Then

Let fn(t) be the characteristic function of

Ifn(t) 12 = J~oo eitx dFns(x) =

J:

cos(tx) dFn 8 (x).

(8) If I ant I :::;; c-t, then there is a constant a

> 0 such that for lxl

~

c-t,

Therefore,

J: I

cos(ant.~) I dFn 8 (x)

~

J

lxi;Sc

cos(antx) dFn 8 (x)

+J

lxl>c

dFn 8 (x)

Inserting this into (8) we get (7). This proves the theorem. It must be noted that "l..

a ·n 2 a n 2 -+

0

as

I tl-+oo.

la,.tl::>c- 1

Therefore,

f(t)

has an estimate of order higher than

exp -at2 •

Now we can show that f(t) cannot be O(exp -at2 ) as ltl -+ oo.

Theorem 13.3.3. Under the conditions in Theorem 13.3.2, with Xn(w) having symmetric distribution functions and the condition that I: an 2 < oo, the characteristic functionf(t) of S(w) cannot be O(exp -at2 ) for large I t I for any a > 0.

13.3.

MAGNITUDE OF THE CHARACTERISTIC FUNCTIONS

549

Proof. Let F(x) be the distribution function of S(w). Take G(x) = ! J=~! F(u) du, which is a distribution function. This can be written F(x) • B(x), where 0, X~ -1, -1 ~X~ 1, B(x) = { l(x + 1), 1, 1 ~ x, whose characteristic function is sin tft. Then G(x) has the characteristic function q;(t) = f(t)(sin tft). (9) As a matter of fact, since G(x) has the density g(x) = ![F(x - F(x- 1)],

q;(t) =

J: eitzg(x) dx.

+ 1) (10)

Now suppose that f(t) = O(exp -at2) for some a> 0. Then

q;(t) = O(exp -at2) for large l t l· From Theorem 13.2.2, 1- F(x) = O(exp -lx2) as x---. oo and F(x) = O(exp -lx2 ) as x---. -oo, for all A.> 0. Then we see that g(x) = O(exp -lx2) for large l x l· We now apply Hardy's theorem (Theorem 8.1.1). Choose A. so that a~ (4A.)- 1 • Then we must have q;(t)

= K

exp -at2,

-oo

< t < oo,

(11)

for some constant K. But this is impossible, since from the form of (11), q;(t) has no zero at all on ( -oo, oo), while from (9), q;(t) has an infinite number of zeros. This proves the theorem. Wintner ([3], p. 38, Example 6) has shown that there is a series of independent random variables for which S( w) = 0(exp[ -a l t 12 -•]) for any e > 0. He constructed a series

(12) for which the above estimate holds, where t E [0, 1] and {rn(t)} is a sequence of Rademacher functions. More precisely, we may show Theorem 13.3.4.

550

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

First, recall that Rademacher functions rn(w), wE [0, 1] (Q is the space on [0, 1], .91 is the class of measurable sets, and P is the Lebesgue measure), have an identical distribution for n = 1, 2, ... ,

l

X< -1,

0,

a(x) =

I xl <

!,

1,

X< 1,

1,

and the characteristic function cost.

Theorem 13.3.4. Let p(t) be any increasing function such that

f~ [p(t)ft3] dt <

(13)

oo.

Then there is a Rademacher series wE [0, 1],

(14)

which converges almost surely such that the characteristic function f(t) of the sum of (14) satisfies

J(t)

=

O[exp -p(l t I)]

as

I t I --+ oo.

(15)

Proof. We may suppose that p(t) increases to oo as t--+ oo. Otherwise we may consider p(t) + t in place of p(t). Define q(t) =

s:

for

[p(t)ft] dt

> 1.

t

q(t) is increasing for t > 1. Since

J~ [p(u)ju 3 ] du ;;;::; p(t) J~ dufu 3 = p(t)f2t 2 , we have

p(t) = o(t 2 )

(16)

from (13). Now, using integration by parts,

ft [q(u)fu 1

3]

du = - [q(u)f2u 2 ]

It + z1 Jt 1

1

[p(u)/u 3 ] du,

from which we have

f~ [q(t)ft 3 ] dt <

00

(17)

13.3.

551

MAGNITUDE OF THE CHARACTERISTIC FUNCTIONS

because

=

q(t)jt2

t- 2

J: [p(u)fu] du = I: o(u) du = o(1) t- 2

by (16). Let

A = log[cos l]- 1 • There is a t 0 such that

q(3t)- q(t)

r

=

[p(u)fu] du ~ p(t) log 3 ~ p(t) +A

(18)

for t > t 0 • Let s(t) be the inverse function of q(t) and write q;(t) = 1/s(t) for t ~ 1. From q(t)-+ oo, obviously q;(t)-+ 0 and furthermore,

tq; 2 (t)

= tfs 2 (t) = q[s(t)]/s2 (t) = o(1)

as

t-+ oo.

Therefore, noting that q;(t) is absolutely continuous,

INq; (t) dt = 1

2

Nq; 2 (N)- q; 2 (1)- 2

= o(1)-

q; 2 (1)- 2

I

IN tq;(t)q;'(t) dt 1

1/tp(Nl

[q(w)fu 3 ] du.

1/tp(l)

Hence from (17),

I~ q; 2 (t) dt < oo.

(19)

Define an= q;(nA).

Then from the fact that q; 2 (u) is monotonic, (19) gives us L an 2 < oo. With this definition of an , we consider the series

which converges almost surely. The number of n such that ant > c(>O) for any positive constant c is easily seen to be [A- 1q(tfc)], where [] means as usual the integral part. Thus the number of n such that 1 > ant ~ ! is

[ q(3t) ] _ [ q(t) ] > q(3t) _ q(t) _ l > p(t) + A _ 1 = p(t) A A =A A =A A

552 for t

XIII. ~ t0 ,

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

using (18). Thus

I/(t) I = ~

I fi cos(ant) I~ n-1

f1

cos(ant)

l>a,.tii:l/3

(cos !)Pi.-t = e-P

for

t

~

t0 •

Since /( t) is an even function, we have the required proof. In connection with Theorem 13.3.4, we prove the following theorem (Kawata [1 ]), which concerns the magnitude of the characteristic function of the sum of a series.

Theorem 13.3.5. Let p(t) be any increasing function such that

I~ [p(t)ft 2 ] dt < oo.

(20)

There is a Rademacher series (14), converging almost surely, such that the characteristic function f(t) of the sum of (14) satisfies

f(t) = O(exp -p(i t I))

as

I t I --+ oo,

(21)

and the distribution function of the sum has a bounded spectrum.

Proof. The proof is carried out in a way similar to that of the foregoing theorem. Let q(t) =

J:

t > 1.

[p(u)fu] du,

As before, we may suppose that p(t) increases to oo. Otherwise, we may take p(t) t 112 • From (20), we have p(t) = o(t) as in (16). Also we have

+

I~ [q(t)ft 2 ] dt < oo as in (17). Define A, s(t), and
I~ dt <

oo.

Writing an=
13.3.

553

MAGNITUDE OF THE CHARACTERISTIC FUNCTIONS

We introduce the function (22) Under condition (3), N 1 (t) is, in fact, well-defined for t > 0 since

Nl(t) ~ t 2

L

an 2an 2 < oo.

la,.tlii:;l

Obviously N 1(t) is a nondecreasing function oft. If :L~1 an 2 = oo, then N 1(t) - oo as t - oo.

Theorem 13.3.6. Under the conditions in Theorem 13.3.2, j(t)

= O(exp{-a[N1(Pt)- N 1(t)]})

for some a and any fixed positive constant

p>

(23)

1.

Proof. This is an easy consequence of Theorem 13.3.2. We may assume without loss of generality that C = 1. We then see that 00

'\' an2,.n2 >= '-' v

la,.tl~l

'\' '-'

m-[tl-1

'\' '-'

1/Cm+ll;;ila,.l
an 2,. 2 vn

00

~

L (m + 1)- [N (m + 1)- N m-[tJ-1 2

1

1

(m)]

Then from (7), we obtain (23 ). The following is a special case of Theorem 13.3.6.

Corollary 13.3.1. If an ~ a > 0, n = 1, 2, ... , holds for some constant a> 0 in addition to the conditions in Theorem 13.3.2, then j(t)

= O[exp{ -a[N(pt)- N(t)]}]

for some constant a > 0 and for any fixed constant is the number of an such that I tan I ~ 1, t > 0.

(24)

p > 1, where N(t)

554

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Suppose again that (2), (3), and (6) hold and that (25)

n = 1, 2, 3, ....

Then condition (3) turns out to be (26)

Lemma 13.3.1. Suppose that

I an I :; ; ; a

for some constant a > 0.

Using the above notation, (i) Ln= 1 I an I = oo is equivalent to f;"' x- 1 dN(x) = oo, which is, in turn, equivalent to f;"' x- 2 N(x) dx = oo, and (ii) Ln- 1 an 2 < oo is equivalent to f;"' x- 2 N(x) dx < oo. (ii) is not needed later but we state it just for comparison.

Proof.

We may suppose a= 1. Since

we have 00

00

L (m + 1)- [N(m + 1)N(m)] :;;;;; L I an I m-1 · m-1 1

00

~

L

m- 1 [N(m

+ 1)- N(m)]

(27)

m-1

and

L m- [N(m + 1)- N(m)] = m-1 L m- Jm+l m oo

1

1

dN(x).

The convergence of the right-hand sum in the last equation is equivalent to the convergence of f;"' x- 1 dN(x). The same is true for the left-hand side of (27). This shows the first equivalence in (i). Integration by parts gives us

f~ x- 1 dN(x)

= [N(T)JT]- N(1)

+ f~ x-

2

N(x) dx.

(28)

If J~ x- 2 N(x) dx---+ oo as T---+ oo, then obviously the left-hand side of (28) diverges to infinity. If J:"' x- 2 N(x) dx < oo, then

f~ x- 2 N(x) dx ~ N(t) (t-1- T-1)

13.3.

555

MAGNITUDE OF THE CHARACTERISTIC FUNCTIONS

and hence

I~ x- 2 N(x) dx ~ N(t)ft. Therefore,

N(t)

=

as

o(t)

t---+

oo.

Then from (28), the left-hand side of (25) should converge. This shows the second equivalence in (i ). (ii) is shown in a similar way. We now give the following theorem.

Theorem 13.3.7. Suppose the conditions given in Theorem 13.3.2, (25 ), and (26 ). If (29)

then the spectrum of S(w) = L~= 1 anXn(w) is (-oo, oo), provided that N({Jt) - N(t) is a nondecreasing function of t for some {J > 1.

Proof.

We shall show that

I~ t- 2 [N({Jt)- N(t)] dt

(30)

= oo.

We have

I;

t- 2 [N({Jt)- N(t)] dt

= (

t- 2

dt

I:'

= IfJ1 dN(x) Ix1

dN(x) t- 2

dt

+IT{J dN(x) Ixxi{J t- 2 dt

+ I {JTT dN(x) ITx/{J t~ IT{J dN(x) Ix

t- 2

dt

2

dt

=

({J - 1)

~

IT{J x- 1 dN(x).

Hence from Lemma 13.3.1 (i) and (28), we get (30). Write O(t) = N({Jt) - N(t). This satisfies the condition given in Section 11.5, Eq. (13) in Theorem 11.5.4. Then the conclusion is true from Theorem 11.5.4.

556

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

13.4. Distribution Functions of the Sums of Rademacher Series; Characteristic Functions of Singular Distributions Suppose that an > 0, n = 1, 2, ... , and

(1) and consider the Rademacher series wE

[0, 1],

(2)

which is almost surely convergent. Let the sum of (2) be S(w). We deal here with the distribution functions of S(w) for special choices of {an}· We then have very interesting examples of distribution functions. This sort of thing was extensively studied by Jessen and Wintner [1] and Wintner [3]. Let a(x) be the distribution function of S(w). We remark that the distribution function of anrn( w ), an > 0, has the spectrum {-an, an} consisting of two points. Hence the spectrum of a(x) is given by the vector sum of {-an, an}· EXAMPLE 1. Let an= 3-n. Then the spectrum S(a) of a(x) becomes the Cantor ternary set in I xI ~ !. The function a(x) is a singular function. This is shown from Theorem 13.1.5 (the pure theorem) and Theorem 13 .1.1. The characteristic function f( t) of S ( w) is 00

f(t) = ncos(t/3n).

(3)

n-1

Since the maximum saltus p of the distribution function of r n( w) is ! and ll::'=1 ! diverges to zero, a(x) is continuous by Theorem 13.1.1. Now, f(2n 3m) = ll::'=1 cos(2n/3n-m) = llk:1 cos(2nf3k), which is independent of m and not equal to zero. Hence lim,,-+oo f(Zn 3m) =/= 0. This shows that a(x) is not absolutely continuous, because if it were, f(t)--+- 0 must hold as I t! --+- oo by the Riemann-Lebesgue lemma. Hence by the pure theorem (Theorem 13.1.5), a(x) must be singular. We may of course show directly from the construction of the ternary set that a'(x) exists and is zero almost everywhere. This example of a(x) is interesting in the sense that it gives an example of a characteristic functionf(t) of a continuous distribution function that does not converge to zero as I t I --+- oo. Recall that limT-+oo (2T)-1

r:.TJ dt = o.

13.4. RADEMACHER SERIES; SINGULAR DISTRIBUTIONS

557

EXAMPLE 2. Let a 211 = a 2 ,_ 1 = 3-n. Then u(x) is [ll::'= 1 (t/3"))2 and is the convolution of the distribution function in Example 1 and of itself. u(x) is again a singular distribution since f(t) does not converge to zero as It 1- oo. For this example, S(u) = [-1, 1]. This is seen from the fact that S( u) is the vector sum of two Cantor ternary sets. EXAMPLE 3. If a .. = 2-n, then S(u) continuous because

= [-1, 1]

and u(x) is absolutely

00

f(t) =

fl cos(t/2") =

sin tft

n=l

and this can be written t J~ 1 e'xt dx; that is, the probability density is given by p(x) = ! for I xI < 1 and p(x) = 0 for I xI > 1. By suitable choices of {a.. }, we can also construct the distribution functions of S( w ), which are infinitely differentiable, or even analytic on (- oo, oo ). Such examples are found in Wintner [3]. We mention without proof the following theorem of Wintner [3].

Theorem 13.4.1. Let l: a 11 < oo, a .. > 0 and r 11 = :Lk:n+l ak. If a 11 > r11 , n = 1, 2, ... , then for the distribution function u( x) of the sum of a Rademacher series (2), the following statements hold. (i) S(u) is nowhere dense. (ii) lim 11 _,. 00 2"+lr" = mS( u ), m being the Lebesgue measure. (iii) If mS( u) > 0, then u(x) is absolutely continuous and for any patr x 1 , x 2 , Finally we also remark without proof that if we take

1

a ..

= 2"n -

1 --=z,.--n---=+1:-:-(n-+____,.,1),.--

'

then S( u) is obtained in a way similar to the method by which the Cantor ternary set is obtained, the characteristic function converges to zero, and u(x) is singular. This function u(x) was given by Menchoff [1], who by this example showed the existence of Fourier-Stieltjes coefficients of a singular function that may tend to zero. The question concerning the rapidity with which the Fourier-Stieltjes coefficients or the characteristic function of singular distribution functions approach zero was studied by several authors, for instance, Kershner [1], Hartman [1], and Littlewood [2].

558

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Wiener and Wintner [1, 2] gave a decisive result that will be shown in Theorem 13.4.2. First we remark that if the characteristic function f(t) of a distribution function F(x) satisfies J(t) = O(l t I)-"),

I tl- oo,

for some a> t, thenf(t) E £2(-oo, oo) and hence by Theorem 11.6.1, F(x) is absolutely continuous.

Theorem 13.4.2. For any e > 0, there exists a distribution function F(x) which is singular on [-n, n] and dF(x) = 0, I xI

> nand such that (4)

In order to show this theorem we need the following lemma (called Vander Corput's lemma).

Lemma 13.4.1. Suppose that g(x) has, in [a, b], the continuous second derivative g"(x) and that g"(x) -=I= 0 in [a, b]. Then

II:

cosg(x) dx

I~ 6[ 4~!~1 g"(x) 1]-112.

(5)

Proof.t Let

e=

min I g"(x)

a:iiz~b

I·

We may suppose that g"(x) > 0 and hence g"(x) ~ e [if g"(x) < 0, then we may consider -g(x)]. Then g'(x) is increasing. Suppose that the sign of g'(x) is constant and, for instance, g'(x) ~ 0, then for any y, a< y < b, m [y, b]. (6) g'(x) ~ (y - a)e Now by the second mean value theorem,

I,bcosg(x) dx = Ib, =

1 d g'(x) dx [sing(x)] dx

g'ty)

I: !

' See Zygmund [5], Vol. 1, p. 197.

[sing(x)] dx,

y <~
13.4.

559

RADEMACHER SERIES; SINGULAR DISTRIBUTIONS

Hence from (6),

IJ: cosg(x) dx I~ 2/(y- a)e.

Then

IJ: cosg(x) I~I J: 1 I+ IJ: cosg(x) I dx

dx

~

y- a

dx

+ [2/(y -

a)e].

Taking y - a= (2/e)1' 2,

IJ:

cosg(x) dx

I~ 2VZ(l/e)1t2 <

(7)

3fe1t2.

If the sign ofg'(x) is not constant andg'(x) < 0 for (a, c) andg'(x) > 0 for (c,b), then we split f!cosg(x)dx into two parts: f~cosg(x)dx cosg(x) dx, and apply (7) to each of the integrals. We then get

+ J:

IJ:

cosg(x) dx

I~ 6fe 1' 2,

which gives us the required proof of the lemma. Proof of Theorem 13.4.2. We prove Theorem 13.4.2 following the proof by Wiener and Wintner [1, 2]. We split the proof into five stages.

(i) We first show that iff(t) = integer p = p( e) such that

TI~ 1

cos(tj3k), then there is a positive

(8) for any e > 0. Write n = [t] and represent n in a 3-adic expansion (9) where 15k = 0, 1, or 2, k = 1, 2, ... , "+ 1, and 15.+1 =/=- 0, some nonnegative integer. Then

t= 0

+151 +1523 +... +15,+13•,

0~0<1.

The fractional part {J - [{J] of {J = t j3k is

~+ 3

15k-1 32

"= P(n) being

+·. ·+ ~+ 151 +{) 3k-1 3k .

560

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

If k;:;;:; 3, and the pair (!5k, !5k_ 1) is one of (0, 1), (1, 0), (1, 2), (2, 1), (2, 0 ), and (0, 2 ), then the fractional part of fJ lies between 1/9 and 4/9 or between 5/9 and 8/9. In these cases,

I cos(3-knt) I ~ cos(n/9)

= e-c < 1,

k;:;;:; 3.

Hence we have

I f(nt) I ~ exp -p.nc or

fn

n+l

I f(nt) IP dt ~ exp -p. pc 11

(10)

for every integer p > 0, where P.n is the number of k such that !5k -=1= !5k_ 1 • Let 0 < t ~ N = [T]. Among 1 ~ n ~ N, the number of n with P.n = m [that is, !5k -=1= !5k-l takes place m times in (9) with 11 = 11(N)] is 6m[11(N)]! f [m!(11(N)- m)!] [the number of pairs of two distinct integers out of 0, 1, 2, is 6], which is not greater than [611(N)]m/m!. Therefore, for any integer p > 0,

J: I

n~l

f(nt) IP dt = ~

J:_ IJ(nt) IP dt 1

N

oo

L exp -p. pc < L exp -mpc [611(N)]m/m!. 11

n-1

m-o

Since 11(N) = O(log N) as is easily seen, the last inequality is 00

L [e-pc(A log N)]m/m! m-o

for some constant A. We then have

J: I

f(nt) IP dt

~ exp[e-PcA log N].

Take an integer p so large that for an arbitrarily given e > 0, Ae-pc < e. Then

J: I

f(nt) IP dt

~ exp(e log N)

= N•.

Hence

(ii) We now define the distribution function F(x) in the statement of the theorem.

13.4.

561

RADEMACHER SERIES; SINGULAR DISTRIBUTIONS

Let ~(x), -:rc ~ x ~ :rc, be a strictly increasing continuous function with ~( -n) = -:rc, ~(n) = :rc. Let G(x) be any continuous nondecreasing function on [-:rc, :rc], G(n) = 1, G( -n) = 0. Define

F(x) =

G[~-l(x)],

(11)

where ~- 1 (x) is the inverse function of ~(x). With suitable choices of G and ~. this function F(x) provides the required distribution. We have

J:, exp itx dF(x) J:, exp it~(x) dG(x). =

Now write

.i.(t, s) = (2n)- 1 ( , exp[it~(x)- isx] dx. Since exp it~(x) is of bounded variation over [ -:rc, Fourier transform, exp it~(x) = lim

a-+oo

r

-a

:rc],

by inversion of the

.i.(t' s) exp ixs ds

(boundedly ). Hence

J:, eitx dF(x) = J:, dG(x) !~r:! J:a .i.(t, s)eixs = lim a-4-oo

= lim a4-oo

where

tp(s) = (iii)

fa-a .i.(t, s) ds f"

r

-a

ds

eisx dG(x)

-n

.i.(t, s)tp(s) ds'

(12)

J:, eisx dG(x).

Choose

Hx) = i(3x

+ n- x

1 2

sgn x),

-:rt ~X~ :rt.

This is continuous and increasing, H-n) = -:rc, We shall show that I .i.(t, s) I ~ cl It 1-112

~(n)

=

:rc.

(13)

for all t -F 0 and all s, and that (14)

if t - s is an integer and

Is I > 21 t 1.

where C 1 and C 2 are constants.

562

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

We easily see that

.A.(t, s) = n-1

I:

cos[it(3x

+ n- x

1 2) -

sx] dx.

(15)

By Lemma 13.4.1, we obtain (13) immediately. The function g(x) = it(3x + n-1x2 ) - sx is decreasing in 0 ~ x ~ n if s > 2t > 0, and increasing in 0 ~ x ~ n if s < 2t < 0. Anyway it is monotonic in 0 ~ x ~ n for I s I > 21 t I· Suppose that t > 0, s > 2t, and s- tis an integer. Making the change of variable

in the integral of (15), we have

.A.(t, s) = n-1

I

lt--Bln

0

h(s, t; u) cos u du,

where h(s, t; u) = 2[(!t- s) 2 + n- 1 tu]- 112 • Using integration by parts and noting that t - s is an integer,, we see that

.A.(t, s) = n-1

I

lt-1)11

0

[oh(s, t; u)fou] sin u du.

Since oh(s, t; u)fou is seen to be monotonic, by the second mean value theorem, for some 0 < ~ < (s __:_ t)n,

I .A.(t, s) I ~ n- I oh(s, t; u)fou l·u-(H)" 1

= n- 2 tf{(!t- s) 2

II,,---4,,. sin u du I E

+ [t(s- t)/n]}3' 2

= O[tf(s- !t)3] = O(s-2). For the cases < 2t < 0, we have the same result in a similar way. This shows (14). (iv) We shall show that

I,.

_,.

eitz

dF(x) = 0(1 t

l-1) + o(l t l- 112

I

2111 -21tl

I cp(s) I ds).

(16)

Suppose t > 1, and consider, for a positive integer N, that

It+N .A.(t, s)cp(s) ds = It .A.(t, s)cp(s) ds + L It+k N

0

0

k-1

t+lk-1)

.A.(t, s)cp(s) ds.

(17)

13.4.

563

RADEMACHER SERIES; SINGULAR DISTRIBUTIONS

Since .A.(t, s) is a function of s of bounded variation (actually differentiable), by the second mean value theorem,

f t+k t+k-1

.A.(t, s)tp(s) ds =- .A.(t, t

+

+ k-

.A.(t, t

f'1

1)

t+k-1

+ k) f11

t+k

tp(s) ds.

Hence if k- 1 ;;;;; t, then from (14) and using I tp(s)

I fl+k

t+k-1

and if k- 1

.A.(t, s)tp(s) ds

I

~

C3k- 2

tp(s) ds

I ~ 1,

f'+" t+k-1

< t, then from (13),

If

l+k t+k-1

.A.(t, s)tp(s) ds

I ~ C t- Jt+" 4

1' 2

t+k-1

ltp(s)lds .

Here and hereafter, C 3 , C4 , • • • are constants. The first term on the right-hand side of (17) is, from (13), equal to or less than C 5 l t l- 112 f~ I tp(s) Ids. Thus, using the above estimates, we see that as n--+oo, s~+N .A.(t, s)tp(s) ds is absolutely convergent and

IJ~ .A.(t, s)tp(s) ds I ~ Cst- 112 k~t ~

C6t- 112

s:::1

J: I

I tp(s) Ids+

tp(s) I ds

+ C t7

Cak'Jr+1k- 2

1•

In the same way, we have a similar estimate for f:O .'.(t, s)tp(s) ds and also, for negative t, we have similar results. Altogether we have

f

.A.(t, s)tp(s) ds

I>

1. From (12), (16) is now obtained.

oo

-oo

for I t (v)

= 0 [ It l- 112 J21tl I tp(s) Ids ] -21tl

f(t) =

+ 0(1 t 1-

1)

00

f1 cos(t3-k),

k=1

which was considered in (i ), is the characteristic function of Ln= 1 3 -nr 11 ( w) and as was shown in Example 1 at the beginning of this section, the corresponding distribution function a(x) has a spectrum contained in I xI ~ t. Hence, f(nt) is the characteristic function of a(xfn), whose

564

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

spectrum is contained in I xI ~ :rc/2. Therefore, [f(2:rct)]P is the characteristic function of B(x) = a(xf2:rc) • a(xf2:rc) • · · · • a(xf2:rc), the pfold convolution of a(xf2:rt) with itself. B(x) = 0 for x < -:rep and B(x) = 1 for x >:rep. B(px) has n points of increase in I xI > :rt. Let us take B(px) for G(x) in (ii). Then F(x), defined by (11), is a distribution function. The characteristic function rp(t) of B(px) is [f(2:rtt fp )]P and therefore, T I -T

I rp(t) I dt =

IT -T f(2:rctfp) IP dt

I

= (p/2) I2T/p _2Ttp I/(:res) IP ds = O(T•)

from (8). Then by (16),

J:,. eitzdF(x) = O(j t j-I) + O(j t j-lj t j•) = O(j t j-l+•), which is (4). From Example 1, a(x) is singular and hence B(x) or B(px) is singular in [-:rc, :rc] because the characteristic function of B(x) is [f(2:rtt}]P, which does not converge to zero as t - oo as in Example 1 and is of pure type. Now F(x) = B[p~- 1 (x)] and therefore F'(x) =

B'[p~- 1 (x)]p d[~- 1 (x)]/dx

a.e.

But d[~- 1 (x)]/dx = 1/~'(y), y = ~(x), for which 3/4 Hence F'(x) = 0 a.e. in ( -:rc, :rc) and F(:rc) = 1, F( -:rc) is singular. This completes the proof of Theorem 13.4.2.

~ ~'(y) ~

11/4.

= 0. Thus F(x)

With similar methods, Schaeffer [1] generalized Theorem 13.4.2 in the following stronger form.

Theorem 13.4.3. Let r( n ), n = 1, 2, ... , be an increasing sequence such that r(n)- oo as n- oo. For n ~ t ~ n + 1, let r(t) be linear. Then (i) F(:rc)

there is a singular distribution function F(x) with F( -:rc) = 0,

= 1 such that

J:,. eitz dF(x) = as I t

I-

oo; and

0[1 t j- 1' 2r(l t I) log! t I]

(18)

13.5.

565

FURTHER THEOREMS ON RADEMACHER SERIES

(ii)

there is a singular distribution function in ( -:n, :n) such that

J:,.

einx

dF(x)

= O[J n J-112r(J n J)]

(19)

as an integer I n I - oo. For the proof, the reader is referred to the original paper (Schaeffer [1]). For a further result see Note 3.

13.5. Further Theorems on Rademacher Series In this section we shall deal with the Rademacher series O~t~1,

(1)

and show some theorems concerning this series. (1) is almost surely convergent if and only if 00

L an2 <

(2)

oo.

n-1

We now prove the following theorem (Kaczmarz and Steinhaus [1] ).

Theorem 13.5.1. The necessary and sufficient condition for the series (1) to converge everywhere in (0, 1) is that 00

L I an I< oo. n=1

(3)

Proof. The sufficiency is obvious. So it is sufficient to show th t (3) is necessary. Suppose that (3) does not hold. Then there is a sequence for each n, n = 1, 2, ... , such that

"'n =

1 or - 1

(4)

Define

En

by if TJn = 1, if TJn = -1,

566

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

and consider the number

eto be represented by the dyadic expansion (5)

Then from the definition ofrn(t) or from Section 12.8, Eq. (2), TJn = rn(E). Hence if (5) is not dyadic rational, that is, {en} contains an infinite number of 1's and an infinite number of O's, then (6) which contradicts the everywhere convergence of (1 ). If E is dyadic rational, r n( E) is zero from some n 0 on because of the definition of rn(E) and (6) does not hold. However, in this case, we choose a subsequence {an~:, k = 1, 2, ... } of an such that n 1 < n 2 < · · ·---+- oo, and L I an~: I < oo. This is possible since an---+- 0 (from the convergence of ( 1) for an w which is not dyadic). Now in the sequence {en}, we change {en~:} (if necessary) to {e~~:}, where , e~1 = 0, e~. = 0,

...

alternately. Then {e~k} contains an infinite number of O's as well as an infinite number of 1's and therefore

is not a dyadic rational. Furthermore, 00

L an
oo

(7)

n-1

because

(7) is the contradiction we need for our proof.

Before going further, we give a lemma which is needed in the proof of the subsequent theorem.

<J:

Lemma 13.5.1. Let M,.(f) = I f(x) I" dx) 11", a> 0. If a= a 1t 1 + a 2t 2 with a 1 , a 2 > 0, t 1 , t 2 > 0, t 1 + t 2 = 1, then (8)

When (8) holds, log M,.(f) is seen to be a convex function of log a. We suppose that M,.1 ( / ) , M,.,(/) are finite.

13.5. Proof.

567

FURTHER THEOREMS ON RADEMACHER SERIES

The Holder inequality gives us

J: ~ (s: I

Mrx"(f) =

IJ(x)

I" dx =

/(X)

I" dx 1

J:

IJ(x)

r(s:

l"'t' IJ(x) 1"

111

I/(X) I" dx 2

r

dx

We now prove the following theorem (Khinchin [1], Kaczmarz and Steinhaus [1], and Paley and Zygmund [1]).

Theorem 13.5.2. Suppose that (2) is satisfied for a Rademacher series (1). Then the sum belongs to LP(O, 1) for p > 0 and (9) where AP and BP are constants depending only on p. As a particular case of Theorem 13.2.1, we know that if L~ anrn(t) exp(A.S 2 ) dt < = for any A. > 0, which is, of course, much stronger than S(t) E LP and actually, the inequality relation corresponding to the right-most inequality in (9) for such a class of S is known [see Note 5]. = S(t), then

J!

The right-hand inequality (with p = 2m and p an even integer) is a particular case of (10) in Section 13.2 and has already been shown. For 0 < p < 2, from the Holder inequality, Proof of Theorem 13.5.2.

which is equal to 00 (

Lan

2)p/2

n-l

.

For p ;;;; 2, we have, choosing 2m such that 2m > p,

[

( oo

~ B2m ];lan2)

[in view of (9) with p = 2m].

)mjp/2m

=

( oo

B~m ];lan2

)p/2

568

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

The first inequality of (9) with p

I: 1.~~

anr ..

~

2 is immediate, for

lp dt ~ ( s: [Jl a"rn f dtr

2

= ( L an2)p12, We may prove the case where 0 < p < 2 in the following way. Choose positive numbers t 1 and t 2 in such a way that t 1 + t 2 = 1, 2 = pt 1 + 4t 2 • Then by Lemma 13.5.1,

M 22 (S) where S = S(t) =

L~=l

~ M~1 t(S)Mtt•(S),

a 11r 11 (t) as before. Here 00

M 22 (S)

=

L an n=l

2

and

which is by (10) and (9) in Section 13.2, equal to or less than

where C is a constant (independent of a 11 ). Therefore,

which turns out to be

C 1 and C 2 being constants depending only onp. This shows the theorem. Another important theorem is the following, which is due to Marcinkiewicz and Zygmund [2].

Theorem 13.5.3. For a Rademacher series, we have

for any n ~ 1 and any oo only on p.

~

p

~

1, where Ap is a constant depending

13.5.

569

FURTHER THEOREMS ON RADEMACHER SERIES

For oo > p > 1, AP is given by Ap

= 2(pf(p- 1))P.

(11)

Proof. Since the case p = oo is obvious (essential bounded ness), we suppose 1 ;;:;; p < oo. Let t:5m be an interval (of length 2-"') on which r m(t) is constant. Then from the definition of Rademacher functions rn(t), for 1 ;;:;; k;;:;; m rk(t) = constant on t:5m

and k

for

~ m

+ 1.

Let t be any point in t:5m. We then see that

where n ~ m, and I t:5m I = 2-m is the length of t:5m. Writing Sn(t) = L~=I aA.rt(t), we have, from (12),

If

akrk(t)

k=I

or

~~~;" I

J

1

I; :; max r I Sn(t) I dt, o
akrk(t)

;

I; :; ~~~~I: I S,.(t) I dt. e,

(The maximum is taken for all 'YJ such that 0 < e < 'YJ < 1, and e< t < 'YJ for a given t.) From the Hardy-Littlewood inequality (Theorem 1.10.2),

r[

maxi'1 I Sn(t) dt]p dt;;:;; 2(pf(p- 1))P

0

;
;

II I Sll(t) IP dt,

p > 1,

0

we obtain (10) for the case where p > 1. The case p = 1 is shown in the following way. Let A. be a constant which is to be determined. Write E = {t : I Sn(t) I > ATn}, where

By the Chebyshev inequality, mE;;:;; A.- 2 T7j 2

I:

[Sn(t))2 dt = A.- 2 ,

(13)

570

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

where mE is the measure of E. Since on Ec,

I Sn(t) I ; :;:; A-rn,

Now noting that rk 2 (t) = 1 a.e.,

J!

rk(t)ri(t) dt = 0, k =F j. where we used the fact that Consider the sequence {rk(t)r;(t), k =F j} arranged in a single sequence. This constitutes an orthonormal sequence on (0, 1) and JE rk(t)r;(t) dt is the Fourier coefficient of an indicator q>E(t) of E [q>E(t) = 1 or 0, according as t E E or t E Ec] with respect to {rk(t)r;(t) }. Then using the Bessel inequality (Theorem 6.1.2), we have

which, from (13), is not greater than -rn 2/A. Hence we have from (15),

JeC [S (t)] 2 dt ~ (1- mE)-rn 11

~

'rn2(1 _

2 -

).-2 _

(2-rn2f).) 2).-1).

Choosing ). = 4, we get

Therefore from (14 ), (16)

13.5.

571

FURTHER THEOREMS ON RADEMACHER SERIES

Now

by (10) with p = 2 (which we have shown above). Hence (16) gives us

which completes the proof. Now we consider the case which corresponds top = oo and prove the following theorem.

Theorem 13.5.4.

For a Rademacher series we have

where A is an absolute constant and A is any positive constant. First we give the following lemma.

Lemma 13.5.2. Let Sn(t) = :L~- 1 a1Jk(t), ak being real, k = 1, 2, .... Let An= L~- 1 ak2 , Bn = :L~- 1 ak 4 and let A be any real number. Then

The left-hand inequality in (18) is not needed right now but for completeness it is given.

Proof.

Using the independence of {r n(t) },

n f1exp[ASn(t)] dt = iln f1 exp Aah(t) dt = il t(exp Aak + exp -Aak) k-1 k-1 0

0

(since rk(t) = 1 or -1 on exclusive sets of measure

!), and thus (19)

572

Xlll. PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

which is, by 2Ppl

~

(2p)l, equal to or less than

n L= 1. Pa~Pj2Ppl = n exp(!A.2ak2) = exp(!A.2An), n

n

2

k-1 p-0

k-1

which is the right-hand inequality of (18). From (19), we have

which is, because of the elementary inequality 1

+ u > exp(u- lu2),

u > 0,

equal to or greater than n

fl exp(lA.2ak2-

l1.4ak4)

~

exp(lA.2An- ),4Bn)·

k-1

This is the left-hand inequality of (18). Proof of Theorem 13.5.4.

In place of (17), we show (20)

where n(t) is any measurable function, assuming integers such that 1 ~ n(t) ~ n. In the following proof, we could proceed with exp[;, max

f akrk(t)]

1:Sm;:;!n k-1

in place of the integrand on the left-hand side of (20), but just for convenience we will prove (20), which is of course sufficient for the proof of our theorem. Let 1 ~ k ~nand let (a, f:J) be any dyadic interval (11/2•, (11 11 = 0, 1, 2, ... , 2k- 1. Writing n

Sn(t) =

L airi(t),

i-1

we have, as in (12),

Sk(t) = (f:J- a)- 1

J:

Sn(u) du

for a< t < {:J.

+ 1)f2k),

Hence for all t (except dyadic points),

I Sk(t) I ~

({J- a)- 1

tI ~

Sn(u) I du

~

max (7J -

e
~)- 1

rI e

Sn(u) I du.

Since k is any integer, we have

for all t except dyadic points. From the Hardy-Littlewood inequality referred to before,

for every p > 1. If p

~

2, then (21)

Now

I:

exp[J. I Sn
I] dt ~I: {exp[J.Sn
From (21 ), this is not greater than

+8

I:

exp[ -ASn(t)] dt,

which is, from Lemma 13.5.2, not greater than 16 exp(li. 2An)·

13.6. Sums of Independent Random Variables The purpose of this section is to generalize Theorems 13.5.2 and 13.5.3 of the preceding section to the case of a sequence of general independent random variables. But in so doing, we make use of these theorems and of some very interesting devices used by Marcinkiewicz and Zygmund [2].

574

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

We give first a simple lemma.

Lemma 13.6.1. Let {Xk(w), k = 1, 2, ... , n} and {Xk'(w), k = 1, 2, ... , n} both be sequences of independent random variables such that Xk(w) and X~:'(w) have the same distribution function for each k = 1, 2, ... , n. Let f(x 1 , • •• , xn) be a Borel function of (x 1 , • •• , Xn)· Then f[X 1(w), .. . , Xn(w)] and f[X 1'(w), .. . , Xn'(w)] are random variables having an identical distribution. Proof. It is obvious thatf(X1 , ••• , Xn) andf(X1', ••• , Xn') are random variables. LetS be any Borel set. Write {(x 1 , ••• , Xn);f(x 1 , •• • , xn) e S} = En, which is a set in the n-dimensional space Rn. By independence of {X~:}. P[X1(w), .. . , Xn(w)

E

En]=

J ··· t . dF (x) · · · dFn(x), 1

(1)

where F~:(x) is the distribution function of X~:(w) and of X~:'(w). We have the same integralfor P[X 1'(w), .. . , Xn'(w) e En]· This proves the lemma. We need another lemma.

Lemma 13.6.2. Suppose that g:' I X(w) IP < oo, p > 1, and g:'X = 0. Let X•( w) be the symmetrized random variable of X( w ). Then (2) where AP and BP are constants depending only on p.

Proof. X•= X-X', where X' is independent of and identically distributed with X. The right-hand side inequality of (2) is obvious from the elementary I y IP) for p ~ 1. inequality I x y IP ~ 2P- 1 (1 x IP We now show the left-hand inequality of (2). We may suppose without loss of generality that (3)

+

+

Otherwise we may consider X/ g> I X be supposed to hold. Then it is sufficient to show that

IP instead of X;

g> I X

IP >

0 may

(4) with some constant AP depending only on p.

13.6.

575

SUMS OF INDEPENDENT RANDOM VARIABLES

Let A. be a positive number, 0 < A. < 1, which will be determined later. We classify two cases: CASE

(i). (5)

and CASE

(ii).

WI XI Suppose case (i). Write E

=

~A..

{w: I X(w)

(6) I~

1}. Then

JE" I X(w) IP dP ~ JE" I X(w) I dP ~A..

(7)

By the Minkowski inequality given in Section 11.1, Eq. (4),

Since X' and X are independent and Ec depends only on X,

by (3). Since

from (7), (9), and (8),

u

E" 1

x

_X' lp dP

rp ~

(1 _ A.)l/p _

A_ liP.

the right-hand side of which is a constant (!)P - (!)P if A. is chosen to be !. This shows (4). Next consider case (ii). Write F = {w: X(w) ~ 0}. From WX = 0,

and

576

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Hence

Accordingly,

t

1

x<w> 1P dP IrXPdP

Suppose P(F)

w I X-

~

t.

~

(t

1

x<w> 1 dPr

~ (tcXdPrP ~

Then, writing F'

X' IP ~ I

Fr'IF'

I X-

= P(F) I

F'

~

u;.>p,

(l.'.)P.

= {w : X'(w) > 0},

X' IP dP ~ I

Fr'IF'

X'P dP

X'P dP.

(Since if Y(w) and Y'(w) are the indicators ofF and F', then

I Fr'IF' X'P dP

=

I YY' X'P dP

= P(F) I

F'

=

I y dP I Y' X'P dP

X'P dP.)

Thus

With ;.

= t, we have

This again shows (4 ). If P(F) ~ t, then P(Fc) ~ t and hence P(F') ~ !. So if we interchange the roles of X and -X', then we get (4) again. This completes the proof. We now give the following generalization of Theorem 13.5.2.

Theorem 13.6.1. Let {X,.(w), n = 1, 2, ... } be a sequence of independent random variables. Let p ~ 1. Suppose that ff I X 11 (w) IP < oo and (10) n = 1, 2, ....

13.6.

SUMS OF INDEPENDENT RANDOM VARIABLES

Then for every n

577

= 1, 2, ... , we have (11)

where AP and BP are positive constants depending only on p. Proof.

Let

be the symmetrized random variable of X,.(w). Let {r,(t), n = 1, 2, ... } be the sequence of Rademacher functions. X, s( w) 1~ symmetrically distributed. Consider O~t~l.

(12)

rk(t 1 )Xk"( w) and rk(t 2 )Xk'( w) have the same distribution for every pair t 1 and t 2 that are not dyadic rationals, since rk(t 1 } and rk(t 2 ) are 1 or -1 and Xk'(w) has a symmetric distribution. Hence by Lemma 13.6.1, the series (12), with t = t 1 and t 2 , are identically distributed; or, for almost every t, (12) has the same distribution. Therefore,

has the same value for almost every t, and hence

for almost all t. Hence

J dtr.J' I A~. 1 rk(t)X/(w) lp =ir I ,. Xk"(w).lp l

0

A"f 1

(13)

From Lemma 13.6.2, with X= L~- 1 rk(t)Xk> X'= L~- 1 rk(t)X/, we have

(14) with constants AP' and Bp' depending only on p.

578

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Now from Theorem 13.5.2 [with

ak

= Xk( w )],

Hence (15) From the left-hand side of (14) and from (15),

[ ..

AP'A~' W ];1 Xk 2 (w)

]p/2 ~ J1 dt W Ik'2;/k(t)Xk"(w) . I 0

=

wl t Xk (w) lp· 8

k-1

which is, by Lemma 13.6.2, not greater than B~' a constant B~'. So we get

WI I::= 1 Xk(w) IP with

with AP = Ap'A~/B~'· This shows the left-hand side of (11). We have the right-hand side in exactly the same way. This completes the proof. The theorem is true if n

= oo, or more exactly we may state:

Corollary 13.6.1. If WI X,.(w) IP < oo for p ~ 1, {Xn(w)} being a sequence of independent random variables with expectation given by (10), if I::'= 1 X,.(w) converges almost surely, and if the sum belongs to LP, then I::. 1 X,. 2(w) converges almost surely, belongs to LP12 , and

Proof. If I: X,. converges almost surely, then LX,.' does too, where {X,.'} is a sequence of independent random variables and X,.' has the same distribution as does X,.. This is obvious, since I: X,. and I: X,.' have the same characteristic function. Let xks be the symmetrized random variable and as before xks = xk- Xk'· Then L xk· converges almost surely, Xk 8 (w) and rk(t)Xk8 ( w) have same distribution for every nondyadically rational t, and Lt=I rk(t)Xk8 (w) converges almost surely for such t and has the same

13.6.

SUMS OF INDEPENDENT RANDOM VARIABLES

579

distribution as that of I;t., 1 Xt( co). Hence I;t., 1 rJ:(t)Xt•( co) e LP for almost every t and (13) holds with n = oo. The proof in Theorem 13.6.1 holds with n = oo. This shows the corollary. Next we generalize Theorem 13.5.3.

Theorem 13.6.2. Let {XJ:(co), k = 1, 2, ... , n} be a sequence of independent random variables. Let 1 ~ p ~ oo. Suppose that WI XJ: I'P < oo and WXJ: = 0 fork= 1, 2, ... , n. Then (16) where A'P is a positive constant depending only on p. Proof. Let 1 ~ p < oo. Suppose first that each of the XJ:( co), k = 1, 2, ... , n, has a symmetric distribution. Let (17)

where 11( co) is some integral-valued function of co, 1 ~ 11( co) ~ n. Consider, as in the proof of the foregoing theorem, the sequence {rJ:(t)Xt( co), k = 1, ... , n}, where rJ:(t) is a Rademacher function. Since Xt(co) is symmetrically distributed, as we supposed, for each t except dyadic rationals, rJ:(t)XJ:( co) is identically distributed with XJ:( co) for every k. Let 11(t, co) be defined by (18) For those t for which rJ:(t) = 1, k = 1, 2, ... , n, 11(t, co)= 11(co). Since 11(t, co) is a Borel function of XJ:( co), k = 1, 2, ... , n, from Lemma 13.6.1, for each t, I:i.:~:", rJ:(t)XJ:(co) has the same distribution as I:i.:<:1 Xt(co). Therefore, (19)

for every nondyadically rational t, which is, if we integrate with respect t, (20)

580

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Now from Theorem 13.5.3, with ak = Xk(w),

Integrating both sides on D,

J:

dt

wL~::J ~/k(t)xk(w> rJ ~ AP

s:

dt

WI kt rk(t)Xk(w) r= 1

AP

WI

t

1

Xk(w)

r

since (19) is true with n in place of v(t, w ). The extreme left-hand side of the last inequality is the same as the left-hand side of (20) and therefore (21)

This is (16), as is seen from (17). Thus we have shown (16) for the case in which Xk( w) is symmetrically distributed, k = 1, 2, ... , n. Now we remove the condition that Xk(w) be symmetric. In order to do this, we introduce the product space Q X Q. The a-field in this space is the minimal a-field containing all the sets of the form A X B with A E ~. B = ~. This a-field is denoted by~ X~ [A X B is the set of elements (w, w') such that wE A, w' E B.] The probability of a set of .91 X .9.J ts determined so that for A X B, P(A X B)= P(A)·P(B). Define for all w' ED, for all

wE

Q.

Then on Q X D, Xk(w, w') and Xk'(w, w') are independent and equally distributed. Let v( w, w') be defined by

v(w, w')

=

v(w)

for all

w' ED',

where v( w) is the function defined before; that is, 1 all wE Q and

~

v( w)

~

n for

13.6.

581

SUMS OF INDEPENDENT RANDOM VARIABLES

Then [ Wl

v(w)

L

Xk(w)

lp]I/p

=

[

Ww

k-1

lp(w,w')

L

Xk(w, w')

lp]l/p

(22)

k-I

for all w' E !J, where Ww means the integral over Q with respect to the first element w. The terms W w' W w,w' given in (23) which follows, have meanings similar to Ww· The right-hand side of (22) is

(23)

by the Minkowski inequality [see Section 11.1, Eq. (4)]. We apply (21) to the first member of the right-hand side of (23). Actually (22) is true for any integral-valued function 'l'(w). We also note that Xk(w, w') - Xk'(w, w') is symmetrically distributed and independent from one value of k to another. Then the right-hand side of (23) is equal to or less than

The first term of (24) is, by the inequality I x equal to or less than

(where W w,w' to

= J J · dP dP',

W wWw'

+y I

P

< 2P- 1 (1 x IP + I y IP),

= JdP J · dP'),

which is equal (25)

582

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

The second term of (24) is, from Theorem 13.6.1,

(26) Hence (22)-(26) put together show (16). The case p = oo is obvious with essential bound for (~I ·IP)11P. The theorem is true if n = oo, or more precisely, the following corollary is true.

Corollary 13.6.2. If ~I Xn(w) IP < oo, 1 ~ p ~ oo, {Xn(w)} being a sequence of independent random variables with ~ Xn = 0, n = 1, 2, ... , and if ::E:=l Xn( w) is almost surely convergent and belongs to LP, then

where A 11 is a constant depending only on p. Proof.

From Theorem 13.6.1 and Corollary 13.6.1,

~I

tl

Xk(w)

lp ~

Bp

~[I

tl

Xk2(w)

~ B ~[~~1 Xk2 (w) 11

r'2]

r'

~ B 11 '~1 k-1 EXk(w) 111 ;

2 ]

13.7.

583

CONVERGENT SYSTEMS

that is, (28)

Hence from Theorem 13.6.2,

Since the left-hand side of the last inequality never decreases as n increases, we have Corollary 13.6.2, letting n-+ oo.

13.7. Convergent Systems We have seen that for the Rademacher series

almost everywhere convergence and the convergence of :L an2 are equivalent. We are now interested in the relationship between the convergence of (1)

and the convergence of :L an2 , where {Xn( w ), n = 1, 2, ... } is a sequence of independent random variables. A fundamental theorem in this connection is the following theorem of Marcinkiewicz and Zygmund [1] ; the proof is also due to them.

Theorem 13.7.1. Let Xn, n = 1, 2, ... , be independent random variables such that n = 1, 2, ....

(2)

In order that every sequence {an, n = 1, 2, ... } such that (1) converges almost surely should satisfy (3)

it is necessary and sufficient that lim inf W I Xn I > 0. n-+oo

(4)

584

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

If Xn is uniformly bounded almost surely, then without (4), the series (3) implies the almost sure convergence of (1) [Theorem 12.5.2 (ii)]. In order to show the theorem we give the following lemma.

Lemma 13.7.1. Let X(w) be a random variable. If

WI X(w) I ;;;;; A{WX2 (w)} 112

(5)

for some constant A> 0, then, writing u = {WX2}1'2,

P(! X(w) I ;;;;; Au/2);;;;; A2 /16. Proof. Write E 1 = {I X(w) I < !Au}, E 2 E 3 = {I X(w) I ;;;;; 4u/A}. Then

(6)

= {luA
that is, Hence

I

.

Ea

I X(w) I dP ~

[WX2(w)]li 2 [P(E3 ))1' 2 ~ Au/4.

Therefore from (5), (7), and (8),

I

I X(w) I dP = WI X(w) I

-I

~

I X(w) I dP-

;;;;; AU - tAU - fAU = fAU.

Hence P(E2 )(4u/A);;;;;

I

Ea

(8)

.

I

I X(w) I dP

I X(w) I dP;;;;; !Au,

which gives us (6).

Proof of Theorem 13.7.1. Necessity.

It is sufficient to show that if condition (4) is not satisfied, there be a sequence {an} such that (1) converges almost surely while 2: an 2 = oo. Suppose that W I X m,. I - 0 for some sequence {mn} of subscripts. We may choose another subsequence {nd of {mn} such that

"! 3. 7.

585

CONVERGENT SYSTEMS

Now consider the sequence {an} defined by

0, an= { 1, then I: an 2

n n

=I= nk>

=

nk>

k

=

1, 2, ... ,

= oo. On the other hand, 00

00

L WI anXn I = k-1 L WI Xn1c I < n-1

00 •

Hence by Fubini's theorem, L anXn converges almost surely. Sufficiency. In view of Egorov's theorem, if Sn(w) = L~- 1 a~k(w) converges almost surely, then there is a set E of w, for a given e > 0 such that P(E) < e, and Sn(w) converges uniformly on Ec. Hence there is a constant C such that I Sn(w) I ~ C, on Ec, n = 1, 2, .... Using the method employed in S~ction 13.5, Eq. (15) and subsequent inequalities,

f Sn (w) dP ; : ; Eak J Xk (w) dP

C 2 P(Ec) ;;::;;

E"

2

2

c

k=l

-2

2

E

L

1;:;;-i
jaiak

J

E

c

Xi(w)Xk(w) dP

I

(9) Now from (4), there is a constant A.> 0 such that WI Xk I >A.> 0, k = 1, 2, .... Write Fk = {I xk I ; : ; lA.}. Then by Lemma 13.7.1, P(Fk);;::;; A2 /16. Since P(Ec) ;;::;; 1 - e, Ec and Fk have a common part for e < A. 2 /32, and actually Therefore,

586

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

Inserting this into (9), n

C2P(Ec);?; [(A.4/128)- 2e112]

L ak2·

(10)

k=l

Taking e so small that (A.4/128)- 2e 112 > 0, we have (3). This completes the proof of the theorem. The connection between the convergence of ( 1) and (3) is of interest even in the more general case in which {Xk(w)} is not a sequence of independent random variables and condition (4) plays an important role. A sequence {Xn(w)} of random variables is said to be a convergence system if every series ( 1) with I: an2 < oo converges almost surely. From the viewpoint of analytical interest, we consider [0, 1] in place of D. If a sequence Un(x ), n = 1, 2, ... } of measurable functions on (0, 1) satisfies lim inf n-+oo

f ifn(x) 0

1 dx

>

o,

we say that Un(x)} satisfies the condition A. The problem, when Un(x)} has a subsequence Unk(x)} that is a convergence system, that is, when {/n(x)} contains a convergence subsystem, is of interest and the following result was given by Gaposhkin (although we are not going to prove it). (See Gaposhkin [1].)

Theorem 13.7.2. For a sequence of functions Un(x)} over (0, 1) with J!fn2 (x) = 1 to contain a convergence subsystem, it is necessary and sufficient that it satisfy the condition A.

13.8. Integrability of Sums of Series; Strong and Weak Convergences of Series Let Xn(w) E LP, that is,~ l Xn(w) iP < oo for n = 1, 2, .... We often say that the series :E:=l Xn(w) converges strongly in LP to S(w) E LP if it converges in LP; that is, as n--+ oo,

(1)

where Sn( w) is the partial sum :E~- 1 Xk( w ), in contrast to saying that the series converges weakly to S(w) in LP, p > 1 if, for any random vari-

. 13.8.

INTEGRABILITY; STRONG AND WEAK CONVERGENCES

able Y(w) e Lq, p- 1

+

q-1

587

= 1,

gosn(w)Y(w)- gos(w)Y(w)

as n-oo

(2)

(not the weak convergence of distribution, of course). Let {Xn(w), n = 1, 2, ... } be independent. The LP-theory of a series of independent random variables depends basically mostly on Theorems 13.6.1 and 13.6.2 or Corollary 13.6.1. The results in this section as a matter of fact are due to Karlin [1]. We begin with the following lemma.

Lemma 13.8.1. Let p ~ 1. Suppose that {Xn( w)} is independent with Xn(w) E LP, and goxn(w) = 0, n = 1, 2, .... The necessary and sufficient condition that the series (3) converge almost surely and the sum S( w) e LP is that

(4) for all n, where C is a constant and Sn( w) is the nth partial sum of (3 ).

Proof. Necessity. This is no more than (28) in Section 13.6 with C = Bp' go I S(w) IP. Sufficiency. From Theorem 13.6.1, we have

where C 1 , C2 here, and C 3 , ••• , in what follows, are constants. Since 1:::'=1 Xn 2 increases with m, 1:~ 1 Xn 2 converges almost surely to a random variable belonging to LP12 • Hence by the monotone convergence theorem, as n- oo.

(5)

Write Tn,m(w) = ma""n+ 1 :;;~::;;ml Sm- S~; IP. Then Tn,m is, for each w, increasing monotonically for m. Hence limm-+oo Tn,m exists as a finite value or an infinite value. Since this is again decreasing monotonically for n, limn-+oo limm-+oo Tn,m = T exists as a finite value or infinity. From the monotone convergence theorem,

go T

=

lim n-+oo

w Tn = nlim m-+oo lim go Tn,m ;;;;; Ca lim lim go I sm- SnIP n n,.-+oo -+co

-+oo

(6)

588

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

(by Theorem 13.6.2) and then

(7)

which is zero by (5). Since T ~ 0, 'if'T = 0 gives us T = 0 almost surely. This means that Sn converges almost surely. If Sis its limit, then from Fatou's lemma, 'if' I s lp ~ lim inf 'if' I sn lp ~ C; n-+oo

that is S

E

LP. This shows the lemma.

The proof of Lemma 13.8.1 substantially implies the following lemma.

Lemma 13.8.2. Let p ~ 1. Let {Xn} be independent with Xn E LP, 'if'Xn = 0, n = 1, 2, .... Also, (4) is the necessary and sufficient condition that L~=l Xk(w) converge strongly in LP. Actually, if sn converges strongly to s E LP in LP, then

(by the Minkowski inequality) with some constant C. Hence (4) is necessary for LP convergence. (6) and (7), with (5) show the sufficiency of (4). The above two lemmas show that for a series of independent random variables, the almost sure convergence to a random variable belonging to LP is equivalent to LP-convergence (strong). We state this more completely in the following theorem.

Theorem 13.8.1. Let 1 ~ p. For the series L.::'= 1 Xn(w) of independent random variables, with Xn(w) E LP, 'if'Xn = 0, n = 1, 2, ... , the following five statements are equivalent to each other: (i) Sn(w) converges almost surely to a random variable belonging to LP. (ii) Sn(w) converges strongly in LP. (iii) Sn(w) converges weakly in LP. (iv) Sn(w) converges in probability to a random variable belonging to LP. (v) Sn( w) converges in distribution, with the limit distribution function having finite absolute pth moment.

13.8.

INTEGRABILITY; STRONG AND WEAK CONVERGENCES

589

Proof. The equivalence of (i) and (ii) is no more than Lemma 13.8.2. If Sn(w) converges weakly in LP, then from the basic result on weak convergence, 8" I Sn(w) jP ~ C for some constant C. Then Lemma 13.8.1 shows the equivalence of (ii) and (iii). The equivalence of (i) and (iv) or (i) and (v) is obvious from the equivalence theorem (Theorem 12.4.2). This shows the equivalence of all five statements.

In the above proof we used the equivalence theorem in order to show the equivalence of (i) and (iv). But as a matter of fact we can avoid the use of it and show directly that (i) and (iv) are equivalent if we use Theorem 13.6.1. It is of course sufficient to show that (iv) implies (i). The convergence of Sn( w) in probability implies that a subsequence {Snk(w)} of {Sn(w)} converges almost surely (Theorem 12.1.10). Hence the following lemma finishes the proof of the implication (iv) =- (i).

Lemma 13.8.3. Let p

~

1. If a subsequence Snk( w) of the sequence {Sn(w)} of partial sums of the series (3), with independent Xn(w), Xn(w) E LP, and 'if'Xn = 0, n = 1, 2, ... , converges almost surely to a random variable belonging LP, then Sn( w) converges almost surely to the random variable.

Proof.

Write k = 1, 2, ....

Then { Y k, k = 1, 2, ... } is a sequence of independent random variables for which 'if' Yk = 0, Yk E LP from the assumption that Yk( w) --+ S( w) ask--+=, where S(w) E LP. Hence from Lemma 13.8.1,

(8) with some constant C. For every n, we choose nm ~ n ~ nm+1· Then from Theorem 13.6.1,

(Theorem 13.6.1 was used twice in the first inequality and the last inequality), which is equal to

c2 wl f

k-1

Yk

lp ~ c

590

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

by (8), where c)' c2 are constants. Hence 13.8.1 gives us the conclusion.

w I SnIP~ c. Then Lemma

The combination of this lemma and Theorem 13.8.1 gives us the following theorem.

Theorem 13.8.2. The weak, strong, or almost sure convergence to a random variable belonging to LP (p ;;:;: 1) of any subsequence Snt of Sn = L~=l X,., with independent random variables X,. E LP, W X,. = 0, implies the same conclusion for the entire sequence Sn. Finally we shall give the proof of Theorem 12.10.8; that is, if :L~ 1 Xn converges almost surely to a random variable belonging to LP, p ;;:;: 1, then it is unconditionally convergent. W I XnP I < oo, WXn = 0 are supposed. Proof of Theorem 12.10.8. Let {p 1 , p 2 , ••• } be any rearrangement of {1, 2, 3, ... }. For any n, choose N so that {p 1 , •• • ,pn} is a subset of {1, 2, ... , N}. By Theorem 13.6.1,

where C1 and C 2 are constants. Since the last one is bounded by a constant because of the assumption that L Xn converges to a random variable belonging to LP almost surely (Lemma 13.8.1 ), again from Lemma 13.8.1, the series L XP 1 converges almost surely. This shows the unconditional convergence of L Xn.

13.9. Vanishing of the Sum of a Series We consider series of the form (1) where {Xn(w), n = 1, 2, ... } is a sequence of independent random variables. Suppose that (1) converges to S(w) almost surely. We are interested in the uniqueness property of (1). Suppose S(w) = 0 almost surely. Then for

-oo
(2)

13.9.

591

VANISHING OF THE SUM OF A SERIES

where fn(t) is the characteristic function of Xn. This implies that fn(ant) = 1, -oo < t < oo, for every n. Unless Xn(w) = 0 almost surely, an= 0. In other words, if all Xn(w) are not degenerate, then S( w) = 0 a.s. implies an = 0, n = 1, 2, .... We ask the question: Do we have the same conclusion if P[X(w)

> 0? This is answered negatively. Let us take a sequence of Rademacher functions and consider = 0]

t

E

[0, 1],

(3)

for which :L an2 < oo. Then (3) converges almost everywhere. Now choose an so that (4)

where m is any fixed positive integer. Take the interval (0, 2-m) in which rk(t) = 1 fork= 1, 2, ... , m. Hence for

0


2-m:

but the ak (1 ~ k ~ m) are not necessarily zero. In this example an+l = am+ll = · · · = 0. Actually, we may show that if :L~ 1 akrk(t) = 0 in some interval of the form [kf2m, (k 1)/2m] for some integer m (k = 0, 1, 2, ... , 2m- 1), then an= 0 for n > m. This is implied by Theorem 13.9.1 which will follow. Let {Xn(w), n = 1, 2, ... } be a sequence random variables. Let .5fln,m (m > n) be the 0'-fi.eld generated by Xn, Xn+I•· . . , Xm, that is, the smallest 0'-fi.eld which contains all the sets {w : Xn( w) < Xn, . .. , Xm(w) < xm}. {xn•· .. , Xm} being any set of m- n 1 real numbers. We write .5flm for .5flm,m· Also we agree to write

+

+

t

X(w) dP = W(X; B)

(5)

for a random variable X(w) and a set B e.5fl. We now show the theorem.

Theorem 13.9.1. Suppose that {Xd is a sequence of independent random variables with O'k2 = WXk 2 < oo, k = 1, 2, .... Suppose also

592

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

= 1, 2, ... , and

that wxk = 0, ak 2 -=1= 0, k

00

L ak2ak2
(6)

k-1

Let m, n be fixed positive integers such that m > n. If the sum of the series (1) is zero almost surely on a set of positive probability in -~n.m• then we have

k< n

for

k > m.

and for

(7)

If (6) holds then ( 1) converges almost surely. We may replace (6) by the assumption that ( 1) converges almost surely to a random variable belonging to £2, in view of Theorem 13.8.1.

Let Sn(w) be the partial sum of (1). From Theorem 13.8.1 or directly from the fact that

Proof.

W[Sk-

slp

l

=

6"

I

L aiajxixj = i-t+l L ai Ui i-t+l 2

2 '

l

>k

(because WXiXi = W Xi W Xi = 0, i -=/= j), St converges in £2 to the sum S of the series (1 ). Hence Sk converges weakly in £2 to Sand thus as Since akak2

=

W(S.vXd for N

Suppose that S Now

N--->- oo.

> k, we see that

= 0 a.s. on B, where B is a set of ~n.m with P(B) > 0. N

akak2 = W(SXk; Be)=

L ai l
(8)

j-1

for N > k. Suppose k > m. If 1 Xn,. . . , Xm, and

in view of wxi = 0, i If n ~ j ~ m, then

~

j

~

n- 1, then XJXk is independent of

= 1' 2, ....

13.9.

593

VANISHING OF THE SUM OF A SERIES

Finally, if m

+

1

~

j

~

N, then

for j -::f::- k, for j = k. Hence altogether, from (8), for

akak2 = a,J'(B 2 )ak2

k > m.

Since ak -::f::- 0, and P(Bc) = 1 - P(B) < 1, we have ak Similarly we ll.ave ak = 0 for k < n.

=

0 fork> m.

Next we ask how large P(S = 0) can be. We easily have the statement for a Rademacher series. There is an absolute constant A ( < 1) such that, for any ak such that 0 < :Lk:t ak2 < oo, m{t: S(t) = 0} ~A, (9) where S = L~=I a 11T 11 (t). Actually (9) is true with A= 2 (VZ- 1). This is seen from Section 13.5, Eq. (15) and the subsequent inequalities or from Section 13.7, Eq. (9). Actually we have seen there, that for any measurable set E,

where S 11 (t) is the partial sum of that S 11 (t) converges in £2 to S,

2: a,r (t). Letting n --+ oo and noting 11

(10)

Let Ec

=

{t; S(t)

=

0}. Then we have

mEc

~

2(mE)ll2,

which gives us mEc ~ 2 (VZ - 1). We now show that the same result is true for a more general sequence of random variables.

Theorem 13.9.2.

Let {Xn, n = 1, 2, ... } be a sequence of independent random variables with W X,. = 0 and

I Xn I ~ 1

a.s.,

0 < a2 ~ 'if' X"2'

(11) (12)

594

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

for n = 1, 2, ... , where a 2 is a constant. Let {an, n sequence such that

= 1, 2, ... } be any (13)

Then for any a > 2, there is a constant A which depends only on a such that P(S = 0) < 1 - a" exp( -Aa2), (14) where S(ro) = L:, 1 anXn(w). It is emphasized that A depends only on a and is independent of a and {an}· The convergence of L anXn is obvious from (13). Proof. We use the method employed in the proof of Theorem 12.11.3. Let {bn, n = 1, 2, ... } be a sequence of numbers, which will be determined later. Then as in Section 12.11, Eq. (10),

w[s fi (1 + b~,.>] = w[s(1 + t, b,.x,. + L b,.b;X~; + k-1

k-1

k<j

+ b1b2 · · · bnX1X2 · · · Xn)],

(15)

where for k =I= j, we see that ff(SX~;)

=a,. WX,. 2X; +a;

ffX~l

=a,. WX,. ffX; +a; WX,. ffXl = 0, 2

and also that the expectation of S multiplied by each term after the third term in the parentheses in ( 15) vanishes. Therefore, we have

where a,.2 = ff X,.2 • Now

w[ S ,.a (1 + b~,.)] w[ S ,.I] (1 + b~,.): {S =I= 0}] =

~

[P(S =1= 0)]11"(ffS2)112

. ( ff

,.a 11 + b~,.,prp

by the Holder inequality, where (1/a)

+ (1/fJ) + (1/2) =

(17) 1.

13.9.

595

VANISHING OF THE SUM OF A SERIES

Let y be an integer such that {J

~

y. Then

(18) Now choose b,. = ba,., where b is a constant such that

I ba,. I ~ t. Since by (11) and (19), I b,.X,. I ;:;;; 1, we have 11 + b,.X,. 1,

=

=

(1 + b,.X,.)Y

(19)

1 + yb,.X,. +

t (~ ) J

J-2

and

w l1+h,.X,. 1, =

1+

t (~) J

J-2

Wb/Xr!

~

1+

b,/X,/

f (J~) b,.

2

i=2

wx,.~

[because of I b,. I ;:;;; 1, and (11 }],

W 11 + b,.X,. 1, = 1 + b2Ca,. 2 a,.2 where

t()

c = i-2 J~ =

;:;;;

exp b2Ca,.sa,.2 ,

(20)

2, - 1 - 'Y.

Hence from (16)-(18) and (20}, we have

[P(S =f=. 0))1'"

~ b[ ~~ a,. ak 2

2

r

2

1

exp[ -(b 2 C/y) "'~ a,.2 a,. 2 ].

Now choose

We see that 1 ba,. 1 ;:;;;

aa,./(2, a,.sa,.2)112 ;:;;; a,.a,./(2, a,.~a,.2)112

that is, (19) is satisfied. Then (21) turns out to be [P(S =f=. 0)]1 1" ~ a exp( -Ca2 /y),

which shows (14).

~

1,

(21)

596

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

13.10. Summability of Series Let

... ...

)

(1)

.be a matrix of infinite order with the condition for

lim bnk = 1

k

= 1, 2, ....

(2)

n->-oo

Suppose that for a given series

:L%: 1 xk,

an=

l: bn,kxk k-1

the series

00

(3)

converges for each n = 1, 2, .... If lim

(4)

an= s,

n->-oo

then :L%: 1 xk is said to be summable T* to s. There is a small difference between this definition of summability and the T -summability defined in Section 4.5, Theorem 4.5.1. If we write cn,k = bn,k - bn,k+l and consider 00

Tn

where From

sk

=

L:=

1 xi,

=

then when .v

L Cn,!.Sk> k=1 T 11

-

s,

:L xk

(5)

1s called T-summable.

.V-1

L bn,!.Xk = k=1 L Cn,ksk + bn,.vs.v, k=1 if, for instance, b 11 ,k = 0 for k > n and consequently cn,k = 0 for k > n, then the T -method and T*-method are the same. In particular, if bn,k = 1 - [(k- 1)/n] (k ~ n), h11 ,k = 0 (k > n), then cn,k = 1/n for k ~ n, Cn,k = 0 for n > k, and both reduce to (C, 1)-summability. We are here concerned with T*-summability. Let us first consider, for convenience, a series of the form (6)

13.10.

597

SUMMABILITY OF SERIES

We shall show the following theorem.

Theorem 13.10.1. Let {X,(w), n = 1, 2, ... } be a sequence of independent random variables with (7) Suppose that

lim inf W I X,

I>

0.

(8)

n-+oo

If the series

(9) is divergent, then almost surely the series (6) is not summable for any T*-method. That is, if (6) is T*-summable with positive probability, then (9) must converge and by Theorem 13.7.1, (1) converges almost surely. Proof. Suppose that (6) is T*-summable, 00

an( w)

=

L bn,kakXk( w)

k-1

is almost surely convergent, and an(w)- S(w) on E with P(E) > 0 for some random variable S( w ). From Theorem 13.7.1, Lk: 1 b~.kak2 < oo and 1111 (w) E £2. Since a,(w) converges to S( w) on E there is a set F of w with the property that a11 (w) converges uniformly to S(w) onE n pc and P(F) < e, e being an arbitrarily given positive number. Hence there is a constant C such that lan(w) I ~ConE n pc, n = 1, 2, .... Then

C 2P(E n Fe)~

I

EnF"

a,. 2 (w) dP =lim s -+oo

I

c

Er'IF

a! ..v(w) dP,

(10)

where a,,.v( w) = Lf= 1 bn,kakXk( w ). The last equality follows from the fact that a 11 ,.v(w) converges strongly in £2 to a 11 (w) by Theorem 13.8.1. As in the proof of Theorem 13.7.1 (9),

I

EnF"

a~ ' .v(w) dP ~ k'~=l bn '1flk 2 IEr'IFc -2

L

I;;;i
Xk 2 (w) dP

Ib,.,ibn,kaiak I

Er'IF

c

Xi(w)Xk(w) dP

I

598

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

From this and (10), we get, as we did in Section 13.7, Eq. (10), C 2P(E

n pc)

N

~

[(A.4j128) - 2e 112 ]

L b~.kak2 ' k-I

where A. is a positive number such that

n = 1, 2, .... Taking e so small that (A-4/128)- 2e 112 > A-4/130,

C2 ~ (A-4/130)

N

L b~.kak2

k-I

for all N and n. Letting n --+ oo, we have, using (2), N

L ak2 ~ 130C2JA.4, k-I

which implies the convergence of I:k: 1 divergence. This shows the theorem.

ak2 ,

and thus contradicts its

In particular, we have the following theorem.

Theorem 13.10.2. If the Rademacher series

is T•-summable on a set of positive measure, then I: an 2 < oo and I:::'= 1 anrn(t) converges for almost every t (0 ~ t ~ 1). In other words, if I: an 2 = oo, then for almost all t, the Rademacher series is not summable by any T•-summability method. We are now going to show that T*-summability is equivalent to essential convergence.

Theorem 13.10.3. Let (11) be a series of independent random variables. If (11) is T*-summable almost surely, then it is essentially convergent, that is, there is a sequence {A.t} of numbers such that 00

L

k=I

{Xk(w)- A.d

13.10.

599

SUMMABILITY OF SERIES

converges almost surely. If Xk( w) is symmetrically distributed, k = 1, 2, ... , then Ak is chosen to be zero, k = 1, 2, .... The theorem is due to Marcinkiewicz and Zygmund [2]. Proof. Note that the assumption that (11) is T*-summable implies the almost sure convergence of :L~ 1 ai.~k(w) and Si(w) = L~ 1 ai,kXk(w) converges almost surely to some random variable S(w). Let fk(t) be the characteristic function of Xk(w). Let Xks be the symmetrized random variable of Xk(w) and be Xk(w)- Xk'(w) (where Xk' is independently and identically distributed with Xk)· Then :L~ 1 ai ~k, ( w) also converges almost surely and hence 00

sis(w) =

L ai,kxks(w) k-1

also converges almost surely and Sis( w) converges in distribution as i--. oo since Si(w) converges almost surely by assumption and S/(w) = L~ 1 ai,kXk'(w) has the same distribution as Si(w) and is independent of Si( w ). Let rp be the characteristic function of the limit distribution function of sis( w). Then rp is n~1 I it( ai,kt) 1 which is nonnegative. Hence rp( t) > 0 in some neighborhood [- 15, 15] of the origin.

2,

nift(ai,kt) oo

0

< rp(t) =lim

i-+oo k-1

12

~lim

nlfk(ai,kt) n

12

i=oo k=1

for every positive integer n, which is equal to

by condition (2). Since n is arbitrary, we have

o < rp(t) ~ k=1 nl~t(t) 1 00

2

over [ -15, 15]. Hence from Theorem 12.9.2, llk: 1 I ft(t) 12 must be a characteristic function and consequently n~1 xks converges almost surely (by Lemma 12.9.1 and the equivalence theorem), which is equivalent to the conclusion, by Theorem 12.12.1, that :Lk:, 1 Xk is essentially convergent. The second statement of the theorem is obvious.

600

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

We now prove the converse theorem.

Theorem 13.10.4. If a series (11) of independent random variables is essentially convergent, then there is a T*-summability method according to which (11) is summable. This theorem is due to Tucker [2]. The following is his proof. Proof. Suppose that there is a sequence {An , n = 1, 2, ... } of numbers such that

(12) converges almost surely. We are going to define a matrix T* such that (2) holds,

= (bn,k)

00


=

I, bn,kXk

(13)

k-1

converges almost surely, and a11 (w) converges almost surely as n- oo. We assume that L A11 does not converge almost surely, that is, L An is divergent. Otherwise b,,k is taken to be 1 for all n, k. We split the proof into three cases. (i)

Suppose that {A 11 } is unbounded. Then for every n there 1s an m~ > n such that I L~~~ An/ Am,. I ~ 1. Define for

1

~

for

k

= mn,

k

for k > n,

~

n,

k =I= mn .

Then (2) obviously holds. We have

an=

f Xk -( ·•-1t AdAm,.) Xmn

k=1

=

t

k-1

(Xk- Ak)- (

t

A;/Am,.) (Xm,.-

i=1

~).

From the almost sure convergence of (12), the first series on the right of the above series converges almost surely. Since we also have Xm,.Am,.---+ 0 a.s., the second term on the right converges to zero almost surely. Thus <111 converges almost surely to (Xk- Ak), which shows case (1 ).

Lr

13.10.

601

SUMMABILITY OF SERIES

(ii) Suppose that {A.,} is bounded and there is an increasing sequence {m,, n = 0, 1, 2, ... } of subscripts such that

is not bounded. Then, denoting by Yn,

which converges almost surely. Case (i) applies for I: ( Y n - 1-'n) and there is a sequence {b~.d such that b~.k- 1 as n- oo for each k and a,'(w) = I;k:, 1 b~.kyk converges almost surely and O'n'(w) converges as n- oo almost surely. Let bn,k = b~.z for m1_ 1 + 1 ~ k ~ m1• m

m,_

m

1

I

m

L bn,kXk= k=l L b,.,kXk+ k-m,_L +1 bn,kXk= k=l L b~,kYk+b~,l k=mz_ L +1 (Xk-Ak) k=l 1

1

for m1_ 1 < m < mz, which converges as m-oo to an'(w) almost surely. Since an'(w) converges almost surely, and b,,k- 1 as n- oo is obvious, the required proof is shown for this case. (iii) Suppose that {}. 11 } is a bounded sequence and there is no increasing sequence {m,} of integers such that {p,, }, as defined in (ii ), is unbounded. Write v, = A1 + · · ·+ A,. Then the Vn have a finite upper bound because, if this were not true and the v, had no upper bound, then we might find an unbounded sequence {p,, }, choosing m, in such a way that

For vn having no lower bound, the same reasoning applies. Hence {vn} is bounded. Then there are at least two limiting points of {v,} since if there were only one, this would mean that :L An would converge, which contradicts the assumption that L A, diverges. We may suppose without loss of generality that the smallest limiting point is 0 and the largest is a (>0). Let e > 0, 3e > a. Then there is a sequence of subscripts {m,} such that a- e

<

'~~man-1

<

a

+ e,

602

XIII.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

and -B

<

'Vn


for all n

~

m1 •

Define

Then

and

1~k

~ mn,

mn < k ~ mn+I• k > mn+I·

Thus

Since Kn is bounded from (14), Gn converges to :Lk.,1 (Xk- A.k) almost surely. This completes the proof. NOTES 1. Theorem 13.1.3 is an improvement of the results obtained by Kawata and Udagawa [1] (Theorems 10 and 11).

2. In connection with Theorem 13.3.4, the question arises as to whether or not the characteristic function of a Rademacher series could be O(e-PIII) for large I t I if J'; [p(t)/t 3 ] dt = oo. For this question, we have the following answer. If I:-a,.1 log a;;1 < oo, then the characteristic function of any series I: a,.r,.(w), wE [0, 1], cannot be O[exp{ -PC It I)}] for p(t) with J';' p(t)ft3 dt = oo, If I: a,. 1 log a;; 1 = oo, then for the characteristic function f(t) of I: a,.r,.(w ), J';' [log If(t) l/t8 ] dt = - oo, (See Kawata [3].) 3. As a generalization of Theorem 13.4.2, Ivasev-Musatov [1] has shown that for the Fourier-Stieltjes coefficients c,. = J:'" r'""' dF(x), of a monotone singular functionF(x), I::C, I c,. 11 can diverge as slowly as desired, in a certain sense. Precisely, he proved the theorem: If x(y) is a positive monotonically decreasing function over [0, y], which is differentiable for small y ~ 0 and satisfies

I:

(i) 'l'(y) = x•(y) dy -+ 00 as y -+ oo, (ii) yl+•x•(y) -+ oo as y-+ oo for any e > 0, (iii) YX1 (y) -+ 0 as y -+ oo, (iv) there is a Yo such that for ally > y 0 and any 8 ~ 1, (y)fx(8y) ~ 8314 ,

603

NOTES

then we can construct a continuous nondecreasing singular function F(x) on [0, ln], such that c,. = o[x(l n !)]. 4. In connection with Theorem 13.5.2, Zygmund [2] showed that if ~ a,.r,.(t) is a Rademacher series and if 1 < p ~ 2, r;;;; 1, and ~I a,. I" < oo, then the sum f(t) of ~ a,.r,.(t) (which must converge almost everywhere) satisfies

(J1I

o f

I' dt

)1/t ~ rll'' ( •~o00 I a. I~~ )1/P ,

where (1/r) + (1/r') = 1. He also showed that under the same conditions, exp(.A. 1/(t) for every .A., (1/p) + (1/p') = 1. If 2lep'c"' ~ 1, where c" exp(.A. If I>" dt < 2.

J!

I"') e =

~

L'(-oo, oo) then

I a,. I",

5. The differential properties of a function /(t), which is represented as the sum of a Rademacher series ~:;': 1 a,.r,.(t), 0 ~ t ~ 1, are studied. The function/(t) has a derivative at least one point if and only if lim,._,.00 2"a,. exists and is finite (Balasov [1 ]). If ~ a,./2" < oo, then f(t) is of bounded variation and, in particular, if a,. = 0(2"n log n log1 n · · · (log, n)l+•)- 1 for some e > 0 and some positive integer p, where logk n denotes the kth repeated logarithm of n to the base 2, then f(t) is differentiable almost everywhere (McLaughlin [1]). On the other hand, it can be shown that for any integer p ;;;; 2, there exists a sequence {a,.} satisfying a,. = 0(2"n log1 n log3 n · · · log, n)- 1 with the property that f(t) = ~ a,.r,.(t) is differentiable almost nowhere (McLaughlin [2]). 6.

Lemma 13.6.2 is true with the assumption that X has a median zero instead of

'i!'X = 0, Lemma 12.9.3 being used. 7.

Theorem 13.6.1 ceases to be true if p

< 1. (See Marcinkiewicz and Zygmund [1].)

8. In connection with Theorem 13.8.1, we remark that a uniqueness theorem of this type holds in a complete form for a lacunary trigonometric series: If {nk} is a sequence of positive integers with nk+llnk ;;;; 8 > 1, 8 being a constant, and the trigonometric series ~~ 1 (ak cos n~ + bk sin n~) converges to zero in a set of positive measure, then ak = bk = 0 for all k. This is Zygmund's theorem. (See Bary [1], Vol. 2, p. 265 and Zygmund [3], Vol. 1, p. 206.) 9. In Theorem 13.8.1, the equivalence of (i) and (ii) was also shown by Marcinkiewicz and Zygmund [1]. 10. We may show that an orthonormal system of functionsj,.(x) satisfies condition A in Section 13.7 and from this, we see by Theorem 13.7.2, that every orthonormal system has a convergent system. This was given by Menchoff and the problem was studied also by Marcinkiewicz and Senchishchev. (See Gaposhkin [1], pp. 3, 4.) 11. Theorem 13.9.1 was shown by several authors (Levy [3], Zygmund [2], Tucker [2], and Arnold [1]). The proof here is due to Arnold [1]. 12. Let T = (a,.k), n, k = 1, 2, ... , be a given infinite matrix. Let T,. = ~~1 Clni:Sk, where sk is the partial sum of ~~. X 1 • If T,. ~ l almost surely when X;= 0 or 1, respectively, with probability i, then (T) is said to have the Borel property (BP). The

604

XII I.

PROPERTIES OF SUMS OF INDEPENDENT RANDOM VARIABLES

study of BP has been made by Hill [1, 2]. All the Toeplitz properties in Section 4.5 are not needed. One of Hill's results is the following: In order that T have the Borel property, it is sufficient that 1:~ 1 ank converge for each nand tend to 1 as n __.. oo, and that 1::;':, exp(-15 2 /An) < oo for each t5 > 0, where An= 1:~, ank· In connection with this, Chow treated the convergence problem concerning the almost sure convergence of 1:~ 1 an~k, where {Xk} is a more general sequence of independent random variables (Chow [1]).

13. Theorems 13.7.1, and 13.10.1 were extended to the case in which the Xn are not necessarily independent by Gundy [1] and Chow (2].

XIV Some Special Series of Random Variables

14.1. Fourier Series with Rademacher Coefficients Let {rn(t), n = 1, 2, ... }, 0 functions. Consider 00

L rn(t)An(x) =

n-o

~

t

~

1, be a sequence of Rademacher

00

!a0

+ n=l L rn(t)(an cos nx + bn sin nx),

(1)

where r0 (t) = 1, 0 ~ t ~ 1. We suppose an and bn to be real. Let {rn(t)} be thought of as a sequence of independent random variables, where D = [0, 1], the a-field is the class of Lebesgue measurable sets, and the probability is the Lebesgue measure. We then often use "almost surely" in place of "almost all t." We begin with the following theorem. 605

606

XIV.

Theorem 14.1.1.

SOME SPECIAL SERIES OF RANDOM VARIABLES

(i) If the series (2)

converges, then the series (1) converges almost surely for almost all x, -:n ~X~ :n.

(ii) If series (2) diverges, then series (1) is almost surely not a Fourier senes. Before proving the theorem, we remark that if a proposition P, depending on two parameters t and x in a ~ t ~ p, a ~ x ~ b (a, b, a, p are supposed to be finite), holds for almost all t for each x E E, where E 0 has measure zero, then it holds for almost all x for each t E F, where F is a set such that the measure of F 0 is zero. This is obvious from the Fubini theorem because, if we write P(x, t) = 1 when P holds and P(x, t) = 0 when P does not hold, then P(x, t) = 1 for almost all t if xe E. Hence

I:

I: I:

Then

for each

P(x, t) dt = (P- a)

dx

dx

P(x, t) dt

E

E.

P(x, t) dt = (b - a)(a - a)

since mE= b- a. Now since P(x, t)

I: I:

x

=

I: I: dt

~

0, by the Fubini theorem,

P(x, t) dx

=

(b - a)(e - a),

J!

which implies that P(x, t) dx is a measurable function of t and is 1 for almost every t, or that there is a set Fin (a, p), with mF = P- a such that for t E F, P(x, t) = 1 for almost all x. This shows the above statement. Proof of Theorem 14.1.1. (i) Since :L (an 2 cos 2 nx + bn2 sin 2 nx) converges for every x by the convergence of (2), series (1) converges for every x for almost all t. Then the conclusion of (i) follows from the above-mentioned statement. (ii) The divergence of (2) implies the divergence of 00

L (an cos nx + bn sin nx)

n-1

00

2

=

L An (x) 2

n-1

(3)

14.1.

607

FOURIER SERIES WITH RADEMACHER COEFFICIENTS

for almost all x. This is shown in the following way. Suppose that :E;:'= 1 An2 (x) converges in a set G, mG > 0. Then there is a set G' c G with mG' > 0 such that :E An2 (x) converges uniformly in G' and hence there is a constant C such that :E;:'=1 An2 (x);;;;; C on G'. Write An(x) = f!n cos(nx + En)· Then

r

!?n2

n-1

f ,cos (nx + ~n) dx;;;;; CmG'.

(4)

2

G

Now

f

G'

cos 2 (nx

+ En) dx = =

1 2

J

G'

[1

+ cos 2(nx + ~n)] dx

21 mG' + 21 cos Un

+~

sin Un

f

G'

cos 2nx dx

JG' sin 2nx dx.

When n --+- oo, the last two terms converge to zero, because the integrals there are cosine and sine coefficients of a function that is one on G' and zero outside G' and the Riemann-Lebesgue lemma can be applied. Hence from (4 ), for a sufficiently large N, CmG' ~

L f!n n-N 00

2

f

,

G

cos 2 (nx + ~n) dx ~

[taking N so large that fa, cos 2 (nx mG' > 0, this gives us

+ En) dx >

1 L !?n mG' 4 n-N 00

2

!mG' for n ~ N]. Since

Hence :E f!n 2 = :E (an 2 + bn2 ) < oo, which contradicts the assumption. Once (3) diverges, we have, from Theorem 13.9.2 that (1) is almost surely never T•-summable for almost all x, that is, almost surely (1) is for instance not (C, 1 )-summable for almost every x. But if (1) is a Fourier series on a set of t of positive measure, it must be (C, 1 )-summabie. This is a contradiction and shows the theorem. The following important theorem was given by Paley and Zygmund. (See Zygmund [5], Vol. I, p. 219.)

Theorem 14.1.2. If (5)

608

XIV.

SOME SPECIAL SERIES OF RANDOM VARIABLES

for some e > 0, then series (1) converges uniformly for x, almost surely, and hence (1) is the Fourier series of a continuous function almost surely. This theorem was generalized in the following form by Salem and Zygmund [1].

Theorem 14.1.3. Let 00

Rn

L (a,.a + b,.2).

=

(6)

k-n

If 00

L Ri.'

2

fn logll 2 n < oo,

(7)

n=2

then (1) is uniformly convergent for x almost surely and hence the sum of (1) is continuous almost surely. Before giving the proof of Theorem 14.1.3 we first show that (7) is more general than (5). Suppose that (5) holds. Noting that n

L (log• k/k)

~

n

L k-

log• n

k-2

~

1

logi+• n

k=2

and 00

Rn logl+• n

~

L (a,. + b,. 2

2)

logl+• k,

k-n

we see that N

N-1

L (Rn log• nfn) = L (Rn n-2

n-2

~

N

n

Rn+I)

L (log• kfk) + R.v L (log• kfk) k-1

k-2

N-1

L (an + bn2 ) 2

oo

logl+• n

n-2

+ n=N L (an + bn 2

2)

log 1 +• n

00

=

L (an + h 2

11 2 )

logl+• n.

n=2

Hence

Rn log• n n

)1'2(

oo

L

n-2

1 nlog 1 +•n

)1/2 < oo.

We shall show the proof of Theorem 14.1.3 (following Salem and Zygmund) but first we need some lemmas.

14.1.

609

FOURIER SERIES WITH RADEMACHER COEFFICIENTS

Lt=o (ak cos kx + {Jk sin kx) be a trigonometric polynomial of order n (ak, fJA-> real). Let Mn = maxrl P 11 (x) I and let 0 be any positive constant 0 < 0 < 1. Then there is an interval with length equal to or greater than (1 - O)fn, in which I P 11 (x) I ~ M 110.

Lemma 14.1.1. Let P11 (:~) =

Proof. Suppose that Mn = I P 11 (x 0 ) I and that there is an x 1 , which is the first point to the right of x 0 , such that I Pn(x) I = MnO. Then Mn(1 - 0)

I Pn(Xo) I - I Pn(x.) I ~ I Pn(x 0 ) - P 11 (x 1 ) I ~ (x 1 -

=

z

By the well-known Bernstein theorem, maxzl Pn'(x)

Since Mn

I·

x0 ) maxi Pn'(x) '

I ~ nM

11 •

Hence

> 0, x1

-

~

x0

(1 - O)fn.

(8)

Hence (x 0 , x 1 ) is the required interval. If there is no x 1 with I P 11 (x 1 ) I = M 11 0 for x 1 > x 0 , then this means that I Pn(x) I ~ M,O for any x > x 0 and hence (8) holds, taking x 1 large enough. This shows the lemma.

Lemma 14.1.2. Let P11 (t, x) =

J: ~~:n Proof.

:L~ 1

akrk(t) cos kx. Then

mxaxiP11 (t, x) I dt

~ C(A 11 log n) 112 ,

(9)

Let A> 0. Then from Lemma 14.1.1, for 0 < 0 < 1,

J:,. cxp[A I P (t, x) IJ dx ~ [(1 11

where M 11 (t) = max,l P 11 (t, x)

I·

O)fn] exp[OAM11 (t)],

Since this inequality is true for every n,

J:, exp[A I Pn
0)/n(t) exp[OAM11 (t)(t)],

where n(t) is a measurable integral-valued function such that 1 ~n(t)~n. Then we have

J:, exp[A I Pn
O)fn] exp[OAMn
(10)

XIV.

610

SOME SPECIAL SERIES OF RANDOM VARIABLES

On the other hand, from Section 13.5, Eq. (20) (using elul

J~ exp[A.I Pn(t, x) I] dt ~A exp( ~ A kt1ak2cos 2

~

2

~ eu

+ e-u),

kx)

A exp( !A. 2An),

where A is an absolute constant. Hence

J~ dt

J:,. exp[A.I Pn(t, x) I] dx ~ 2:nA exp(!A. An)· 2

Combining this with (10),

J~ exp[OA.Mn(t)(t)] dt ~ [n/(1 - 0)] 2:nA exp(tA. 2An), from which we have

r

exp[OA. max Mm(t)] dt

0

1~m~n

~ [2:nA/(1 - 0)] exp(tA.2An + log n).

Since eu is a convex function for u > 0, we have by Jensen's inequality,

Therefore,

r

max Mm(t) dt

0 1~m~n

~ 0- 1 A.- 1 log[2:n/(1 - 0)]

+ t0- 1A.An + 0- A.1

1

log n.

Taking A.= (2A;;? log n) 112, we have

r

max Mm(t) dt

0 1~m~n

~

0- 1 log[2:n/(1 - O)](Anf21og n) 1' 2

+ 0- 1(2An log n )

1' 2

=

0- 1(2An log n) 1' 2 [1

~

C(An log n )1' 2 ,

+ log[2:n/(1- 0)](2log n)-

where C is an absolute constant, 0 being taken to be We now turn to the proof of the theorem.

1]

t, for example.

14.1.

611

FOURIER SERIES WITH RADEMACHER COEFFICIENTS

Proof of Theorem 14.1.3.

We may deal with the cosine series

t anrn(t) cos nx because, for the general case,

+

with f!n 2 = an2 bn 2 • For this type of series, it is easily seen that Lemma 14.1.2 continues to hold and the following proof depends on Lemma 14.1.2. Let n 0 = 1, nk = 2' 2t 1, k = 1, 2, .... Write

M~r.'*'(t) =

max

[maxi

n~c+l
x

~

anrn(t) cos mx

n-n~:+l

I].

From Lemma 14.1.2,

where C is an absolute constant and Rn = Lk=n a~r.2 • From (7), :L n-1 (log n)- 112RU 2 converges. Then by the well-known Cauchy theorem in the theory of series, :L k- 1 1 2 (R~)1 12 converges. Again by the same theorem, the convergence of :L (2kR 2~: ) 112 is implied. Hence from (11 ),

L

I:

M~r.'*'(t) dt <

oo,

which implies that :L Mk •(t) converges for almost all t. Let t be a point of the set on which :L Mk •(t) converges. Let :Lk:N Mk '*'(t) < e for a given e for such a t. Then for 11 > n ;;;:; nN, maxi

t

a;r;(t) cos jx

=

L M~r.'*'(t) <E.

J-n

I

00

k=N

Hence fortE {t: :L~ M~r.• < oo }, the series :L anrn(t) cos nx converges uniformly. This completes the proof.

612

XIV.

SOME SPECIAL SERIES OF RANDOM VARIABLES

14.2. Random Fourier Series It is quite natural to extend series (1) in the preceding section to a series of the form 00

L (an cos nx + bn sin nx)Xn(w),

(1)

n-1

where {Xn(w)} is a sequence of independent random variables. Paley et al. [1] (See also Paley and Wiener [2], Chapter 9) investigated the series co

L an[Xn(w) cos nx + Yn(w) sin nx],

(2)

n-1

where both {Xn(w)} and {Yn{w)} are sequences of independent random variables depending on normal distribution functions and a11 ~ 0. Series (2) is called the Fourier-Wiener series. (1) and (2) are special cases of co

L Rn(w) cos[nx + 1Pn(w)],

(3)

n-1

where {Rn(w)} and {IPn(w)} are both sequences of independent random variables. (Rm and IP n are not necessarily independent.) (3) can be written as co

L

[An(w) cos nx + Bn(w) sin nx]

(4)

n-1

with

Write (1) as Len cos(nx + On)Xn(w), en2 = an2 + bn 2 , en> 0 and define Rn(w) = enl Xn(w) I. IPn(w) =nO+ i:n:[1- sgn Xn(w)]. Then (3) reduces to ( 1 ). If Rn(w) =I an I [Xn 2 (w) + Y112 (w)] 112 and IPn(w) is defined so that cos IP" = anXnfRn, sin IP11 = anYnfR11 , then (3) reduces to (2). The series of the form (3) was investigated by Kahane [1] and the following theorems in this section are due to him.

Theorem 14.2.1. Let {Rn(w), n = 1, 2, ... } with Rn(w) ~ 0, n = 1, 2, ... , and {IPn(w), n = 1, 2, ... } be sequences of independent random variables. If the series co

L W min(1, Rn 2 ) n-1

(6)

14.2.

613

RANDOM FOURIER SERIES

converges, then (3) converges for almost every x almost surely and (3) is a Fourier series. If (6) diverges, then (3) is not a Fourier series almost surely. Proof.

Suppose that (6) converges. Then 00

L go[min(1, Rn

00

2 )] 2

n-1

~

L go min(1, Rn

r-1

2)

< oo,

hence from Theorem 12.5.3, I;:,1 Rn2 (w) converges almost surely. Hence for every x, (3) converges almost surely. Then (3) converges almost surely for almost every x (by the statement immediately after Theorem 14.1.1). From (4), we have 00

00

L [An (w) + Bn (w)] = L Rn (w). 2

2

n-1

2

n-1

(7)

Consequently (3) is almost surely the Fourier series of a function F(x, w) belonging to £2( -n, n). Thus we have first part of the theorem. Suppose that (6) diverges. Then I; Rn2 does not converge almost surely (the probability that I; Rn2 converges is less than one). Since {Rn 2 } is independent, I; Rn2 diverges almost surely (by the zero-one law). Hence in view of (7), (3) is almost surely not a Fourier series.

Theorem 14.2.2. Under the condition of the preceding theorem, the series (3) converges absolutely for all x, if the series 00

L go min(1, Rn)

n-1

(8)

converges. Proof. As in the proof of the preceding theorem, the convergence of (8) implies the almost sure convergence of I; Rn (using Theorem 12.5.3), which in turn obviously implies the absolute convergence of (3) for all t.

We now discuss uniform convergence. For this problem, a more elaborate argument is needed as is expected from Section 14.1. Actually, we are going to employ a method similar to the one in Section 14.1. In order to do this, we introduce (following Kahane) a class of random variables. Let F(x) be the distribution function of a random variable X.

614

XIV.

If there is an a

~

SOME SPECIAL SERIES OF RANDOM VARIABLES

0 such that

= J~oo exp 2yx dF(x) ~

Mx(2y)

exp 2a2y 2

(9)

for all real y, then X is called semi-Gaussian. The minimum value of a ~ 0 such that (9) holds, is denoted by T(X) and is called the Gaussian error of X. If X depends on the Gaussian distribution N(O, u 2):

T(X) is given by u 2. That is, T(X) = var X. Since Mx(2y) = W(e 2Yx),

Mu(2y) = W(exp 2yAX);;;;; exp 2a2A2y2

for

A> 0

if X is semi-Gaussian. Hence,

A> 0.

T(AX) = A.T(X),

Lemma 14.2.1. If there are h Mx(2y) then for every I

~

~

~

(10)

0, t > 0 such that, for all y > 0,

exp(2t 2y 2

+ 2h),

(11)

e- 21 ,

(12)

0,

1 - F[2t(h

+ /)112]

;;;;;

where F(x) is the distribution function of X.

Proof.

From (11 ), W(exp 2yX) exp(-2y 2t 2 - 2h- 2/);;;;; exp -2/;

that is

W {exp[2y(X- yt 2

-

[(h

+ 1)/y])]} ~ exp -2/.

If y > 0, then by Theorem 11.1.3, Eq. (7) with g(x) = exp x and with X in place of I XI,

P{X ~ yt2

+ [(h + 1)/y]} ;;;;; W {exp[2y(X- yt ~

+ /)

2 -

[(h

+ 1)/y])]}

exp -2/.

1' 2, we have (12). Taking y = t- 1 (h Let X( w) be a given random variable. Take a random variable e( w) which is independent of X( w) and assumes the values 1 and -1 both

14.2.

615

RANDOM FOURIER SERIES

with probability equal to t. The existence of such e(w) follows from the following consideration. Let Y( w) be a random variable assuming ± 1 each with probability !. Consider the product probability space !J' = !J X !J and define the random variables X 1(w) and e(w) for w = (w 1 , w2) E {J' by X 1(w) = X(w 1 ) and e(w) = Y(w 2 ). Then X 1(w) and e(w) are independent as random variables on !J'. X 1(w) can be identified with X( w ). Consider now X<SI(w) = e(w)X(w). (13) Then X<SI( w) has a symmetric distribution function because, for x > 0, P[X<SI(w)

=

>

x]

P[e(w)

=

= !{P[X(w)

P[X<SI(w)

<

1]P[X(w)

>

x]

>

x]

+ P[e(w) =

+ P[X(w) <

-1]P[X(w)

<

-x]

-x]},

-x]

=

P[e(w)

=

=

!{P[X(w)

1]P[X(w)

<

-x]

<

-x]

+ P[e(w) =

-1]P[X(w)

>

x]

+ P[X(w) > x]}.

Hence for an odd integer m > 0, for an even integer m

~

0,

(14)

(15)

when they exist. If a sequence of independent random variables {Xn , n = 1, 2, ... } is given, then choosing a sequence of independent random variables {en( w ), n = 1, 2, ... }, which are also independent of X"' k = 1, 2, ... , we may construct a sequence of independent random variables {X~SI(w),

n

= 1, 2, ... }

having symmetric distribution functions. We call x<SI the symmetric random variable derived from X. Using this notion, we prove the following lemma.

Lemma 14.2.2. for y > 0,

Suppose that X is semi-Gaussian and W X = 0. Then Mx(2y)

~

Mx<s>(4y)

(16)

and (17)

XIV.

616

SOME SPECIAL SERIES OF RANDOM VARIABLES

Proof. I

wX2n+l I =

I g" xn xn+l I

~

(

wX2n)112( g'X2n+2)112

=

{[(2y)-1 WX 2n](2y 8"X2n+2)}1 12

~

l-[(2y)-1 g" X2n

for every y

+ 2y g"X2n+2].

Writing the moment WXk = ak, we have (18) and 00

Mx(2y) = 8"e211X = W n~o

(2yX)n n!

=

1

1

1

+ 2f(2y) 2a 2 + 3f(2y) a + .... 3 3

Using (18),

(2n

~ 1)!

Mx(2y)

(2y)2n+II a2n+II

~ 1+ [

i! + /

~

(2n

~ 1)! ~

[(2y)2na2n

+ (2y)2n+2a2n+2],

31 ] (2y) 2a 2

+ n-2 }: [ (2n +1 1)!

_!_ 2

+ _1_ + (2n)!

1 [2n- 1]!

1 ] (2 )2n Y a 2""

2

The right-hand side is easily seen to be equal to or less than (4y)2fta2n = ro CP exp 4y X<Sl , (2n)! by (14) and (15). This shows (16). Using (13), for every realy [X<Sl is easily shown to be semi-Gaussian],

W exp 2yX which gives us T(X)

~

~

W exp 4yX<Sl

~

exp 2[2T(X<Sl))2y2,

2T(X<Sl).

Lemma 14.2.3. If X and Y are independent and semi-Gaussian, then (19) This is obvious from

W exp 2y(X + Y) = Wexp 2yX Wexp 2yY ~ exp 2[T2(X) + T2(Y)]y 2. We shall now give a sufficient condition for the uniform convergence of series (3 ).

14.2.

617

RANDOM FOURIER SERIES

Theorem 14.2.3. Suppose that {R,., n = 1, 2, ... } and {(1),., n = 1, 2, ... } are both sequences of independent variables and R,. ~ 0, I (1),. I ~ n, n = 1, 2, .... Suppose also that W cos (/),. = W sin (/),. = 0,

(20)

and that R,. is semi-Gaussian, n = 1, 2, .... Let 2 t+l

tk2

L

=

-c,.a,

n-ak+l

where -c,. = -c(R~Sl), from R,.. If

R~Sl

being the symmetric random variables derived (21)

then the series (3) converges uniformly almost surely for x and the sum is continuous.

Proof. We use a method similar to the one used in the proof of Theorem 14.1.3. Let M,.. = max maxi P,.,m(x) I, n:>m:>v

:r.

where

P,.,m(x)

=

m

P,.,m(x, w)

= L

Rk cos(kx

k=n+I

+ (J)k)·

(22)

Let t-t be an integer such that

M,.,.

=

maxi P,., 11 (x) z

I

(t-t depends on w). Then from Lemma 14.1.1 (taking(}= !), for each w, (23) on an interval I of x, with length not less than 1/2'11. Let M,.,.(x) be the function defined by

M

n,v

(x)={ M,.,.,

Now from (18), 'i!:R,. cos(nx

Since R,.

~

0,

+ (/),.) =

XE X E

I, Ic.

0. By Lemma 14.2.3,

0,

W exp[2yR,. cos(nx

+ (/),.)] ~ W exp 21 y IR,..

(24)

618

XIV.

SOME SPECIAL SERIES OF RANDOM VARIABLES

Hence

by Lemma 14.2.2. Therefore we have (25) On the other hand, from (24 ),

J:,. exp[yMn,.(x)] dx [from (23)] ~ W J:,. exp[2y I Pn,,..(x) IJ dx ~ W J:,. {exp[2yPn,,..(x)] + exp[ -2yPn,,..(x)]} dx ~ 2 J:,. exp{2y [Pn,,..(x)]} dx

(2v)- 1 W exp yMn. ~ W

2T 2

[from (25)]

Hence we have obtained

or

W exp 2yMn,•

~ exp[32y2

t

k-n+l

Tk 2

+ log(Snv)].

Then from Lemma 14.2.1,

Choosing l = log(Snv) for v so large that

P[Mn,v > 8

vz(

t

k-n+l

Tk2

! log(Snv) < 2 log v,

log v) 1' 2 ]

~

(Snv)- 2 •

(26)

14.3.

RANDOM POWER SERIES, CONVERGENCE

619

We have from (21), (27) Since L (8.nn;+l)- 2 < oo, from the Borel-Cantelli lemma (Theorem 12.2.1), the set of won which Mn;,n;+l > Ai holds for an infinite number of i's has probability zero. In other words, Mn;,n;+' ~A; fori~ i 0 almost surely, where i 0 is some integer which may depend on w. Hence from (27), L Mn;,n;+' converges almost surely. This implies the almost sure uniform convergence of (3 ). The continuity property of the sum is the consequence of almost sure uniform convergence. This completes the proof. Theorem 14.2.3 implies Theorem 14.1.3. Take Rn = enrn(w), en = an 2 bn 2 , where {rn(w)} is a sequence of Rademacher functions, 0 ~ w ~ 1. We see that in place of Tn 2 we may take cen, c being a constant. Then it is easily seen that condition (21) implies condition (7) of Theorem 14.1.3. 2

+

14.3. Random Power Series, Convergence We shall investigate the properties of series of the form (1) where z is a complex number and {Ak( w ), k = 1, 2, ... } is a sequence of complex valued random variables (that is the real and imaginary parts are random variables); (1) is called a random power series. An extensive investigation of random power series was made by Arnold [1, 2] and the results here are due to him. We shall adopt his methods. For each w E D, a random power series (1) has a radius of convergence R( w)

= [lim supj

An( w) j11n)-1.

(2)

n-+oo

R( w) is a random variable which can assume the value oo; that is, a measurable function on (Q, ~, P) for which {w : R( w) < oo} may be a genuine subset of Q and P[R( w) < oo] < 1. If {An(w)} is a sequence of independent random variables, then R(w) = constant almost surely because R( w) is a tail function and the zero-

one law is applicable (Theorem 12.3.1 ). Let the partial sum of ( 1) be (3)

620

XIV.

SOME SPECIAL SERIES OF RANDOM VARIABLES

We now deal with the circle of convergence of ( 1). We do not necessarily suppose that {An(w)} is a sequence of independent random variables. We consider as usual the several methods of convergence of Sn(w) as n---+- oo. If Sn(w) converges at all z inside some circle I z I = r according to some method of convergence but does not converge outside this circle, then the circle is called the circle of convergence of (1) according to that method of convergence or, the circle of almost sure convergence, the circle of LP-convergence, and so forth. For instance if (1) converges almost surely for all z such that I z I < r, and the almost sure convergence of (1) is not true for any z such that I z I > r, then I z I = r is the circle of almost sure convergence. The radius of convergence will be called the radius of almost sure convergence, the radius of convergence in probability, etc., as appropriate. We begin with the following theorem.

Theorem 14.3.1. For a given arbitrary random power series, (i) there exists a circle of almost sure convergence and the radius of almost sure convergence is given by r 0 = sup{x; P[R(w)

~

x] = 1};

(4)

(ii) there exists a circle of LP-convergence, p > 0, and the radius of LP-convergence is given by rp = {lim sup[W I An(w) IPJlln}-liP;

(5)

n-+oo

(iii) there is a random power series that has no circle of convergence in probability; and (iv) there is a random power series that has no circle of convergence in distribution. Proof.

Let ( 1) be the given random power series.

(i) Let r 0 be defined by (4), where R(w) is given by (2). Let x < r 0 • Then R(w) ~ x almost surely. Hence (1) converges almost surely for I z I < x, which means that (1) converges almost surely for I z I < r 0 • Let x > r 0 • Then by definition (4) of r 0 , P[R(w) ~ x] = 1- a, where 1 ~a> 0 and a may depend on x. Hence P[R(w) < x] =a> 0 and on the set {w: R(w) < x}, (1) diverges at z with I z I = x. This shows (i).

14.3.

621

RANDOM POWER SERIES, CONVERGENCE

(ii) First we shall show that if the random power series ( 1) converges in LP at z = z 1 , then it converges in LP at every z, with I z I < I z 1 I· {This statement is the crucial part which does not hold for convergence in probability and in distribution [cases (iii) and (iv)].} Write

p;;:;:; 1,

(6)

0


(7)

where the random variable X belongs to LP. Suppose (1) converges in LP at z 1 • Then as

m---+-oo,

n---+-oo,

(8)

From this, we have II A 11 (w)z 1n liP---+- 0 as n---+- oo and hence there is a constant C such that for all From the Minkowski inequality, for

n.

(9)

p ; :;:; 1,

m

=

L II Ak(w)zliiP I zfzl lk k-n+l

~ C

m

L I zjz1lk' k=ll+l

(10)

from (9), which converges to zero as m---+- oo, n---+- oo since I z/z 1 I < 1. Hence (1) converges in LP for I z I
~ C

m

L I zfzl lpk' k-n+l

which also converges to zero as m, n ---+- oo. Hence the above statement concerning the convergence of (1) in LP at z = z 1 has been shown. From this statement we have that there is a number rP such that for I z 1< rP, (1) converges in LP and, for I z I > rP, it diverges in LP.

622

XIV.

SOME SPECIAL SERIES OF RANDOM VARIABLES

In order to show (5), we first mention the elementary and well-known fact (it is also seen from the Cauchy-Hadamard formula for the radius of convergence) that the radius of convergence of an ordinary power series is the same as the radius of absolute convergence. We now show that the radius of convergence of a random power series in LP is the same as the radius of convergence of the power series (an ordinary power series) 00

L II An(w) iiPzk k=O

for p;;;; 1.

(11)

If (11) converges at z = z1, then L II Ak(w) liP I z Jk < oo for any z : I z J < J z1 J from the above-mentioned fact. Therefore, as in (10), for p;;;; 1,

as m---+- oo, n---+- oo. Hence the series (1) converges in LP; that is, Tp is not less than the radius of convergence of (11 ). Next suppose that (1) converges in LP at z = z 1 • Then as before II Ak(w)z1k liP~ C, k;;;; 0, and 00

00

L II Ak(w) liP I z Jk ~ k-o L II Ak(w)zlliP I zjz1ik k-o 00

~ C

L J zfz

1

Jk

< oo,

k=O

Hence ( 11) converges for I z J < z 1 ; that is, rP is not greater than the radius of convergence of (11 ). Hence the radius of convergence in LP, p ~ 1, is the same as the radius of convergence of (11 ), which is given by lim sup[JJ An(w) JJP]-11n, n-+oo

which is the same as (5). For 0 < p ~ 1, we may also show that L II A~c(w)zk liP= L IJAk(w) liP X I z JPk and that the radius of convergence in LP of (1) is equal to (/ 1P, where (! is the radius of convergence of L II Ak(w) liP w" (as a power series of w), w being J z JP. We then see that the radius of convergence in LP of (1) for 1 ;;;; p > 0 is

[li~_!'P(Ji A~(w) lip)-1'n] 1'P. From definition (7) of II · liP for 0 < p ~ 1, this is (5).

14.3.

623

RANDOM POWER SERIES, CONVERGENCE

(iii) Let {Sn(w ), n = 1, 2, ... } be a sequence of independent random variables such that P[Sn(w) = nn] = n-1, (12)

P[S11 (w) = 0] = 1 - n-1.

(13)

Since L n- 1 = oo, the probability that Sn( w) = nn occurs infinitely often is one, by Theorem 12.2.2. Hence Sn(w)zn does not converge to zero with probability one for any z =F 0. Therefore, almost surely the series L:,o Sn( w )zn does not converge for z =F 0. Now define

A 0 (w)

= S 0 (w) =

0.

Then

n-1

= Sn(w)zn

+ (1- z) k=1 L

Sk(w) zk.

(14)

Now for any e > 0 and any I z I > 0,

P(l Sn(w)zn I > e)= n- 1 --+ 0

as n--+ oo.

Therefore, in probability

(15)

for any z =F 0. If z =F 0, then, since L~= 1 Sn(w)zn does not converge almost surely, it does not converge in probability because of the equivalence theorem. Hence from (14) and (15), the series (16) does not converge in probability for any z =F 0 or 1. (Note the remark below Theorem 12.1.10.) However, (16) converges in probability for z = 1 or 0 because from (14),

for z = 1, from which the convergence in probability of (16) for z = 1 follows and the convergence of (16) for z = 0 is obvious. Hence for the series (16), there is no circle of convergence in probability.

XIV.

624

SOME SPECIAL SERIES OF RANDOM VARIABLES

(iv) The same example (16) gives us a counterexample. Actually, (16) converges in distribution at z = 0 or 1 since it does so in probability. For any z ::/=- 0 or 1, (16) does not converge in distribution because, if it does, then from (15), L~=o Ak(w )zk - Sn(w )zk converges in distribution, which is seen from the remark at the end of Section 12.1. Hence L~=l Sk( w )zk should converge in distribution and then converge almost surely, because of the equivalence theorem, which is a contradiction. REMARK. The distribution function of the radius of convergence R( w) in (2) does not have a particular property. More precisely, if G(x) is a given distribution function such that G(x) = 0 for x < 0, then there is a random power series whose radius of convergence has the distribution function G(x). Or more generally, if G(x) is any nondecreasing function such that G(x) = 0, x < 0, and G(oo) ~ 1, then there is a random power series whose radius of convergence R( w) has the property that P[r(w) < x] = G(x) for 0 ~ x < oo.

Let [0, 1], in the Let

Q = [0, 1], :31 be the class of all Lebesgue-measurable sets on and P be the Lebesgue measure. We show the latter statement remark. G(O+) = 0. Define, for w E [0, 1],

B(w) where G-1(w)

= { ~~(w)

= inf {x; G(x) >

for for

0 ~ w G(oo)

< G(oo), ~ w ~

1,

w} and define

n = 0, 1, 2. Then

R(w) =lim inf[An(w)]- 1/n = B(w). n-+oo

Hence, P[R(w) < x] = P[B(w) < x] = P[w < G(x)] = G(x) for x For the case G(O+) > 0, we take

Gn(x) = {

> 0.

~(x),

[Note that if G(O+) > 0, then G- 1 (w), and therefore An(w), is not defined for 0 < w < G(O+ ).] Defining An(w) = [G- 1 (w)]-n, we may show again that R(w) = B(w).

14.4.

IDENTICALLY AND INDEPENDENTLY DISTRIBUTED COEFFICIENTS

625

14.4. Convergence of Random Power Series with Identically and Independently Distributed Random Coefficients We suppose in this section, that the coefficients of a random power senes (1)

are identically and independently distributed or, slightly more generally, we suppose that the sequence

{I An(w) I,

n = 0, 1, 2, ... }

(2)

is a sequence of identically and independently distributed random variables Then the radius of convergence (3) n-+oo

is a tail function of {I An(w) 1. n = 0, 1, 2, ... } and hence R(w) is constant almost surely. We ask what values this constant may assume. If we let the distribution function of I An(w) I be F(x) (independent of n) and F(x) = 0 for x < 0, the answer is given by the following theorem (Arnold [1, 2] and see also Sparo and Sur [1]).

I An(w) I are identically and independently distributed random variables and the distribution function F(x) is not degenerate. Then R(w) = 1 almost surely when and only when

Theorem 14.4.1. Suppose that

W log+l An(w) I = J~ log x dF(x) < oo

(4)

and R( w) = 0 almost surely when and only when

W log+l An(w) I = J~ log x dF(x) = oo.

(5)

When F(x) = 1 for x > 0, and F(x) = 0 for x < 0, the corresponding random variable is called degenerate. In other words An( w) is not degenerate if and only if F(O+) < 1. It is the purpose of this section to show Theorem 14.4.1. First we shall show that if I An( w) I is identically and independently distributed, then (6) R(w) ~ 1 a.s., provided that F(O+) < 1.

For x > 0, {R(w)

~

x} = {limsupl An(w) 111n

~

x- 1 }

n-+oo

=

{I An( w) I ~

(x-I - e )n infinitely often for all 0 < e < n-I},

that is,

n

{R(w) ~ x} =

lim sup{l An(w) I ~(x-I- e)n}

O<e
n~oo

= lim lim sup{l An(w) I ~(x-I- e)n}.

(7)

n-+oo

e~o+

Hence with x = 1, P[R(w)

~

1] = lim P{lim sup[l An(w) I ~ (1- e)n]}. e-+o+

(8)

n-+oo

Since the I An(w) I are independent, by Theorems 12.2.1 and 12.2.2, (8) is one if and only if for all

1 > e > 0.

This always holds since P[l An(w) I ~ (1 - e)n] =I- F[(1- e)n]---+ 1 - F(O+) > 0 for e > 0 as n ---+ oo. Hence the required statement is shown. We now show the theorem. Proof of Theorem 14.4.1. From the statement shown above, R(w) =con1 almost surely. [Note: R(w) = r 0 , r 0 being the number defined in Section 14.3, Eq. (4).] We have

stant~

{R(w) < x} = {R(w) = [

~

x}c =

{x-I~

n {x-I+ e
lim supl An llln}c n-+oo

An l11n,

finitely often}]c

•>0

U[ e>O

U f) {x-I+ e ~I An l11n}]c

n-I

k-n

u n u {x-I+ 00

00

e


e


o>O n-1 k-n

- U lim sup{x-I + e>O

n-+oo

14.4.

IDENTICALLY AND INDEPENDENTLY DISTRIBUTED COEFFICIENTS

Since the set under {R(w)

U.>o

< x} =

is a nonincreasing set as e increases, lim lim sup{x- 1 e-+-o+

If R(w) = 1 a.s., then P[R(w) all e > 0,

P[lim sup{(1 , n-+-oo

627

n-+-oo

+ e
1'"}.

(9)

< 1] = 0 and hence (9) shows that for

+ e)n < I A, I}] ~ P[R(w) <

1] = 0.

(10)

> 0.

(11)

Hence from Theorems 12.2.1 and 12.2.2, 00

L P[l A, I > n=1

+ e)"] <

(1

for all

oo

e

Conversely, if (11) holds for all e > 0, then from (9), P[R(w) < 1] = lim P[lim sup{(1

+ e)"< I An I}]= 0.

e-+0+

Since from (6), R(w) ~ 1 a.s., we must have P[R(w) = 1]. Thus we have shown that R(w) = 1 a.s. is equivalent to (11), which turns out to be 00

L [1

- F(cn)] < oo

c > 1.

for all

(12)

n=l

Next we shall show that R( w) = 0 a.s. is equivalent to the divergence of the series in (12). Suppose that R(w) = 0 a.s.; then P{limsupl A,

11/n =

oo} = 1.

(13)

n-+oo

That is, P {I An

111"

~

c

infinitely often for all c > 0} = 1

or

P( n lim sup(l An I ~ c"}) c>O

=

1.

(14)

n-+-oo

Hence for any c, in particular for any c > 1, P(lim sup{l An I > c"}) = 1.

(15)

n-+oo

Then from Theorem 12.2.2, 00

00

2: P(l A, I ~ c") = n-1 L [1 n-1

- F(c")] = oo.

(16)

XIV.

628

SOME SPECIAL SERIES OF RANDOM VARIABLES

Conversely, if (16) holds, then (15) holds for all e > 1. Hence lim P(lim sup {I An I >en})= 1. c-+oo

n-+oo

The left-hand side is

P( lim lim sup{l An I >en})= p( n lim sup{l An I >en})= 1 n-+oo

c-+oo

c>O

n-+oo

(since lim supn-+oo{l An I >en} is nonincreasing as e increases), which gives us (13). After all, what we have shown is that R( w) = 1 a.s. or 0 a.s., accordmg as 00

L [1 -

F(en)] < oo

or = oo

for all e > 1.

n-1

Now since fore> 1,

the convergence or divergence of

L [1 - F(en)] is equivalent to that of

I~ [1- F(ez)] dx = (log e)- 1 I~ {[1 - F(t)]ft} dt. Integration by parts gives us, for T > 1,

I; {[1 Hence,

F(t)]ft} dt =log T [1- F(T)]

I;

{[1 - F(t)]ft} dt

~

r

+I; log t dF(t).

(17)

log t dF(t).

If J;"' log t dF(t)=oo, then J;"' {[1-F(t)]/t} dt=oo; if J:' log t dF(t)
14.5. Analytic Continuation of Random Power Series A property of random power series will be suggested from the theory of power series having large gaps. Series of independent random variables have many properties similar to those possessed by trigonometric series

14.5.

ANALYTIC CONTINUATION OF RANDOM POWER SERIES

629

or by power series with, for instance, Hadamard gaps or

> 1 (0 being a constant). We mentioned a few of these properties in Chapter XII, Note 6 and Chapter XIII, Note 7. (See also Gaposhkin [1].) We shall not go into details in this matter. One of the most significant results on power series with Hadamard gaps is that the circle of convergence is the natural boundary; that is, the function defined by the power series cannot be analytically continued beyond the circle of convergence (Hadamard's gap theorem). (See, for instance, Dienes [1] and Titchmarsh [3].) This suggests that a random power series with independent coefficients will have the circle of convergence as its natural boundary. Actually, this result has been shown in special cases by Steinhaus [2], Paley and Zygmund [1 ], and Arnold [2]. A general result in connection with this was given by Ryll-Nardzewski [1] (see Theorem 14.5.3 below), although the statement is not always true in its full generality. Let nk+ 1/nk ~ ()

(1) be a random power series, where we suppose that the complex random variables A 11 (w) are independent; that is, writing A 11 (w) = X 11 (w) iY11 (w), X 11 (w) and Y 11 (w) being real valued random variables, {X11 (w), Y11 (w)} constitutes a sequence of independent two-dimensional random variables. As we mentioned, in this case the radius of convergence R = [lim sup I A 11 ( w) 111 11 )-1

+

n-+oo

is a constant almost surely. In the domain I z I < R, (1) converges almost surely and represents a function f(z, w ). The constant R is denoted by

R(f). Since the property that series ( 1) can be analytically continued or cannot be does not depend on any finite number of A 11 ( w ), the zero-one law is applicable and so we have immediately that the probability that (1) is not analytz"cally continuable is either one or zero. Now let Z(w) be a complex-valued random variable. As in the realvalued case, we may define the symmetrized random variable Z•(w)

=

Z(w)- Z'(w),

where Z( w) and Z' ( w) are identically and independently distributed

630

XIV.

SOME SPECIAL SERIES OF RANDOM VARIABLES

random variables (as the two-dimensional random variables of the real and imaginary parts). When

f(z, w) =

L an(w) zn,

(2)

n-o

we agree to write 00

jB(z, w)

=

L A,.•(w) zn.

(3)

n=O

We first prove a lemma.

Lemma 14.5.1. The power series (3) with independent {An(w)} cannot be analytically continuable at the real point Z 1 = R(f•) almost surely. Proof. Let E be a subset of Q such that for w E E, the series (3) converges in I z I < R(js), P(E) = 1. For an w E E, expand js(z, w) into a Taylor series at a point u, 0 < u < R(js):

d:

f

m=o dz

[f"(z, w)]l z=u

(v -,u)"' m.

(4)

It is sufficient to show that this series has a radius of convergence equal to R(f•) - u for w E E 1 c E, with P(E 1 ) = 1; that is, if v > R(f•), then (4) diverges. So we may suppose v to be real. Since the series (4) converges if and only if its real and imaginary parts converge, we may, without loss of generality, suppose that An(w) is real valued. Since

0 < u < R(jB), the series (4) is given by

00

=

lim

L t,,k[Ak•(w) vk],

i-+oo k-o

(5)

14.5.

ANALYTIC CONTINUATION OF RANDOM POWER SERIES

631

where

t· •,k

=

mlnli,kl (

(.!!...)k-m ( v _

k )

L

m

ti,k

= 1

m-o

v

v

u

)m.

(6)

From this, if i

~

k.

(7)

Suppose tentatively that (4 ), and hence (5 ), converges for some v > R(/8 ) (v may depend on w) on some set E' c E with P(E') > 0. Then it does so almost surely because of the zero-one law. Hence the series I:~o Ak8 (w)vk is T*-summable (Section 13.10) with matrix (t;,k)· Then from Theorem 13.10.3, I:~o Ak8 (w)vk converges for v > R(jS), v = v(w), almost surely. This is a contradiction, since R(/8 ) is the radius of convergence of/8 • Hence P(E') = 0 and hence (4) diverges onE- E', whose probability is one. This completes the proof of the lemma. We shall prove a theorem (Ryll and Nardzewski [1], Arnold [1]) concerning the connection between R(/) and R(jS).

Theorem 14.5.1. (i) For any fixed ordinary (not random) power senes 00

q>(z) =

L Cnzn,

n-o

we have

R(f- q>)

~

R(jS),

wherefis given by (2) with independent {An(w)}. In particular, R(f) ~ R(jS).

(8)

(9)

(ii) For the case R(/) < R(jS), there exists an ordinary power series fPo(z) = I::.o cW>zn such that

R(f- fPo) = R(jS).

(10)

Proof. (i) First we show (9). If I: An(w)zn and I: An'(w)zn converge for some w, then I: An8 ( w )zn converges, where An8 ( w) = An( w) - An' ( w), An(w) and An'(w) being identically and independently distributed. Hence (11)

Since the radius of convergence is equal to the radius of absolute convergence, the radius of absolute convergence is equal to the radius of

XIV.

632

SOME SPECIAL SERIES OF RANDOM VARIABLES

convergence in probability or to the radius of convergence in distribution, because for absolute convergence, we deal with the nonnnegative (and hence real-valued) random coefficients and the equivalence theorem applies. Therefore, which shows that min[R(L Anzn},

R(L An'z")] = R(f).

This with (11) gives us (9). For a fixed numerical power series (nonrandom power series) 9'(z}, f(z, co) - 9'(z) has the symmetrized random power series j•(z, co). This is obvious. Then from (9}, we obtain (8). This proves (i). (ii) This case is assured by Theorem 14.5.2, which follows. First for convenience we define the median /-'z of a complex-valued random variable Z = X + iY by f-tz = f-tx

+ ipy,

where 1-'x and py are medians of X and Y, respectively.

Theorem 14.5.2. Let {An(co)} be independent, and let f(z, co)= :L::o An( co }zn, 00

pJ(z) =

L f-tA,.zn,

n-o

where /-'A,. is a median of An(co). Then (12)

Proof. Let An( co)= Xn(co) + iYn(co), / 1 (z, co)= Ln~o Xn(co)zn, and (z, co)= :L:=o Yn(co)zn. Then it is easy to see that R(/) = min[R(/1}, R(/2}]. Therefore we may suppose that the An( co) are real, n = 0, 1,2, .... From Theorem 14.5.1 (i), R(f- p1) ~ R(f•). So it remains to prove

/ 2

the converse, i.e., (13) From Lemma 12.9.3 [with (r- 1 positive numbers], 1

00

2 L P[i An- /-'A,. I > n-o

(r- 1

+ e)n

+ e)n]

~

for x, where e and r are any 00

L P[i An" I >

n-o

(r- 1 + e}n].

14.5.

ANALYTIC CONTINUATION OF RANDOM POWER SERIES

633

Hence from Corollary 12.2.1, if I An 8 I > (r- 1 + E)n holds infinitely often with probability zero, then I An- 1-'A,. I > (r- 1 + E)n holds infinitely often with probability zero. Fix r > 0. Then this statement says that if lim sup! Ans l11n ~ r- 1 a.s., then lim sup! An- 1-'A,. l11n ~ r- 1 a.s. In other words, R(js) ~ r implies R(f- p.1) ~ r. This shows (13). We now show another theorem.

Theorem 14.5.3. Let An(w), n = 0, 1, 2, ... , be independent random variables. Letf(z, c.o) = I::=oAn(w)zn. For any numerical power series
=

R(js),

(14)

the random power series f(z, w) -
00

/ 1

(z, c.o) =

L An(w)ein~zn,

n-o

(15)

the point z 1 = R(f) is a regular point. Now we see that / 18 (z, w) = I::=o An8 (w)ei~zn and hence R(N) = R(/8 ). Since R(/8 ) = R(f), we have R(f) = R(N). Therefore, z 1 = R(N) is a regular point of (15). This contradicts Lemma 14.5.1 and thus proves the theorem. If R(f-


-




We mention also that the conclusion of Theorem 14.5.3 is not necessarily true without any condition on f(z, w) in addition to the independence of {An(x)}. For, let Xn( w ), n = 0, 1, 2, ... , be any sequence of independent

634

XIV.

SOME SPECIAL SERIES OF RANDOM VARIABLES

random variables and define n

= 0, 1, ....

Then f(z, w) = L~=O An(w)zn = L~=O exp[iXn(w)]zn + L~o 2nzn. The first series on the right-hand side converges for I z I < 1 for every w, and the second for I z I < t and we see that R(f) = t. Now :E 2nzn = (1 - 2z)- 1 has its only singularity at z = t. That is, f(z, w) is continuable through points on I z I = t except at z = t. Note that for this example, R(jB) = 1 and R(f) = t. Finally we show one more theorem.

Theorem 14.5.4. Suppose that {An(w), n = 0, 1, 2, ... } are independently and identically distributed and the An( w) do not degenerate to a constant. Then (16) R(f) = R(P) and f(z, w) cannot always be analytically continued almost surely. Proof. It is sufficient to show (16 ). Then the second conclusion is true from Theorem 14.5.3. Fromf(z, w) = (f- p.1) + p.1 , we have R(f);;;:;; min[R(j- p.1), Rp.1 ], where p.1 = L~oP.A,.zn, P.A,. being a median of An(w). Since R(f-p.1) = R(f8 ) by (12), (17)

On the other hand, from p.1 = (p.1 - f)+ f, R(p.1) ;;;:;; min[R(p.1 - f), R(f)] = min[R(jB), R(f)]. Since by (9), R(jB) ;;;:;; R(f) [R(jB) < oo since An is nondegenerate], we have R(p.1) ;;;:;; R(f). Hence R(f) ~ min[R(jB), R(p.1)]. So with (17), we have (18)

Now P.A,. = constant = /-tAo(* 0). Thus p.1 = P.Ao L~=o zn. Hence R(p.1) = 1, while R(jB) ~ 1 by (6), Section 14.4. We therefore have, from (17), R(f) = R(jB). This completes the proof.

14.6. Fourier Series with Orthogonal Random Coefficients Let {Cn( w ), n = 0, ± 1, ±2, ... } be a sequence of complex-valued random variables with WI Cn(w) 12 < oo, n = 0, ±1, ....

14.6.

FOURIER SERIES WITH ORTHOGONAL RANDOM COEFFICIENTS

635

Suppose that for all n

(1)

and

m

n.

-=1=

(2)

In this case {Cn} is called an orthogonal sequence of random variables or a sequence of uncorrelated random variables. We shall investigate series of the form 00

L

Cn(co)einz,

-n

< x < n,

(3)

n--oo

with the condition that (4) n=-oo

where the Cn are not necessarily independent. The series (3) are particularly important in time series analysis or in the theory of stochastic processes in which x is taken for the time parameter and the series (3) represent the phenomena fluctuating as time passes (which we shall not deal with in this book). Let n

S-m,n(x, co)=

L

k--m

Ck(co)eikz.

(5)

Theorem 14.6.1. There is a measurable function X(x, co) belonging to L 2 on the product space ( -n, n) X D such that

J:,. WI S-m,n(x, co)- X(x, co)

12

dx- 0,

(6)

12

-0,

(7)

and

WI for almost all x

Proof.

S-m,n(x, co)- X(x, co)

E ( -n,

n) as n - oo, m- oo.

By (2),

w

I sm,n(x, co) 12 = =

wIt Ck(co)eikz 12 k-m w

n

n

k-m

u,j=m "k*J

2: I ck 12 + 2:

n

=

2: wIck 1 k-m

2•

wckcJei
XIV.

636

SOME SPECIAL SERIES OF RANDOM VARIABLES

and the last sum converges to zero as n

--+

oo, m

--+

oo or m

--+ -

oo,

n--+ -oo because of (4). Hence there is a function X 1(x, w) such that

(9) for every x E (-:n:, :n:) as m--+ -oo, n--+ oo. Integrating (8) with respect to x over ( -:n:, :n:), we also have

which converges to zero as n --+ oo, m --+ oo or m --+ - oo, n Hence there is a measurable function X(x, w) such that

--+ -

J:,. ~I Sm,n(x, w)- X(x, w) 1 dx--+ 0

oo.

(10)

2

as m --+ - oo, n --+ oo. Then there is a subsequence {8--n~:.n~:(x, w)} that converges to X(x, w) for almost all (x, w). From the Fubini theorem (or by the reasoning in the remark below Theorem 14.1.1 ), there is a set A of x with mA = 2.n such that for each x E A, 8-n,.,n~: converges to X(x, w) almost surely. From (9), for each x E A, there is a subsequence {S_m~:,m~:} of {8--n~:,n,.} such that 8-m,.,m~: converges almost surely to X 1(x, w). For such x, X 1(x, w) = X(x, w) almost surely. Then again by the Fubini theorem, there is a set G of (x, w) such that Gc has measure zero and X 1(x, w) = X(x, w) for all (x, w) E G. This shows the theorem. Let n be any integer. Take m so large that -m < n < m. Then for each w,

Hence

J:,. X(x, w)e-int dx 12 ~I (2.n)-l J:,. [S-m,m(x, w)- X(x, w)]e-inz dx ~ (2.n)-1 J:,. ~I S-m,m(x, w)- X(x, w) 12 dx,

~I C,.(w)- (2.n)-1 =

r

14.6.

FOURIER SERIES WITH ORTHOGONAL RANDOM COEFFICIENTS

which converges to zero as m

oo from (6 ). Hence

---+

w ICn(w)- (Zn)-

1

637

J:,. X(x, w)e-inz dx r= 0.

Thus we have the following lemma.

Lemma 14.6.1. For the series (3) with conditions (2) and (4), Cn(w) is the Fourier coefficient of ·X(x, w) for each n almost surely, where X(x, w) is given in Theorem 14.6.1. Since X(x, w) e L2 on (-n, n) X D, X(x, w) E L2 on (-n, n) almost surely. Hence (3) is almost surely the Fourier series of a function belonging to L2( -n, n). From (8) and (7) we have

w I X(x,

00

w)

12

L w I en 1 n--oo

=

2

(11)

a.e.

Hence WI X(x, w) 12 is independent of x for almost all x. We remark that actually we define X(x, w) for almost all wand almost all x, -oo < x < oo, and X(x, w) is periodic almost surely as a function of x; that is, X(x

+ 2kn, w) =

X(x, w)

k = 0, ±1, ±2, ... ,

a.s.,

for almost all -n < x < n. We agree to write the correspondence between the series (3) and the X(x, w) defined in Theorem 14.6.1, in the following way: 00

X(x, w)"'

L

(12)

Cn(w)einz.

n--oo

An important property of X(x, w) is given in the following theorem.

Theorem 14.6.2. Let X(x, w) be given by (12) with Cn conditioned by (1), (2), and (4). Then WX(x, w) = 0 WX(x

+ h, w)X(x, w) =

a.e.

J:

eW. dF(A.)

(13) a.e.,

(14)

XIV.

638

SOME SPECIAL SERIES OF RANDOM VARIABLES

where

F().)

=

L

-oo
8"' I C n

12

(15)

•

In other words, 8"' X(x + h, w) X(x, w) is independent of x for almost all x, and the right-hand side of (14) can be written (16) n--oo

The function F().) in ( 14) or ( 15) is called the spectral distribution function of X(x, w) although F( oo) is not necessarily one. Actually,

F(-oo) = 0,

(17) n=--oo

8"' X(x + h, w) X(x, w) = e(h) a. e. is called the covariance function of X(x, w ), where e(h) is given by the right-hand side of (14 ).

Since 8"' S-n,n = 0,

Proof of Theorem 14.6.2.

I WX(x, w) I =I

I w)- S-n,n(x, w) 12)112 -+ 0

W[X(x, w)- S-n,n(x, w)]

~[WI X(x,

for almost every x, from (7). This proves (13). We have I

wX(x + h)X(x) - ws-m,m(x + h)S-n,n(x) 1 ~ 28"' I [X(x + h) - S-m,m(x + h)] X(x) 1 + 28"' I S-m,m(X + h)[S-n,n(x)- X(x)] 1 ~ 28"' I X(x) 1 WI X(x + h) - S-m,m(X + h) 1 + 28"' I S-m,m(x + h) 1 w I X(x) - s-n,n(x) 1 2

2

2

2

2

(18)

2•

2

Since 00

w I s-m,m(X + h) 1 = w I s-m,m(x) 1 ~ n--oo L w I ck 1 2

2

2

a.e.

and

w I X(x +h)- s-m,m(x +h) 12 = w I X(x)-

S-m,m(x)

12 -+

0 a.e.,

the right-hand side of (18) converges to zero as m-+ oo, n-+ oo. Hence WX(x

+ h)X(x) =

lim 8"' S-m,m(x m-n-+oo

+ h)S-n,n(x)

a.e.

14.6.

FOURIER SERIES WITH ORTHOGONAL RANDOM COEFFICIENTS

639

while n

ws_m,m(x

m

+ h)S-n,n(x) = L

L

ifekejeiCX+hlke-izj

i=-n k=-m

=

N

L g:- I ek 12 eiih, -.V

where N = min(m, n). This converges to ~k:-oo This shows (14) and completes the proof.

WI ek l 2eikh as m, n - oo.

We now prove the following very simple theorem.

Theorem 14.6.3. Under the conditions of Theorem 14.6.2, if

L I n llogl+• I n I if I en 12 <

oo,

(19)

1111~2

for some e > 0, then 00

L I en I< oo

(20)

?l=-00

almost surely.

Proof.

It is sufficient to show that 00

L if en n=-oo 1

1

< oo

because of the monotone convergence theorem. We have 00

L n=3

00

if

I en I = L [ni12log
L (n logi+• n)-I ~[ n-3 00

]

1/2 [

~[ by (19). For ~;;~-oo, a similar expression holds. This proves the theorem. From this theorem we may easily derive the following theorem.

Theorem 14.6.4. Suppose that the conditions of Theorem 14.6.2 are satisfied. If (19) holds then there is a random variable X 0 (x, w) depend-

XIV.

640

SOME SPECIAL SERIES OF RANDOM VARIABLES

ing on x, which is a continuous function of x almost surely and X(x, c.o) = X 0 (x, w)

for almost all x almost surely. Proof. By the foregoing theorem, Therefore,

:E:.-oo I C,. I <

oo almost surely. (21)

converges uniformly for x almost surely. If this sum is denoted by X 0 (x, c.o), then it is a continuous function of x almost surely. On the other hand, (21) converges in L 2 on (-n, n) X D to X(x, w). Hence X(x, w) = X 0 (x, w) for almost all (x, w ), from which the conclusion of the theorem follows. NOTES 1. Theorem 14.1.3 is false if e

= 0. (See Zygmund [1], p. 219.}

2. Condition (7} in Theorem 14.1.3 is not necessary for the continuity of (1} for almost every t, but it is best possible in the sense that there exists a series (1} that is continuous for no value of T and such that

l; [R,!"/n(log n }""w(n}] < oo, where w(n} is a given sequence increasing to infinity with n as slowly as desired. (See Salem and Zygmund [1].} 3. For a .necessary condition for the continuity of (1} in Section 14.1, Salem and Zygmund have shown the theorem: If (1} (Section 14.1} represents a continuous function of x for almost every t, then writing 00

we should have R,. log T,. -+ 0 as n -+ oo. (See Salem and Zygmund [1].) 4.

Theorem 14.1.3 depends on the behavior of

maxi P,.(t, x} I =

maxi ~ (a,. cos kx + b,. sin kx}r..,(t} lt-1

I·

where r,.(t} is a Rademacher function. A much simpler result is

This is obvious since

I

n -n

I P,.(t, x} I dx ~

(f"

-n

I P,.(t, x} I" dx )1/1 (21f)1/2.

641

NOTES

This is true also for :E (at cos mtx + ht sin mtx), where {mt} is a sequence of distinct positive integers. Uchiyama [1] has shown that for any E > 0, there is a positive constant B. depending only on e such that, except for a set of t of measure less than e, for all

n

~

1.

5. In Theorem 14.2.1, if (6) diverges, then the series (3) can converge almost everywhere only with probability zero. This is shown by the method used in the proof of Theorem 13.7.1. Note that

[,. I cos nx I dx =

J:,. I

sin x I dx

=

J:,. I

cosy I dy.

6. In Theorem 14.2.2, if (8) diverges, then (3) can converge absolutely on a set of x of positive measure only with probability zero. This is seen from the fact that if a trigonometric series converges absollutely on a set of positive measure, then the series of the absolute values of the coefficients converges. (See Zygmund [5], Vol. 1, p. 232.).

7. Concerning the series (3) in Section 14.2, Kahane [1] made more extensive studies of the boundedness of the sum and the membership of the class Lip a (Lipschitz class). 8.

In connection with Theorem 14.1.2, Jessen [1] has shown the following: Let {Xn} be a sequence of independent random variables with uniform distribution on ( -:n, :n). Let {An} be a sequence of real numbers. If an are real and if for some G > 0, :E an• X exp G I An I < oo, then :E anXn exp iAnx = o(log x) 112 a.s. as x -+ oo.

9. Let {Xn} be a sequence of independent random variables with EXn = 0, n = 1, 2, .... Suppt'se that :E EXn" < oo. Let 1 < A1 < A2 < · · ·. If :E EXn• (log An )I +a < oo, E > 0, then .S(t) = :E Xn exp iAnx is almost surely continuous. This was given by G. A. Hunt [1].

10. The case where the I An(w) I are not necessarily independent (Theorem 14.4.1) has also been studied with less complete results by Arnold [1].

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OsTROVSKII,

Index A A-summability, see Abel-summability Abel-summability {A-summability), IH for Fourier series, 140 Abelian theorems, 3 01 Abel's continuity theorem, 1 H, 3 01 Abel's transformation, 23 Absolute convergence of Laplace transform, 2 H of series of independent random variables, J03, J26

Absolute moment, 401, 428 Akhiezer, N. 1., 70, 398, 40J, 642 Almost everywhere convergence, 47J Almost sure absolute convergence, J03, J26 Almost sure convergence, 47J Analytic characteristic function, H 6, 4 J 2 Analytic continuation, 32 of Laplace transform, 298 of random power series, 629 Analytic function, 29 Analyticity of distribution function, 439 of Laplace-Stieltjes transform, 242 strip of, 3 J7 Andersen, G., 344, 398, 642 Aonuma, T., 194, 642

Approximate Fourier series, 1 J 7 Arcwise connected set, 28 Arnold, L., 602, 62J, 629, 630, 640, 642 Asymptotically unbiased estimate, 186

B Balasov, L. A., 603, 642 Banach, S., 70, 643 Banach space, 40 Bary, N., 79, 113, 117, 164, 643 Bernoulli convolution, I l l Bernoulli distribution function, 110 Bertrandias, J.-P., 399, 643 Bessell inequality, 197 Bilateral Laplace-Stieltjes transform, 26J Binmore, K. G., 419, 643 Birkhoff, G., 398, 643 Boas, R. P., Jr., 36, 112, 423, 440, 444, 473, 643, 644 Bochner, S., 4J, 79, 1J9, 172, 182, 192, 201, 210, 231, 316, 317, 329, 377, 397, 398, 399, 643, 644 Borei-Cantelli lemma, 486

Borel Borel Borel Borel 661

field, 4 function, 6 measurable function, 6 measure, J

662

INDEX

Bord set, 4 Bord cr-fidd, 4 Bord theorem, 488 Boundary, 28 Boundary function, 213, 22J Bounded convergence theorem, 11 Bounded operator, 42 Boundedness of spectra of distribution function, 431 Breiman, L., J41, 644 Buticka, Y01. B., 28, 6JO

c (C, a) -summability, 192 (C,1) -summability, 133 for Fourier series, 1 H Caratheodory, C., 340, 644 Carleman's theorem, 31 Carlson, L., 164, 644 Cauchy integral, 149 Cauchy sequence, 40 Cesaro-summability, 133 Chandrasekharan, K., 4J, 201, 210, 231, 329, 644 Characteristic function analytic, 4J2 in a half-plane, 4 J 6 characterization of, 372, 376 criteria for, 38J, 392 of distribution function in R.,, 326 entire, H7, 460, 466 of random variable, 89 of set, 8 of singular distribution function, J J7 Characteristic property of Fourier coefficient, 366 Characteristic sequence, 339 Chebyshev's inequality, 403 Chow, Y. S., J34, 604, 644 Chung, K. L., 193, J34, 644 Circle of convergence, 620 Circular distribution function, 8 3 Compact set of characteristic function, 3 62 Complete, 4 in L•, 362 Complete measure, 4 Complete measure space, 4 Completeness of sequence of elements, 19 8 of trigonometric functions, 69

Concave, 27 Concentration function, 44J mean, 44J Conjugate Poisson integral, 149 Conjugate Poisson kernel, 149 Consistent estimate, 189 Continuity interval, 324 Continuity modulus, J4 Continuity property of sum of independent random variables, J3 6 Continuity theorem, 349 on intervals, H 6 Continuous spectrum, 83 Convergence almost everywhere, 47J almost sure, 47 J almost sure absolute, J03 bounded, theorem, 11 circle of, 619 in distribution, 47J dominated, theorem, 11 essential, J 31 of Fourier series, 113 of Fourier-Stieltjes series, 121 general, 167, 174 in L' (in norm L'), 47J in measure, 47J monotone, theorem, 12 in probability, 747 radius of almost sure, 619 in probability, 619 of random power series, 618 with independently and identically distributed random coefficients, 624 of sequence of monotone functions, 332 of series of random variables, J8 3 strong, JBJ unconditional, J20 weak of sequence of monotone functions, 332 of series of random variables, J8 6 Convergent subsystem, J86 Convergent system, J86 Convex, 27, 38J Convolution, 7J, 2J3 Cossar, J., 78, 79, 644 Cramer, H., 99, 380, 398, 473, 64J Cross neighborhood, 316

663

INDEX

D Davis, P. 1., 70, 231, 64J Decomposition theorem, 18 De Finetti, 934 Derivative of Dini, 22 Derived Fourier series, 67 Determ:·ned moment problem, 40J Dienes, P., 16J, 629, 64J Dini's test, 116 Dirichlet integral, 114, 123 Dirichlet kernel, 114, 123 Dirichlet series, 2 33 Discontinuity property of sum of random variables, n 6 Distribution functiqn, J, 7 analyticity of, 439 empirical, 187 marginal, 321 probability marginal, 32J pure, 141 purely discontinuous, 141 in R .., 321 of sum of Rademacher series, H 6 Doetsch, G., 276, 64J Dominated convergence theorem, 11 Domingnez, A. G., 380, 64J Dugue, D., 1H, 391, 39J, 64J Dyson, E. J., 349, 64J

H Empirical distribution function, 187 Entire characteristic function, 317, 460, 466 Equicontinuous, 362, 363 Equivalence theorem, 493 Esseen, C., 112, 619 Essential convergence, J3 1 Estimate, 186 asymptotically unbiased, 187 consistent, 187 unbiased, 187 Exponential entire function, 2 84 Exponential type of entire function, 461

p Fatou's lemma, 11 Fejer, L., 79, 164, 16J, 64J Fejer's integral, 136, 146

Fejer's kernel, 13 6, 146 Feller, W., 302, 328, 646 Fekete, M., 397, 646 Field, 3 First mean value theorem, 23 Fourier coefficient, J 0 characteristic property of, 366 cosine, JO with respect to orthogonal sequence, 19J sine, JO Fourier series, J 0 approximate, H7 of Fourier-Stieltjes transform, 1 J J in inner product space, 196 L~-theory of, 194 multiple, 31 J with orthogonal random coefficients, 634 witn Rademacher codlicients, 60J random, 612 with respect to orthogonal sequence, 19 J Fourier-Stieltjes coefficient, 84 Fourier-Stieltjes series, 84 Fourier-Stieltjes transform, 89 multiple, 326 Fourier transform, J 3 invene, 206 in L1 , 204 theory of, 201 multiple, 3 16 Fourier transformation, 63, 207 Fourier-Wiener series, 612 Fourier's integral theorem, 124 double, 121 single, 124 Fubini's theorem, 12, 17 Function of bounded variation, 16

G Gaposhkin, V. F., 186, 603, 629, 646 Gaussian error, 614 General convergence theorem, 167, 174 for Stieltjes integral, 178 General moment problem, 70 General summability theorem, 167 General Tauberian theorem, 30 J Generalized Riemann-Lebesgue lemma, 47 Girault, M., 391, 641 Gnedenko, B. V., JOJ, 646 Goldberg, R. R., 63, 646

664

INDEX

Gross, W., 397, 646 Gundy, R. F., 604, 646

H

H" class, 210 .j;) 2 class, 2 18 Hadamard gap theorem, 629 Hamburger moment problem, 401 Hardy, G. H., 27, 28, 113, 132, 182, 192, 193, 231, 277, 279, 301, 311, 314, 328, 329, 646, 647 Hardy-Littlewood maximal theorem, 27 Hardy theorem, 279 Harmonic function, 142 Hartman, P., H7, 647 Hausdorff moment problem, 401 Helly-Bray theorem, 333 Helly's first theorem, 33 0 Helly's second theorem, 333 Helly's selection theorem, 330 Hermite function, 162 Hermite polynomial, 161 Hilbert space, 40, 277, 473, 647 Hill, J. D., 604, 647 Hille, E., H, 647 Hirshman, I. 1., 277, 647 Histogram, 187 Hobson, E. W., 22, 24, 363, 647 Holder inequality, 2J Hoffman, K., 231, 647 Holomorphic, 29 Hunt, G. A., 641, 647 Hunt, R. A., 161, 647

I Independent random variables, 7 sum of, 173 Independent set, 487, 490 Indicator, 8 Ingham, A. E., 288, 328, 648 Ingham-Levinson theorem, 288 Inner product, 37 Inner product space, 3 8 Integrability of sum of series, 143 Integrable, 9 Integrable characteristic function, 437 Integral modulus of continuity, 16 Intermediate type of entire function, 461

Inverse Fourier transform, 12 7, 2 0 6 Inversion formula for Fourier-Stieltjes transform, 128 for Fourier transform, 12 3 for Laplace transform, 246 Levy, 128 for Mellin-Stieltjes transform, 270 for Mellin transform, 274 Isometric, 2 0 8 lvasev-Musatov, 0. S., 602, 648 Izumi, S., 79, 172, 193, 648

J Jacob, M., 192, 648 Jensen's inequality, 2 6, 2 8 Jensen's theorem, 31 Jessen, B., 1H, 641, 648 Jordan's decomposition theorem, 18 Jordan's test, 116

K Kac, M., 112, 643, 648 Kaczmarz, S., 70, 113, 134, 161, 167, 648 Kahane, J.-P., 612, 613, 641, 648 Karamata, J., 302, 301, 311, 648 Karlin, S., 124, 648 Kawata, T., 164, 328, 344, 361, 398, 4H, J11, J19, 120, 132, 138, 602, 649 Keogh F. R., 79, 134, 648 Kershner, R., H7, 649 Khinchin, A., 96, 378, 107, JIO, 167, 649, 610 Khinchin criterion, 378 Knopp, K., 122, 610 Kojima, T., 161, 610 Kolmogorov, A., 491, 496, 497, 101, 107, JIO, 134, 610 Korosonel'skii, M. A., 28, 610 Krein, M., 398, 401, 642, 610 Krickeberg, K., 361, 610 Kunisawa, K., 101, 134, 610

L

L•, 2J LP(dp.), 2J

L•(O), 2J Lamperti, J., 473, 610 Landau, E., 301, 329, 610

665

INDEX

Laplace-Stieltjes transform, 232 analyticity of, 242 hila teral, 2 6 J Laplace transform, 78, 232 of convolution, 2 J4 singularity of, 298 Lattice distribution function, 83 Leadbetter, M. R., 473, 64J Lebesgue, H., J 8 Lebesgue decomposition theorem, 17 Lebesgue integral, 9 Lebesgue measurable set, J Lebesgue point, 20 Lebesgue-Stieltjes integral, 14 Lebesgue-Stieltjes measure, J Levinson, N., 73, 79, 80, 288, 292, 293, 6JI Levy, P., 128, 347, 349, 44J, 462, J22, J32, J37, 139, 603, 6JI Levy inversion formula, 128 Lindeliif, E., 36 Linear operator, 41 Linear space, 37 Linnik Yu, V., 394, Lipschitz condition, Littauer, S. B., 193, Littlewood, J. E., 647, 6JI

39J, 396, 4H, 6JI JJ 6JI 27, 30J, 329, H7, 646,

Locally uniform limit, 372 Loeve, M., 101, I6J, 402, JI7, J2J, J34, 6JI Lozinskii, S.M., 79, 6JI Lukacs, E., 3J6, 391, 392, 394, 39J, 397, 409, 4H, 473, 474, 6J2 Lusin, N., H, 6J2

M McLaughlin, J. R., 603, 6J2 Makabe, H., 473, 6J2 Mandelbrojt, S., 231, 613 Mapping, 41 Marcinkiewicz, J., 4J6, 474, J02, J68, J73, J83, 199, 603, 6J3 Marginal distribution function, 32J Markov's inequality, 402 Masani, P., 398, 613 Matveiv, V. A., 79, 613 Maximal type of entire function, 461 Maximum modulus theorem, 3 J Mean concentration function, 44J

Mean value theorem, 23 first, 23 second, 23 Measurable function, 6 Borel, 6 multivariate, 7 Measurable set, 2 Measurable space, 2 Measure, 2 generated by distribution function, 32J Mellin formula, 277 Mellin-Stieltjes transform, 269 Mellin transform, 2 73 Mencholf, D., H6, 613 Minimal o:r-field, 2 Minimal type of entire function, 461 Minkowski inequality, 2J Moment, 400 absolute, 40 I Moment generating function, 403 Moment problem, 40J general, 70 trigonometric, 70 Monotone convergence theorem, 12 Monotone function, 81 Morgan, G. W., 328, 613 Multiple Fourier series, 31J Multiple Fourier-Stieltjes transform, 326 Multiple Fourier transform, 316 Murthy, V. K., 180, 613 Mutually singular, 18

N Nagy, B. Sz., 210, 616 Norm, 38 Normal orthogonal sequence, 194 Normed inner product space, 19J Normed space, 38 Nondecreasing function in R .., 324 Nonnegative definite function, 376 Nontangentially, 143 Null element, 37

0 Offord, A. C., 80, 398, 613, 614 Operator, 41 Orthogonal, 194, 19J

INDEX

666 Orthogonal sequence of random variables, 63 5 Orthogonal series, 19 5 Orthonormal sequence, 194, 195 Orthonormal set, 196 Ostrovskii, I. V., 474, 660 Ostrowski, A., 28, 654

p Paley, R. E. A. C., 228, 231, 284, 328, 543, 567, 607, 612, 629, 654 Paley-Wiener theorem, 228, 284 Papoulis, A., 157, 654 Parseval relation for Fourier series, 198 for Fourier transform, 2 0 5 for Mellin transform, 275 for periodic characteristic function, 104 Parzen, E., 189, 193, 654 Periodic characteristic function, 9 3 Ph ragmen, E., 36 Phragmen-Lindelof function, 2 8 0 Pitman, E. J. G., 414, 654 Pitt, H. R., 329, 654 Plancherel, M., 201, 231, 654 Point of density, 21 Point of increase, 8 3 Point spectrum, 8 3 Poisson integral, 141, 149 conjugate, 149 Poisson kernel, 141, 149 conjugate, 141, 149 Pollard, H., 399, 6H Pblya, G., 37, 389, 6H Porcelli, P., 231, 6H Prasad, B. N., 78, 6H Pringsheim, A., 4 5, 6 H Privalov, J., H, 652 Probability, 3 Probability density function, 82 Probability distribution function, 325 Probability marginal distribution function, 325 Probability measure, 3 Product measure, 8 Product measure space, 8 Product probability space, 8 Product space, 7 Product cr-neld, 8 Projection, 195

Pure distribution function, 541 Pure theorem, 541 Purely discontinuous function, 541

R Rademacher function, 508 Radermacher series, 56 5 Radius of almost sure convergence, 620 of convergence in probability, 620 Radon-Nykodym derivative, 19 Raikov, D. A., 96, 363, 364, 6H Ramachandran, B., 111, 462, 466, 6H Randels, W., 58, 6H Random Fourier series, 612 Random power series, 619 analytic continuation of, 629 Random variable, 6 m-dimensional (random vector), 324 truncated, 496 Random vector, 324 Rearrangement of series, 52 0 Rectifiable curve, 2 9 Regular, 29 Regular point, 299 Regular summability, 133 Relative frequency curve, 187 Renyi, A., 429, 6H Ridge function, 4 55 Riemann-Lebesgue lemma, 43 generalized, 47 Riemann-Stieltjes integral, 15 Riemann summability, 193 Riemann zeta function, 276 Riesz, F., 79, 197, 210, 534, 6H, 656 Rogosinski, W. W., 113, 647 Rossberg, H.-J., 474, 656 Royden, H. L., 20, 42, 485, 656 Rudin, W., 201, 231, 328, 656 Ryii-Nardzewski, C., 629, 631, 656

s Saks, S., 22, 656 Salem, R., 608, 640, 656 Sample covariance function, 3 8 3 Schaeffer, A. C., 564, 565, 656 Schmetterer, L., 347, 656 Schmidt, R., 305, 656

INDEX

667

Schoenberg, I. J., 398, 616 Schur, I., 165, 616 Schwarz inequality, 3 8 Second mean value theorem, 23 Semi-Gaussian, 614 Senchishchev, N. A., 603 Shohat, J. A., 70, 401, 617 cr-field, 1 generated by random variable, 489 minimal, 2 tail, 489 cr-finite measure, 3 Signed measure, 18 Silverman, L. L., 161, 617 Simple function, 8 Singular integral, 167 Singular point, 299 Singularity of Laplace transform, 2 9 8 Slow growth, 302 Smirnov, W., 214, 617 Smithies, F., 440, 444, 473, 643 Smoothness, 408, 418 Sparo, D. J., 621, 617 Spectral distribution function, 63 8 Spectrum, 8 3 continuous, 83 point, 83 Spherical partial sum, 316 Stable distribution function, 391 Standardized, 82 Statistic, 18 6 Steinhaus, H., 70, 397, Jll, J13, 134, 161, 167, 629, 617

Steinhaus function, 1 11 Stratton, H. H., 419, 603 Strip of analyticity, H7 Summability, 131, 196, see also specific type of series of random variables, 197 Summability theorem for Fourier series, 1H for Fourier transform, 141 Summation by parts, 23 Sunouchi, G., 113, 617 Sur, M. G., 621, 617 Symmetric distribution, 83 Symmetrized (symmetric) random variable, HI, 6U Szasz, 0., 18, 391, 617 Szego, G., 37, 214, 617 Szeg~Smirnov theorem, 214

T T-summability, 132 T*-summability, 196 Taibleson, M., 79, 617 Tail function, 489 Tail set, 489 Tail cr-field, 489 Takahashi, T., 193, 617 Takano, K., 391, 618 Tamarkin, J.D., 70, 405, 473, 647, 617 Tauberian condition, 301 Tauberian theorem, 301 general, 3 0 1 Tauber's theorem, 305 Three series theorem, 497 Titchmarsh, E. C., 32, 36, 164, 167, 192, 201, 231, 27J. 277, 299, 474, 629, 618 Toeplitz, 0., 132, 165, 618 T oepli tz matrix, 13 1 Toeplitz method of summability, 132 Tonelli, L., 316, 618 Tonelli's theorem, 13 Total variation, 16 Transformation, 41 bounded linear, 2 0 8 Trigonometric moment problem, 70 Trigonometric polynomial, 50 Trigonometric series, 49 in complex form, 10 Truncated random variable, 496 Tucker, H. G., 132, 600, 603, 618 Typical function, 101

u Uchiyama, S., 641, 618 Udagawa, M., 361, J15, J19, 602, 649 Ugaheri, T., 240, 618 Unbiased estimate, 187 Unconditionally convergent, 120 Unicity (uniqueness) theorem for characteristic function, 3 57 for Fourier-Stieltjes coe11icient, 88 for Fourier-Stieltjes transform, 91 for Fourier transform, 73 Uniform distribution function, 83, J12 Uniformally continuous operator, 42 Unitary, 209

INDEX

668

v Valiron, G., 474, 6f8 Van Bahr, B., 112, 6f' Vander Corput's lemma, H8 Vander Vaart, H. R., H3, 16f, 6U Van Kampen, E. R., f23, f26, f32, f34, 6f' Vanishing of sum of series, f '0 Vector sum of sets, 104 Vivanti's theorem, 2"

w Watson, G. N., 210, 277, 6U Weak convergence of sequence of monotone functions, 332 of series of random variables, f 8 6 Weierstrass singular integral, 174 Whittaker, E. T., 227, 6f' Widder, D. V., 276, 277, 2,, 32,, 40f, 6f' Wiener, N., 7,, ,6, 164, 18f, U3, 201, 228, 231, 284, 3os, 328, 32,, 383, 3n, 612, 6f'

Wiener class, 383 Wiener's formula, 18f Wiener's remark, 288 Wiener's theorem, 7' Window, 1'3 Wintner, A., 108, 111, 328, f02, f03, f20, f22, 143, f47, H6, H7, 6f,, 660

y Young, G. C., 3,7, 660 Young, W. H., 38f, n7, 660

z Zero-one law, 4n Zolourev, V. M., 130, 164, 271, 660 Zygmund, A., 26, 28, 7,, 113, 117, 164, 316, 340, 3U, n7, 411, 417, so2, f42, 167, f68, f73, f83, f,, 603, 607, 608, 62,, 640, 641, 660