Linear approximations in convex metric spaces and the application in the mixture theory of probability theory

h Afc

(3.3.60)

roo

y»oo

//

K3(x,y)f(x)f(y)dxdy, (x,y)f(x)f(y)dxdy, K 3

■oo «/—oo

where # 3 is the third iterate of the kernel function K. ([32], p. 230) *.(«,») = i < £ ^ Mx,y) = £±i*e^k!M., k=l Jfe=l

By a Theorem of Mercer

x ,, yy e£ RR ,,

;

x

k

*

and the convergence is uniform. From here •M(*) = E ^ § ^ . . Jk=l

(3-3.61)

*

Let us introduce the following notations.

/, = ( ^ ( i j j e R .

L e / i . ^ ^ W e E j L

22 {* = (*,-) (»,.) £1III£E£I^= 1^gv, gv,4(x) / = js (*,) €€g 4A g?*(*) EE2 J}},,, 4(x)€e€E where £2 is the Hilbert space of square integrable real sequences. There is a one-toone correspondence between E 2 and £\, as well as between E 2 and £. Thus the sets £1 and £ are convex, and £2 is closed in the metric of the space £2. Now let X (k=l,...,U>) Xkk = = {f,ip {f,¥k) (* = l,...,o>) . k) From Eq. 3.3.59, Eq. 3.3.60 and Eq. 3.3.61 we obtain that the discrepancy function $a?fr will be minimized over the set E 2 by the function h G E 2 defined by h x

() = E 'E,yM*) AW r ^ W

(3.3.62) (3-3.62)

32

INTRODUCTION

if and only if the function F defined by

F(xu...,xw)

= X]^3 k

k=i

is minimized by the vector condition

x = (XJ) G t\

over the set l\

under the auxiliary

£(!^*) It = l .

(3.3.63)

Since F is a convex function over the convex set £i, the system of equations | ^ = 0 (fc = l,...,u>) oxk must be satisfied by the point of absolute minimum in i\, ^(x1,...,x(tf) = ^ ^ | - 2 A ^ f

(1,^),

(3.3.64) where ^ R .

After differentiation from Eq. 3.3.64 we get the system xk = \(l,tpk)\2k

(*=l,...,w).

(3.3.65)

Substituting Eq. 3.3.65 into Eq. 3.3.63 we obtain A£(1,<^)2A* = 1.

(3.3.66)

k=i

On the other hand, substituting the values Eq. 3.3.65 into Eq. 3.3.61 and into Eq. 3.3.62, and using Eq. 3.3.66 we get the statements Eq. 3.3.57 and Eq. 3.3.58 of Theorem 3.3.1. Thus Theorem 3.3.1 is proved. Starting from Theorem 3.3.1 and using the well-known inequality m a , 6 (E 2 ) > m a , 6 (E 2 ) > 0 we get the following Corollary 3.3.1 Keeping the notations and assumptions of Theorem 3.3.1, we have h* G E 2 if and only if 1 m,a,&(E ) Y%=i(l>
*

=

£L,(i,v*) 3 A*"

Finally, the main result of this paragraph can be formulated in the following way.

3. SPECIAL

33

CASES

T h e o r e m 3.3.2 Let A be a totally convex metric space with respect to a 6 A. the function b with values b(t) £ A,

b(t) fa,

Let

teR

be given. Let the elements of the sequence {A*}^ be the eigenvalues of the symmetric positive definite Hilbert-Schmidt kernel function Eq. 3.3.53 in increasing order. Let {(fk}i be the sequence of the corresponding eigenf unctions forming an orthonormal system. Then the element a £ A is decomposable by b(t) £ A, t £ R over E 2 if and only if £(l,V?*) 2 A* = o o .

(3.3.67)

A:=l

In this case the representation

a = I b(t)h(t)dt J—oo

holds, where h x

( ) = ^(htpk^kMx)

e E2,

xeR

Proof. The statements of the Theorem follow immediately from Theorem 3.3.1 and Corollary 3.3.1, respectively. It is obvious that condition Eq. 3.3.67 is satisfied if and only if the kernel function K is non-degenerate.

Chapter II On the decomposability of probability distribution functions In this chapter we deal with the decomposability problem of probability distribution functions using the general results of Chapter I.

4

Formulation of the problem

It is well known that a function F(x), x G R is a probability distribution function (distribution function for short) if and only if a) it is not decreasing, b) it is right continuous, c) F(—oo) = 0, F(oo) = 1. We keep the notation E for the set of probability distribution functions. Definition 4.1 We say that G(z,x) ; x,z G R is a family of distribution with parameter x, if the following conditions are satisfied. a) For each value of x the function b)

G(z,x)

G(z,x)

functions

is a distribution function in

z.

is a Borel measurable function of x.

The following statement holds (Appendix A, Theorem A.l). If H G E, then G{z, x) dH(x) G E.

F(z) = I

(4.1)

J—oo

Definition 4.2 The distribution function mixture with the weight function H G E, G(z,x) , x,y eK.

F(z) defined by Eq. \.l is called the of the family of distribution functions

The Fourier transformation of F G E, i. e. the function oo

/(*)

/

■oo

35

eH*dF(x) (x)

36

CHAPTER II. DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

is called the characteristic function of F. Denote by g(t,x) the characteristic function of the family of distribution func­ tions G(z,x) , a ; , j / 6 R . It is obvious that oo

/

(4.2)

g(t,x)dH(x) ■OO

is the characteristic function of the mixture distribution function Eq. 4.1. Definition 4.3 We say that g(t,x) ; t,x G R, is a family of characteristic functions with parameter x, if a) for each value of x g(t, x) is a characteristic function in t, b) g(t,x) is a Borel measurable function of x. Definition 4.4 The characteristic function Eq. J^.2 is called the mixture with the weight function H G E, of the family g(t,x) , x G R of characteristic functions. Definitions 4.2 and 4.4 are equivalent in the sense that Eq. 4.2 follows from Eq. 4.1, and vice versa. The following definition has a fundamental role in the subsequent paragraphs: Definition 4.5 Let C c E , C ^ 0. The distribution function F is said to be decomposable by the family G(z,x), i G R , of distribution functions over the set C, if there exists a H G C, that satisfies equation Eq. J^.l. In this section we deal with the following two questions corresponding to the questions formulated in Chapter I. A/ What is the necessary and sufficient condition for the decomposability of F G E by the family of distribution functions G(z,x), x G R, with parameter x over the set C ? B / In the case of decomposability, which element of C is the weight function ? In other words, our task is to determine the necessary and sufficient condition for the integral equation Eq. 4.1 being solvable over the set C, and in the case of solvability, to find the weight function. The following decomposition theorem of Lebesgue is well known. If F G E ,

then F(x) = aFa(x) + /3Fj(x) +

1Fs{x)

with a>0, 0>O, 7>0, a + /? + 7 = l , where all functions on the right side are distribution functions. Fj(x) is a jump function the discontinuity points of which coincide with those of F , Fa(x) is a strictly increasing distribution function, absolute continuous with respect to the Lebesgue

4. FORMULATION OF THE PROBLEM

37

measure, and Fs(x) is a singular function, i. e. a continuous, increasing function with F^(x) = 0 almost everywhere. If j3 = 1, then F(x) is a discrete distribution function with a finite or denumerably infinite number of discontinuity points. Let Ej denote the set of these distribution functions. If a = 1, then f(x) = F'(x) almost everywhere is a probability density function of F (density function for short) with respect to the Lebesgue measure. E a denote the set of absolutely continuous distribution functions. The following set of distribution functions has a role. Let a < b be real numbers, where a = — oo and b = oo are also permitted. Let E(a, b) be the set of distribution functions, that are strictly increasing on [a, 6], continuous on the whole real line, have value zero at x = a, and value one at x = b. As we have already said we give answers to the questions A/ and B / by the help of the results of Chapter I. Namely, we treat the cases where the totally convex set of distribution functions is one of the following: 1) The set E. 2) The set E(a,6). 3) The set E a . 4) The set E p C Ej depending on the parameter p, 0 < p < 1, with a denumerable number of discontinuity points. In these four cases the metric will be generated by different scalar products. On the other hand Chapter II is a complement to Chapter I. Namely we did not examine in Chapter I whether the concepts introduced and the connections between them have reality. It comes to light in Chapter II that totally convex sets of distri­ bution functions do exist, we can define a scalar product over these sets so that these sets become totally convex spaces. Thus the theorems and corollaries of Chapter I hold in these more special totally convex spaces of Chapter II as well. The following special case is said to be the decomposability problem in the narrow sense: Let C C Ey be the set of distribution functions which may have jumps only at the prescribed points Xi < ... < xn , n > 2. Let F G E, and let a family G(z,x), x G R, parameter x be given. Using the notation G(z,xk)

= Gk(z)

of distribution functions with

(k = l,...,n),

(4.3)

then Eq. 4.1 can be written in the form n

F(z) = £ > G * ( z ) ,

(4.4)

A;=l

where now the set C is replaced by S n . In this special case the problem of decomposability is the following: what is the necessary and sufficient condition in order that F be representable in the form Eq. 4.4 by the distribution function Eq. 4.3 with a vector of S n ? Moreover, in the case of decomposability we have to determine a vector p^ which satisfies Eq. 4.4. These current problems play an

38

CHAPTER

II. DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

important role in the applications. They are called the problems of decomposability of distribution functions in the narrow sense. So far in the literature the focal point has not been the elaboration of the theoretical background but of numerical methods. The procedures have provided solutions for decomposability problems arisen from different concrete applications. Such procedures are described in the two monographs of Medgyessy ([28], [29]). The main idea is that the distribution functions in the decomposition are deduced from the observations related to the distribution functions. The observations also give estimates of those parameters which appear in the Cauchy, Poisson, normal and other distribution functions. These procedures have an u ad hoc" character (i. e. are specific for a given problem. They are unified by some fundamental remarks of Medgyessy. Bibliographical data, concerning various applications of the decomposition procedures can be found in [29]. Some areas of application: analysis of absorption spectra, separation of albumen by electrophoresis. The latter plays a significant role in the analysis of human blood serum and the therapeutic separation of gamma-globulin. Beyond the above applications to biometry, these decomposition procedures have been used in econometry, biochemistry. In the following we present the solutions of some decomposability problems building upon "ad hoc" methods that are independent of the results of Chapter I. Let C = E, and let the family of distribution functions G(z,x) with parameter x G R be defined by G(z,x) = G(z-x) , where G G E.

xeR ,

Then oo

/

G(z-x)dH(x)

(4.5)

■oo

is the convolution of the distribution functions G and H. Let / , g and h be the characteristic functions of F , G and if, respectively. Then /(*) = g(t)Ht),

t€R

(4.6)

In this case the basic problem can be formulated as follows. Given the characteristic functions / and g, a characteristic function h is to be found which satisfies equation Eq. 4.6. Or in other words, when is the quotient of two characteristic functions a characteristic function ? If the characteristic functions in formula Eq. 4.6 are infinitely divisible, then the question can be answered without difficulty ([28], 168-174). If this condition is not satisfied, then some criteria are available for setting the question (see e. g. [12], [30]). The subject has been treated in [25] and [26]. Let EQO C E denote the set of distribution functions which have moments of all orders; i. e. if F G E ^ , then oo

/

•oo

\x\kdF(x)
(lb = 0 , 1 , 2 , . . . ) .

39

4. FORMULATION OF THE PROBLEM

Let E+, C EQO be the set of distribution functions, which are equal to zero on the negative real line; so if F G E+> , then TOO

Mk(F) =

xkdF(x)
(£ = 0 , 1 , 2 , . . . ) .

Jo

T h e o r e m 4.1 Let f and g be the characteristic functions of F G E ^ G G EQO , respectively. Then there exists a H G E ^ satisfying the equation

/(*) = f0g(tx)dH(x)

and

(4.7)

if and only if \Mk(G)j0 is a Hankelian positive sequence. T h e o r e m 4.2 Let f and g be the characteristic functions of F G E ^ G G E+, , respectively. Then there exists a H G EJ> satisfying the equation

and

/•OO

f(t) = /

g(tx) dH(x)

(4.8)

JO

e/ and only if

(M fMkk(F)}°° (F)r \M (G)f0 \Mkk(G)j 0 is a Hankelian totally positive sequence. Proof. If Eq. 4.7 or Eq. 4.8 is satisfied, then Mk(F) = Mk(G)Mk(H)

(k = 0,1,...) .

From here by Theorems H.4 and H.5 of Appendix H we get the statements of Theo­ rems 4.1 and 4.2, respectively. As it was mentioned above, F G E is said to be decomposable by the linearly independent distribution functions G^GE

(i = l , . . . , n ;

n>2),

if the relation

F = J2a3G3 holds with some vector a = (ctj) G S n . A generalization of this definition is the following:

40

CHAPTER

II. DECOMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

We say that a distribution function F G E is asymptotically decomposable by a sequence of linearly independent distribution functions Gj G E (j = 1,2,...) , if there is a representation oo

F(x) =

^ajGj(x)

that holds uniformly in x G R, where a = (a,) 6 S + . In the usual way, we obtain the following statement: If the distribution function F G E is decomposable by the linearly independent distribution functions Gj G E (j = 1 , . . . , n) , then the decomposition is unique. Now we give a necessary and sufficient condition for a distribution function from E to be decomposable by a finite sequence of linearly independent distribution functions from E. Then a necessary and sufficient condition will be given for a distribution function from E to be asymptotically decomposable by a sequence of linearly independent functions from E. T h e o r e m 4.3 A distribution function F G E is decomposable by the linearly independent distribution functions H3; G E (j = 1 , . . . , n) if and only if there exists linearly independent distribution functions G

G0 = F,

M

€B

(j = 2,...,n-l),

Gn-X = Hn ,

and numbers 0 < 7, < 1 (j = 1 , . . . , n — 1) such that the identities Gj-i = (1 - riHj + fjGj

(j = 1 , . . . , n - 1)

(4.9)

are satisfied. In this case n-l

F = Y,Wi--.

7 i - i ( l - lj)Hj + 7i • • • ln-iHn

(4.10) (4.10)

i=i

with 70 = 1, where n-l

7i---7n-i + 5^7071 ■ • • 7 j - i ( l - 7 j ) = 1 •

(4.H)

3=1

The following Lemma will be used in the proof of Theorem 4.3. L e m m a 4.1 Suppose that the distribution functions the distribution functions Go = F G E ,

G

H

6 E

H3; G E (j = 1 , . . . ,ra) , and

(i = 2 , . . . , n - l ) ,

G _ , = J5f„ ,

satisfy the identities Eq. \.9. Then Hj (j = l , . . . , n ) are linearly independent if and only if Gj-i (j = 1 , . . . , n) are linearly independent.

4. FORMULATION OF THE PROBLEM

41

Proof: Multiplying the identities Eq. 4.9 by the real numbers A i , . . . , An consecutively, summing up these new identities, and, finally, reducing the relations obtained to zero, we find that [—AxGo 4- (A171 — A2)Gi 4- • • • 4 (An_27n-2 — An_1)Grn_2 4 A

n—l7n—1

+ [Aj(l -

7l)i/+

. . . + A n _!(l - 7 n - i ) # n - i + KHn] = 0 .

From here it is easy to deduce our statement. Proof of Theorem 4.3: a) First suppose that the distribution functions G0 = F,

G

H

GE

(j = 2 , . . . , n - l ) ,

Gn.x = Hn

are linearly independent and that the identities Eq. 4.9 are satisfied. In this case the distribution functions Hj (j = l , . . . , n ) are also linearly independent by Lemma 4.1. Multiplying the identities Eq. 4.9 by the numbers 1 ,

7i >

7i72 ,

•••,

7i72---7n-i

one after the other, and adding the new identities, we get the expression Eq. 4.10. Here the coefficients are positive, and it can be easily verified that they satisfy Eq. 4.11. Thus the distribution function F is decomposable by the linearly independent distribution functions Hj (j = 1 , . . . , n) . b) Suppose next that F E E is decomposable by the linearly independent distribution functions F j G E (j = l , . . . , n ) , i. e. there exists a vector a = (ctj) e S n such that F = aiH\ 4 . . . 4 anHn. If 1 — 71 = » i , then F = Go = (1 - 7 i ) # i + 71 Gi, where Gl=±a^H3,

af>0,

J2af

j=2

= l.

j=2

If n = 2 then G\ = H2, and the procedure is finished. If n > 2 and 1 — 72 = or2 , then G 1 = ( 1 - 7 2 ) # 2 4 72G 2 , where

G2 = ±^H3,

af>0,

3=3

±af>

= l.

j=3

If n = 3, then G2 = H3 , and the procedure is finished. If n > 3, the procedure continues until we have Eq. 4.9. The following statement ensues immediately from Theorem 4.3. Theorem 4.4 A distribution function F E E is asymptotically decomposable by the sequence of linearly independent distribution functions Hj £ E (7 = 1,2,...) if and only if there exists a sequence of distribution functions G0 = F,

Gj£E

(i = l , 2 , . . . )

42

CHAPTER II. DECOMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

and a sequence of numbers 7o = l ,

0 < 7 i < l1

(i = 1,2,...)

such that the identities Gj_, = (1 - ii)Hi + -uGj

0 = 1,2,...)

(4.12)

are satisfied, and lim[7i7 2 ...7„] = 0 .

n—t-oo

In this case, oo

FW = ^

7

O 7 I - 7 H ( 1 - # ; W

i=i

uniformly in x G R, where CO

$^7o7i • • • 7 i - i ( l - 7 j ) = 1 » and £/ie distribution functions

Gj (j = 0,1,...) are linearly

independent.

It may seem that Theorems 4.3 and 4.4 do not assert too much. Now we show nevertheless that they are very useful in finding distribution functions by which a given one is decomposable or asymptotically decomposable. In fact, we shall construct distribution functions Go = F G E(a,6) ,

Gj G E(a,6) ,

Hj G E(a,6)

(j = 1,2,...)

which satisfy condition Eq. 4.12. We say that
P(t=j) = <*j (i = 1,2,...) ,

5>i

= 1

-

(4.13)

It is obvious that 0, x G [0,1], y>'(l) > 1. L e m m a 4.2 Let
where £ is the random variable defined by Eq. 4.13.

4. FORMULATION OF THE PROBLEM

43

It is evident that G(a) = 0, G(b) = 1. The function G is continuous because it is a uniform limit of continuous functions. Finally, G(x) is strictly increasing on [a, b]. In fact, if a < a < fl < 6, then G(/9)-G(a) = [F(f3)-F(a)] \ca x + £<*,■ {F^(a)

=

+ F'" 2 (a)F(/?) + . . . +

F^(/3)} >

3=2

> [F(/J) - F ( a M F ( a ) ) > 0 . L e m m a 4.3 Let
1 1-7

[F-7G]€E(a,6),

w/iere G = ip(F) e E(a,6). Proof. Let <£>(#) = E(x^), where £ is the random variable defined by Eq. 4.13. It is obvious that H(a) = 0, H(b) = 1, and H is continuous. We show that H(x) is strictly increasing on [a, 6]. In fact, if a < a < ft < 6, then

H(0) - H{a) = 1

i-y

[F(/?)-F(a)]

1 - £ > ; { f - * ( / ? ) + F'-\p)F{a)

+ ... +

F>-\a)}

i=i

>[F(/3)-F(a))1

™'{1)>0. 1—7

L e m m a 4.4 and

Le£ y?(x) 6e a generating function without zero state. If F G E(a, 6) G ^ y , ^ "

1

)

(jfc = i , 2 , . . . ) ,

then Hk+\ =

X

■[Gfc -

1 - 7*+i

7*+iGiH.i] € E(a, 6)

( i = 1,2,...) ,

tuyere 0 < 7^+1 < 1/2. Proof. Obviously, Hk+i(x) is continuous and Hk+i(a) = 0, Hk+i(b) = 1. We claim that Hk+i(x) is strictly increasing on [a, 6]. Let ?(*) = £ ( * « ) ,

x€[0,l]1

44

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

where £ is the random variable defined by Eq. 4.13. If a < a < /3 < 6, then

Hk+1(p) - Hk+1(a) = 1

(

1

oo

-

OO

oo

oo

1

= r r ^ 7 £ « > K " ' ^ ) - Fi2"1(«)] I1 --»+i (*"'2'~*0*) + *"'2*~'(«))] 1 2 = 71 -- 17*+i — £j r j *; Lk i 2 M (« - F32k~l (")]1 - ^+il ^ o. J

Thus the proof of the Lemma 4.4 is complete. Lemma 4.5 Let a:x €£ [0,1] [0? 1] ^ e a generating function without zero state. Let F G E(a,6). T/ien ^ e distribution functions Gk=V(F2"-1). 2

G0 = F, are linearly

fc J

- \

(jfc=l,2,...)

independent.

Proof: Let
n

oo

Jfc=i

i=i

oy + X > E ^ 2 ' " = 0 >

ye [0,1].

After a simple rearrangement we obtain successively that Xj = 0 (j = 0 , 1 , . . . , n), and this is the statement of Lemma 4.5. By Theorems 4.3, 4.4 and Lemmata 4.1-4.5 we obtain the following Theorem. T h e o r e m 4.5 Let
G0 = F,

( i = 1,2,...)

are linearly independent, and if 7o = l , then the distribution

0

<

7 l

< — - , (p'(l)

1

0<7fc<22

(* = 2 , 3 , . . . ) ,

functions Gn_x ,

Hi =

T^—[F

I—71

[F-- 7.G,] € E(a, 6) ,

4. FORMULATION OF THE PROBLEM Hk = — ^ - [ £ * _ i -

lkGk]

45

e E(a, b)

(k = 2 , . . . , n - 1; n > 2)

are linearly independent, and the representation n-l

F = Yl 7o7i • • • 7.7-1 (1 ~ lj)Hj + 71 • • • 7n-i G n _i i=i

Ao/ds /or n = 2 , 3 , . . . , where n-l

7 ! . . . 7n_! + £

7071... 7,-1(1 -

7i)

= 1.

i=i

Moreover,

00

uniformly in x G R ,

^ 7 0 7 1 • • . 7 j - i ( l ~ Ij)Hj(x) i=i where

= F(x)

^7o7i---7i-i(l-7j) = 1 i=i

This Theorem states essentially that all elements of E(a, b) are decomposable by n > 2 linearly independent distribution functions from E(a, 6) in infinitely many different ways, and that all elements of E(a, b) are asymptotically decomposable by linearly independent distribution functions from E(a, b) in infinitely many different ways. T h e o r e m 4.6 Under the assumption of Theorem 4-5 the identity Hk = F holds if and only if either k = 2, or k = 1. If y>{x), x £ [0,1] , where the random variable £ is defined by Eq. 4-IS, then in the case of k = 2 necessarily ^

= 72(1-72)

ctj = 0

(i = 0 , 1 , . . . ) ,

(4.14)

otherwise.

If k = 1, then c*i = 1. Proof: Substituting y = F(x),

from Hk+i = F we get the identity

00

(1 - 7.4-1)2/ = £

00

a^y^-1^1

+ £ ( a 2 i - 7*+i^V' 2 *~ 1

(4-15)

J=I

valid for fc > 1. If k > 2, the first power of y does not occur on the right side of Eq. 4.15, thus the identity Eq. 4.15, and consequently the assumption Hk+\ = F cannot be true for k > 2. If k = 1, we obtain from Eq. 4.15 that oc\ = 1 - 72 ,

a 2 j-i = °

(i = 2 , 3 , . . . ) ,

46

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

and therefore a2 = 720:1 ,

a4j = 7 2 o 2 j

(j = 1 , 2 , . . . ) .

From here we get Eq. 4.14. For the remaining indices we have ctj = 0, because oo 00

oo 00

( 1 - 7 2 ) ^ ^ = 1k=0 k=o

3=0 j=0

In the case k = 1 we get 00

3

= y-iiJ2 jyJ ajy > a

{i-ii)y

i. e. cti = 1. In what follows we shall deal with some consequence of Theorems 4.5 and 4.6. For y>(x) = x2 we get the following statement. Corollary 4.1

Let F G E(a,6). / / 0 < 7 < 1/2

then the distribution

Hk = - J — \F2k" - 7 F 2 1 € E(a, 6) 1 — 7A:

L

(k = 1,2,...)

J

are linearly independent, and none of them equals to F. tation

In this case the represen­

n-1

F=bnF^ i=i

holds for n > 2, where n-l

K=in-\

A- = ( i - 7 ) y _ i

0 = 1,2,...),

n-l

&» + £ & = 1. Moreover, oo

(l-7)^k-1Hk(x) #*(*) = F(x) k=l

uniformly in x G R . Let now in Eq. 4.13 aj = (l-q)qi-1
functions

0 = 1,2,...),

with 0 < q < 1. In this case, v?(l) = 1, and (f'(l) = 1/(1 - q).

4. FORMULATION OF THE PROBLEM

47

Corollary 4.2 Let F G E(a,6) and 0 < q < 1. 0 < 7 * < 1/2 (A: = 2,3,...) , */*en #1

=

ft = G

1

F-

1-7J i-g

7 l

li1 -

1-7*

,9»»—1

(

» = L g 2n-i ^

/ / 70 = 1, 0 < 71 < 1 - q,

(1-*)F GE(a,6), 1-?F J flF2*"2

7^ 2 *~ 1 - ^F2*"1

eEM)

GE(a,6)

(fc = 2 , . . . , n - l )

(n = 1 ' 2 "")-

These distribution functions are linearly independent, and none of them equals to F. In this case we have the representation n-1

F=

bnGn-1+YiPJHj

for n = 2 , 3 , . . . , where Pj = bn

7o7i---7i-i(l-7j)

= 7i---7n-i

(i = l , : . . , n - l ) ,

(n = l , 2 , . . . )

Moreover, 00

Y,PiBi{x) = Fix) i=i

uniformly in x £ R . uniformly in x G R .

By Lemma 4.3 we have the following statement. T h e o r e m 4.7 Let a sequence {^Jb(^)}i° of generating functions without zero state be given, and assume that
Gk = MGk-i)

(fc = l,2,...)

be linearly independent. If 7o = l ,

0<7A:<

1 ~r7jy

(* = 1,2,...),

then Hk = [G*-! - lkGk] G E(a, b) (k = 1,2,...) ; 1 — 7* these distribution functions are linearly independent, and the representation n-l

F =

7l

. . . 7n_i<2n_i + 5 Z 7071 • • • 7;-i(l ~ 7 j ) # j

(4.16)

48

CHAPTER

II

DECOMP OSABILITY

OF DISTRIB UTION

FUNCTIONS

holds for n = 2 , 3 , . . . . Moreover, if

(4.17)

—S^TT0' then

oo

£oo7 o 7 i • • .7i-i(l - 7;)#;(z) = F(x) ^ 7 0 7 1 . . . 7 i - i ( l -1i)Hi{x)

= F(x)

3=1

uniformly in x E R . Remark 4.1 There are sequences of linearly independent distribution functions sat­ isfying condition Eq. 4.16. Namely let ipk(x) = E(x^k)^ where the random variable £k is defined as follows.

P((k=j) -A m £<#>

=

af

=

1

0 = 1,2,...), (4-18) ^

(fc = l , 2 , . . . ) .

i=i

If we choose probabilities Eq. 4.18 in such a way that af>=0

0 = 1,...,*),

0 = l>2,...),

a[%>0

then it is easily seen that the distribution functions Eq. 4.16 are linearly independent. Remark 4.2 We can see from Eq. 4.16 that the sequence {Gk(x)}^° ing for fixed x G R. Thus the limit distribution G(x) = lim Gk(x) ,

is non-increas­

x e R

k—*oo

exists. If lim 7^ = 7 also exists and 0 < 7 < 1, then &—►oo

lim #*(*) = G(x) .

A;—^oo

If in Theorem 4.7 (^(z) = E(x^k), by Eq. 4.18, and here

where the random variable & is defined

«{5i = (i-«*)«i_1 0 = 1,2,...),

o f = o o' = i.--,*), 0<«<1,

(fc = l , 2 , . . . )

then by Remark 4.1 the distribution functions G0 = F,

(l-ft)gff

Gk={\~qk)^ i - qk^k-\

(* = (k=l,2,...)

5. DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

49

are linearly independent. Moreover,

ri(l) =

(* = 1 , 2 , . . . ) ,

qk l - -qk

thus condition Eq. 4.17 is satisfied. By Theorem 4.7 and Remark 4.1 we have the following result. Corollary 4.3

Let F £ E(a, b), and let the distribution qk) 1 Gk=(\~(l-ft)Off %

Go = F,

,

functions

0
(* = l,2,...)

be given. If

0<7t

7o = 1 ,

#* =

-IW^)

[£*-i - TifeGJ € E(a, 6) -U 1 — 7fc

(* = 1 , 2 , . . . ) ,

(* = 1,2,...)

and £/*e distribution functions Gn-\ , Hk (k = 1 , . . . ,ra—1) are linearly independent. Moreover, the following representation holds: n-\ ?

F = 7 i . . . 7n_iG n_i + ^ 7 o 7 i • • • 7 i - i ( 1 ~ 7j)#? i=i /or n = 2 , 3 , . . . ,

and oo

Y,lo1i-.-lJ-i(l-lJ)Hj(x)

= F(x)

i=i

uniformly in x G R.

5

Decomposability of distribution functions

In this section we deal with the decomposability problems for distribution functions using the concepts and results of Chapter I. We build upon suitable distance con­ cepts defined on certain sets of distribution functions. Specifically we extend our considerations to the following sets of distribution functions: 2.1 The set E of distribution functions. 2.2 The set E(a, b) of distribution functions. 2.3 The set E a of distribution functions. 2.4 The set Ej of distribution functions.

50

5.1

CHAPTER

II. DEC OMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

Decomposability problems on the set E of all distri­ bution functions

It is easy to see that the set E of all distribution functions is totally convex. Namely, the conditions (I)-(IV) of the convexity are satisfied evidently. Condition (IV)* re­ lated to totally convexity is also satisfied, since in this case we deal with an infinite series of distribution functions, i. e. series with positive terms, and these terms can be rearranged. The fulfilment of condition (V) is the consequence of Theorem A.l of Appendix A. Befor introducing a metric on the set E, we mention the following. Let (R, B) denote the measurable space with Borel cr-field B generated by the open sets of real numbers R. Let {Fj} be a finite or infinite sequence of distribution functions. Denote by w{Fj) the measure on the cr-field B generated by the distribution function Fj. Then there exists (Appendix A, Theorem A.4) a cr-finite measure A on B such that each of the measures w(Fj) = u>j are absolutely continuous, and so according to the Radon-Nikodym theorem there exists a function fj > 0 such that UJ(A)=

f fjX(dx), ) ,

A e B .

fj is called the density function, or Radon-Nikodym derivative of the measure Uj with respect to the cr-finite measure A. Let us introduce the following functional on the set E. Let F,G,H e e E. E. Let the measures u>(F), u(G), w(H) be absolutely continuous with respect to the
(G,H)F = [~ ( * - / ) ( * - / ) «/ — oo

J

X{dx) =T9h

£&x{dx) _

J— oo

L

( 8(5.1.1) #L1)

/

T h e o r e m 5.1.1 Functional Eq. 5.1.1 is a scalar product with respect to the distri­ bution function F on the set E . Proof: The conditions a) and b) on scalar products are evidently satisfied. The validity of condition c) is a consequence of the following result. L e m m a 5.1.1

LetF,GeE.

Then 0 < (G, G)F < oo

(5.1.2)

the left hand equality sign being valid if and only if G = F. Proof: The fact that (G, G)F = oo may also occur will be proved later. Let the measures w(F), w(G), be absolutely continuous with respect to the cr-finite measure A, and let / and g be the respective density functions.

5. DECOMPOSABILITY

51

OF DISTRIB UTION FUNCTIONS

Then

(G, G)F = r ^—ft \{dx) > 0 , J—oo

(5.1.3) (5.1.3)

J

which proves the first inequality under Eq. 5.1.2. If G = F, then evidently ( G , G ) F = 0. Since

\F(x)-G(x)\ = I f (f-g)X(dx) \J—oo

- l/-L^ A(Ji)

<

* cc^^r2

\ 1/2

from the condition (G, G)F = 0 we obtain that G{x) = F(x), c) is satisfied indeed. But also condition d) is satisfied, because in our case

x £ R, so condition

rF(G,-(i = !,-•,")) is the Gram matrix of the functions gj — f (j = 1 , . . . , n) with respect to the weight function — X(dx). Now we show that there exists a subset of E, which is uniformly bounded with respect to the scalar product Eq. 5.1.1. Namely, let F £ E be absolutely continuous with respect to Lebesgue measure. If y > 1 then distribution function Fy is also absolutely continuous with respect to Lebesgue measure and has density function yFv"1(x)F'(x). Thus 2

( ^ ^ =2 ^ -

1 =

oo

/

_

E(Vj

-i \ 2k

- 1,

(5(5.1.4) L4)

'

This function is strictly increasing, and unbounded if 1 < y < oo. But if 0 is a fixed number satisfying the condition 0 < 0 < 1, then the function Eq. 5.1.4 is strictly increasing from zero to 02/(l — 02) in the closed interval

1< v< - y -

.

i-o

Thus by Theorem 1.2.5 we have the following statement.

52

CHAPTER

II. DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

T h e o r e m 5.1.2 If F £ E is absolutely continuous with respect to Lebesgue mea­ sure, and if 6 is a fixed number satisfying the condition 0 < 0 < 1, then the set

1

H -*-^} of distribution functions product

is a totally convex metric space with respect to the scalar

(^nF

=^ j ^ ,

v>i,

*>i.

5.1.a In the following we deal with the question of decomposability in the narrow sense for probability distribution functions, as formulated in Section 4; for this purpose, we shall make use of the results of Paragraph 3.1.. Suppose that C C Ej is the set of distribution functions, which may have jumps only at the fixed prescribed points xi < . . . < xn ,

n>2 .

Let F E E , and the family G(z,x), x G R of distribution functions be given. Let the measures generated by the distribution functions F G E, and G(z,xk)

(*=l,...,n)

= G(z)

(5.1.5)

be absolutely continuous with respect to the cr-finite measure A, and let / , gj (j = 1 , . . . , n) be the corresponding density functions. Suppose that f°° 9j / — X(dx)
(j = l , . . . , n ) ,

J — oo J

and let the distribution functions Eq. 5.1.5 be linearly independent, i. e. let the Gram matrix iF(GJ(i = l , . . .

1

n))=fr^A(di)V V/-00

/

(5.1.6)

/ j,fc=l

be positive definite. Obviously LF(Gj(j

= l,...,n))

= TF(Gj(j

= l,...,n))

+ M,

where M — as above — is the matrix with entries one. The discrepancy function of the present decomposability problem is the quad­ ratic form n

n

$ F , G ( ? ) = Yl YL(Gi>Gk>>Fqiqk

'

q =

fai)

G

Qn •

Since Det T F = Det(L F - M) = Det LF - e* adj LF e ,

5. DECOMPOSABILITY

OF DISTRIB UTION

53

FUNCTIONS

and by the Theorem F.l of Appendix F e* adj Tpe = e* adj Lp e , from the assumption of positive definiteness of L we obtain that m F , G (Q n ) = inf $F,G(
1> 0.

The following decomposability Theorem can be derived from Theorems 3.1.1 and 3.1.2. T h e o r e m 5.1.3 Let the distribution function F G E be given, and let Eq. 5.1.6 be the Gram matrix of the linearly independent distribution functions Eq. 5.1.5, where n > 2 is a positive integer. Then the measure of decomposability of the distribution function F by the distribution functions Eq. 5.1.5 is the quantity

-e*U ->1 L~p e~ The relation * ^

(

0

)

)

= ^ -

1

is satisfied by the discrete distribution function which has jumps at the points . . . < xn , the size of the jumps at Xj being

^=7^1^ where aj equality

y-i,-,»),

denote the sum of the entries in the j

ge» = p <°> =

X\ <

t h

row of the matrix

( p <<») e s n

holds if and only if m F ? G (S n ) = ——r 1. e* LF e /n this case H = H0 £ Ej is ^ e distribution function with jumps

po,

' = rik>0 e-1--")

In this case H = H0 £ Ej is the distribution function with jumps at the points x\ < . . . < xn

satisfying the relation

at the points x\ < . . . < xn

1 satisfying the relation *'^) = ?#7-

The main result of this part is the following.

L]?1.

The

54

CHAPTER

II. DECOMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

T h e o r e m 5.1.4 The distribution function F E E can be decomposed by the linearly independent distribution functions Eq. 5.1.5 if and only if e*L~FAe = 1, where Lp is the Gram matrix Eq. 5.1.6 of the distribution functions Eq. 5.1.5 with respect to F. In this case the equation $F,G(P) = 0 is satisfied by the distribution function Ho G E, which has jumps of size

pf = a,>0

0 = l,...,n)

at the points x\ < . . . < xn , where aj is the sum of the entries in the jt]x row of the matrix L~^> . Using Appendix G, the following equivalent of Theorem 5.1.4 can be formulated. T h e o r e m 5.1.5 Let F G E , and let the distribution functions Eq. 5.1.5 be linearly independent. Let L = Lp be the Gram matrix of these distribution functions with respect to F. Then F is decomposable by the distribution functions Gj (j = 1 , . . . ,rc) over S n if and only if (1) L — ee* is a singular matrix and (2) b — (bj) G S n , where b is the vector defined by 66* = adj(L — ee*). In this case

^ E , / \ ,

Gk.

(5.1.7)

From the point of view of computing technique (i. e. regarding the complexity of the algorithms) the representation Eq. 5.1.7 is more advantageous than that given by Theorem 5.1.4. Namely Theorem 5.1.4 requires the calculation of adj Z, i. e. of n(n 4-1)/2 subdeterminants of order n — 1, while representation Eq. 5.1.7 requires the calculation of only n subdeterminants of order n — 1. Proof: Since L is regular and L — ee* is singular, we get Det L = e* adj L e > 0 ,

(5.1.8)

l

i. e. the conditions e* L~ e = 1 and Det(L — ee*) = 0 are equivalent. Applying Theorem G.3 of Appendix G we obtain Det Ladj(L — ee*) = adj Lee*adj L ,

(5.1.9)

from which it follows by Eq. 5.1.8 that e * a d j ( L - e e * ) e = e*adj£e .

(5.1.10)

Using the notations adj L e = (aj) ,

adj(L — ee*) = 66*,

we get e*adj(I,-ee*)e

=

e* adj L e =

(6! + . . . 6 n ) 2 , a\ + . . . + an = a.

(5.1.11)

5. DECOMPOSABILITY

55

OF DISTRIB UTION FUNCTIONS

If we add up the relations Det L bjbk = a,jak

(j = 1 , . . . , n)

obtained from Eq. 5.1.9, and use the equality Eq. 5.1.10, then we get the results a* = (&i + . . . + K)h , and a>k _ (h + . . . + bn)bk e*adjLe ~ (&i + . . . + 6 n ) 2 respectively. In view of the relations Eq. 5.1.11, it follows that ak

~ a

=

h 7T~1 T7TT (6i + . . . + bn)

a. i \ (fc=l,...,n),

which is the statement of Theorem 5.1.5. 5.1.b In this paragraph the results of Paragraph 3.2, and the metric concept, introduced in this Paragraph 5.1 will be used. The aim is to deal with the special case, where the set of weight functions is the set of discrete distribution functions with infinitely many points of discontinuity. We suppose in the following that F G E a is a strictly increasing distribution function. Then Fy is a family of distribution functions with parameter 1 < y < oo. It is obvious that Fy belonges to E 0 , and it is strictly increasing. Let ETV be the set of weight functions that are discrete distribution functions having discontinuities only at the points y = 2 , 3 , . . . . Our task is now to give an answer to the question, what can we say about the representation oo oo

/ 00

where

°° °° +1 x a F»(x)dH(y) F>(x)dH(y) = ^2F' Y,Fi*1((x)a ) i>jt

i=i

oo

aj>0

(7 = 1 , 2 , . . . ) ,

^ a

i=

l

i=i

are the jumps of H 6 E # at the points y = 2 , 3 , . . . . First we define the scalar product of the distribution functions Gk = Fk+1

(*=1,2,...)

(5.1.12)

with respect to F by the expression

(0,.aj,-(-i>«(£2^*-i)-,-irj^

,,U3,

in accordance with the concept of metric introduced in this paragraph. The answer to the question above is contained in the following Theorem.

56

CHAPTER II. DECOMPOSABILITY

Theorem 5.1.6

OF DISTRIBUTION

FUNCTIONS

The following statements are true.

(1) The transsignation of the Gram matrix TF(Gj(j

=

l,2,...))

defined by the scalar product Eq. 5.1.13 is totally positive. (2) The elements of the sequence functions.

{Gk}™

are linearly independent

distribution

(3) The inequalities mF,G(Sn)>0 hold, i. e., the distribution function elements of {Gk}^° • (4) mFiG(Sn) Proof:

(n = l , 2 , . . . ) F

\

is not decomposable by finitely many

0,

n -> oo .

(1) The transsignation of r ( n ) = I>(G*(Jb = l , . . . , n ) )

defined by Eq. 5.1.13 is equal to the transsignation of the matrix /

i

(0)

\

/ 1

)

V

(0)

T(n) =

V

(0)

n

(0)

n J

where C is the Cauchy matrix generated by the values (Appendix C, Eq. C.2) «j=i + l,

bj=j

(j = l , . . . , n )

(2) Since (Appendix C) DetT(n) =

-

n!

+ +1

n(" i )G.

>0

(n = l , 2 , . . . ) ,

the distribution functions {Gk}™ are linearly independent. (3) Since (Appendix C)

K.^))-=E|:c+:+l)C)C+:+1)(:)jTiTT>» (n = l , 2 , . . . ) ,

5. DECOMPOSABILITY

57

OF DISTRIB UTION FUNCTIONS

the statement holds. (4) According to the Corollary C.3 in Appendix C 0< (cT-^nJc)"^

n+ 2 in2

thus the statement (4) holds. Suppose again that F G E a is strictly increasing, and let us start from a sequence Eq. 5.1.12, which has property (1) appearing in Theorem 5.1.6. In the following we deal with the orthogonal system, generated by the sequence Eq. 5.1.12 in the way indicated in Paragraph 3.2. Using the notations of Paragraph 3.2 and the results of Appendix C, we obtain that

where

^-crHC-OO ('- -»•

nmo

Thus ^n

(V'KsfcOC)-

v ^jn

/

(5.1.14)

(5.1.15)

and, consequently, (n + j \ (n)

B(n)

\n-l)\Jj V -/ W

_Jn_

-On

/n>

SC-OG)

/ •

=

l

\

Therefore the orthogonal system of distribution function generated by the sequence Eq. 5.1.12 is given by

¥>»(*) =

tfcOC)*-"'

l

n ,

, .

w

N

SC-OC)

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

(5.1.16)

and the elements of this system are strictly increasing absolutely continuous distri­ bution functions. Since (
n ^ m ,

58

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

by formula Eq. 5.1.16 we immediately get the combinatorial identity

EE(:!0C:0(:-')(::11)^-'

■<-■

Relaying Paragraph 3.2 and on Appendix C, we next deal with the calculation Of (
By formulas Eq. 5.1.14 and Eq. 5.1.15 we have

+

(
n(" r)G) (*") •n(:!0Cj , )[C. +, )it(:!0C)

An_iAn

+

Since

c r')

_ n + j + 1 /n + A n \ra — 1/

Q-^CT) «-•■-"'•

and

yj n+j + 1 n

7=i

=

/ 2n \

V*-1/

"^

we obtain that

nC+ri)(")=(2":I)C-"1)n(:-0(";1) Consequently,

1

{¥n,V>n)F =

FT

(2n + 1)

n

x

.

.v

,

v

l

igC-OG)]

2

(n = l , 2 , . . . )

Using the last formula, we find

1 ™>F,
r

i2

EL!(2A: + l)[iE^(^)(-)]

>0

(n = l , 2 , . . . ) .

(5.1.17)

By the definition of the elements of Eq. 5.1.17, this sequence is decreasing. By the proof of a similar statement in Paragraph 5.4 it can be seen that the limit of the sequence Eq. 5.1.17 is equal to zero. Thus we have raF,v(Sn) \

0,

n —> oo .

By Theorems 3.2.3 and 3.2.4 the following two theorems hold.

(5.1.18)

5. DECOMPOSABILITY

OF DISTRIB UTION

FUNCTIONS

59

T h e o r e m 5.1.7 Let F G E a be a strictly increasing distribution function. Then in the metric generated by the scalar product Eq. 5.1.13 the best linear estimation of F by the first n distribution functions in the orthogonal system Eq. 5.1.16 is equal

to (5.1.19) 3=1

where

[igC-OQ]' i>">[it(:!0G)]' (2fc + 1)

/f> =

(k = l , . . . , n ) .

The error of the estimation is given by Eq. 5.1.17. A consequence of Eq. 5.1.17 and Eq. 5.1.18 is the following statement: T h e o r e m 5.1.8 Let F G E a be a strictly increasing distribution function. Then the sequence {iftn(x)}f> of the distribution functions Eq. 5.1.19 converges to F in the metric generated by the scalar product Eq. 5.1.13. It is interesting that the coefficients ctj , /?^ playing a role in Theorems 5.1.7-5.1.8, and the expression Eq. 5.1.17 are independent of F . It seems that they are universal constants. The statements of Theorems 5.1.7 and 5.1.8 remain valid, if F G E(a, b) is an absolutely continuous distribution function. Starting on from F G E(a, b) below we construct an orthogonal system of distribution functions generated by the linearly independent distribution functions Eq. 5.1.12 by the help of a suitable scalar product with respect to F. L e m m a 5.1.2 tions

Let F G E(a,6).

Then there exists a sequence of continuous func­

cj>k{x) = a[k)F{x) + a{k)F2{x)

+ . . . + a[k)Fk{x)

(5.1.20) (5.1.20)

with a f e R

(j = l,...,k; fc = l , 2 , . . . ) ,

*€[a,6],

which satisfy the conditions (¥>*,¥*)F = 0

of orthogonality with respect to F.

(M=l,2,...;

k^£)

(5.1.21)

Moreover, we have

E(fl k

;=i

2

<

'

S(k)'

(5.1.22)

60

CHAPTER II. DECOMPOSABILITY OF DISTRIBUTION FUNCTIONS

whe

S(k) = l 2 + 22 + ... + k2 = -k(k + l)(2i + 1) , and the coefficients

aS

can be chosen independently of

F.

Proof: Condition Eq. 5.1.22 is evidently satisfied by y>i = F. Since the elements of Eq. 5.1.12 are linearly independent on [a, b] , we can determine the elements of {(Pk+i(x)}'i> by the Schmidt orthogonalization process. Using the relation =

(Gj,Gk)F

jk

j+ k+1

we obtain that 1 4 3

1 2k k +1

2k

k2 2k-1 *(* + *) 2k

F F2

(fc=l,2,...).


fc + 1 2(fc + l) k+2

(5.L23)

FM

Since DetrV = l,...,k + ll)) D e t r F ((F , Fj ' ((jj = l,...,& + )) > 0 , the function 2 we get + ... + a{£V = 0

aW

(5.1.24)

and (af^y

+ ...+

(af+^y

= c*+1Det ( 1 ^ 1 ^ , ) ,

where Tjk+i is the matrix consisting of the first constant Cjt+i can be chosen e. g. as C

Jt+l

_

3

^

+ i)Det(rj + 1 r, + 1 )

k columns of Eq. 5.1.23. The (*=1,2,...).

T h e o r e m 5.1.9 Let F € E(a,6), and let {<£>fc}J° be the orthogonal system with respect to F supplied by Lemma 5.1.2. Then functions 0

Gk{x) = { F(x) + ipk+l(x), 1

x
=

l,2,...)

x >b

are elements of E(a, b) and form an orthogonal system with respect to the distribu­ tion function F.

5. DECOMPOSABILITY

61

OF DISTRIB UTION FUNCTIONS

Proof: Using Eq. 5.1.23 and the definition of F, we obtain that Gk(x) is continuous, G*(a) = 0, Gk(b) = 1. The latter is a consequence of Eq. 5.1.19. It still remains to show that Gk{x) is strictly increasing on the interval [a, b]. Let a < a < 0 < b. Then Gk{P) - Gk(a)

=

F{P) - F(a) + a[k+1)[F((3) - F(a)] + a<*+1) [F2(/?) - F2(a)]

+ ... + a[k+11)[Fk+1(fl)-Fk+1(a)]

+

.

From here we obtain that Gk(/3) - Gk(a) > > F(fi) - F(a) - {\a[k+1)\ [Ftf) - F(a)] + |a<*+1)| [F\P) - F\a)]

+

+ - + | C i 1 ) | [Fk+1(P)-Fk+\a)]} . Since for all positive integers n > 2 Fn{0) - Fn{a)

=

[Ftf) - F(a)] [ F ^ / ? ) + Fn-\p)F{a)

<

n[F\n[F(0)-F(a)],

+ ... + ^ ( a ) ] <

it follows that

G Gkk(ft) (/3)- - GGk(a) [F{fi- - F(a)] F(a)]{ l - [|a<*+1)| + 2 |a<*+1>| + . . . + (* + !) \a{k^\] k{a)>>[F(fi) (A; = 2 , 3 , . . . ) .

}

(5.1.25)

Using Eq. 5.1.22 we see that

|ai fctl >|+2|4* +1 >| + ... + (ib + l)|a£ f 1 1 >|< -,1/2

(«r>)V.. + («C>)s

S1/2(k +

l)
Hence, by Eq. 5.1.25, we obtain Gk(ft) > Gk(a) for a < a < (3 < b. This completes the proof of the fact that Gk 6 E(a, b) (k = 1,2,...) . Finally, we show that {Gk}?* is an orthogonal system with respect to F £ E(a,6). In fact, let k > 1, £ > 1, k±L Then (Gk,Gt)F

=

(F +
=

(F,F)F

partly because (F,F)F Further (Gk,Gk)F

+ (F,
— 0, partly as a consequence of Eq. 5.1.21, and Eq. 5.1.24.

=

(F,F)F + 2(F,(pk+!)F + (
=

(ifc+l)F •

(5.1.26)

62

CHAPTER

IL DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

Since Tp (FJ (j = 1 , . . . , k + 1)) is a symmetric and positive definite matrix, and since there are non-zero numbers among ar- ' (j = 1 , . . . , k -f 1), we obtain that fc+i k+i

(?*+!, ? * + I ) F = ^ ^ ( F i , ^ > a , ( ' : + 1 ) a f + 1 ) > 0 .

(5.1.27)

t'=l j = l

So by Eq. 5.1.26, (G*, Gk)F > 0 . For the sake of brevity let c* = (Gk, Gk)F • L e m m a 5.1.3 Let F G E(a,6), and let {Gk}f* be the orthogonal system of distribution functions defined by Theorem 5.1.9. Then

3g

fcrc'

(n = l , 2 , . . . ) .

Proof: Starting from the identity Eq. 5.1.27 we obtain that fc+i fc+i Ck =

(V*+1)V*+I)F =

l

v - y _ VJ

a (*+D a (*+l)

(k+i)(k+i)

t'=l j = l

Since

"'

<

^

W = i,...,i + D,

i + j - 1 ~ 2k + 1 by the Cauchy identity we have

(ghrl) -

2k+ i

(«r>)2 •

4^\ 3=1

Using Eq. 5.1.22 we get

C*- 1 ' 2 .-).

«*WTi i. e., the statement of Lemma 5.1.3 holds.

T h e o r e m 5.1.10 Let F G E(a,b), and let {Gk}™ be the orthogonal system of distribution functions defined by 5.1.9. Then n

Hn = 2_^PjGj , where

1

Pi

1

Pi = r - « — T

£^ ,=1

Ct

/• , x 0 = l,...,n)

5. DECOMPOSABILITY

OF DISTRIBUTION

with = —n

mFiG(Sn)

1

;=1

1

> 00

J

is the best linear approximation of F by the distribution functions \

mFfG(Sn) i. e. the distribution function F distribution functions {Hn}™ .

5.2

0,

63

FUNCTIONS

{Gk}\-

Further

n -> oo ,

is asymptotically decomposable by the sequence of

Decomposability problems on the set E(a, b) of distri­ bution functions

First of all, we recall the definition of the set E(a, 6). We denote by E(a, b) the set of distribution functions, that are continuous on the whole real line, strictly increasing in the closed interval [a, b] , have value zero at a, and one at b. a = —oo and b = oo are also permitted. T h e o r e m 5.2.1

E(a,6) is a totally convex set.

Proof: Since conditions (I)-(IV) are trivially satisfied by E(a,6), we obtain that E(a, b) is a convex set. In order to show that E(a,6) is a totally convex set it is sufficient to prove that if

G.-eEKfc) (i = o,±i,...),

(«,-)es,

then oo

G(x) = Y,

}Gj(x) G E («> b ).

a

(5.2.28)

ji=—oo

i. e. also condition (V) is satisfied. Since the sum of a series with positive terms is invariant under a rearrangement of the terms, condition (IV)* is satisfied if Eq. 5.2.28 holds. GeE.

Turning to the proof of Eq. 5.2.28, by the Theorem A.2 of Appendix A we have It is obvious that G(a) = 0, G(b) = 1. If a < a < fi < b, then oo

(?(/?) - G(a) = J2 <*> \Gm ~ Gj(ct)] > 0 , j——oo

i. e. the distribution function Eq. 5.2.28 is strictly increasing on the interval [a, b].

64

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

Finally we have to show that the distribution function Eq. 5.2.28 is continuous. Since (aj) E S , for a given e > 0 there exists a positive integer N(e) such that -(n+l)

oo

e

Yl <*j + J2 a* <2 | *

j=-oo

Since the functions Gj(x) 8(e) > 0 such that

i=n+l

^ ^oo •

are continuous on the real line, for e > 0 there exists a

(j = 0 , ± l , . . . , ± n )

\Gj(x)-GJ(x0)\<^ if \x — Xo\ < 6(e).

n

Consequently, if \x — x0\ < 6(e), then

\G(x) - G(x0)\ < —(n+l)

oo

< Y, aj\Gj(x)-Gj{x0)\+ j=-oo

n

Yai\Gi(")-GiM\+Yai\G^x)-Gi^\ J=n+1

j=—n

< £ ,

i. e., the distribution function Eq. 5.2.28 is continuous on the real line. In order to introduce a metric on the set E(a, b) we proceed as follows: F, G € E(a, 6), and let N > 2 be a positive integer. Let the numbers XN,0 < xN,l

be the Nth

Let

< • • • < £7V,N

quantiles of the distribution function F, i.e. let the conditions (* = 0,1,...,7V)

F(*N*) = Jj be satisfied. Let TV

SF,G(K)

= Nj2[G(xN,k) - G(SJV,*-I)]2 , k=i

and let 1 + ( G , G ) F = Hm

SFtG{N).

N-+oo

By the help of a Theorem of F. Riesz ([32], p.68 Lemma 1) it can be shown that 1 + (G, G)F < oo holds if and only if (1) 1 + G(F~x(x)) is absolutely continuous with respect to Lebesgue measure, and (2) £G(F-

1

(X))€L

2

(0,1).

In this case,

1 + (G, G)F = J1 [^ G (F-x(x))J dx .

FUNCTIONS

65

E F ( a , b) = {G G E(a, b) | (G, G ) F < 00} .

(5.2.29)

5. DECOMPOSABILITY

OF DISTRIB UTION

Consider the set

If FGE(a,6),

GiGEHa^)

(i = 1,2) ,

then define (GI,G2)F

=

=

/[i^-'M'i^-'H

dx — \ —

/ i fGi (F"1(:r)) ~ x l i fGs ( F_1 ( x )) - *] rfx -(5-2-30)

Since the set E f ( a , &) can be written in the form

EF(a, b) = { G G E(a, i) I ^ G (F" 1 ^)) € L2(0,1) J , it is easy to see that the functional Eq. 5.2.30 satisfies the properties (a), (b) and (d) of the scalar product. But it also has property (c). Namely (G,G)F > 0 by Eq. 5.2.30, and (G, G)F = 0 if G = F. Suppose now that (G, G)F = 0. Since

\G(F~Hx))-x\ =

\j*±[G{F-\t))-t)dt M

<

<

{jf(s[°cc»-*])'*}""-

=

(CC)F/2,

we obtain that G(F~1(x)) = x for x G [0,1], i. e., G(x) = F(x) The following Theorem is a consequence of Theorem 5.2.1.

for x G R.

T h e o r e m 5.2.2 Ej?(a,6) is a convex set and it is a metric space with respect to the scalar product defined by Eq. 5.2.30. If F€E(a,6),

GJ-GEFM)

(j = 0 , ± 1 , ± 2 . . . ) ,

(a,-) G S ,

then by Theorem 5.2.1

G(x)= £

^(i)e%i),

(5.2.31)

66

CHAPTER II. DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

and on the basis of Theorem A.2 of Appendix A, I

OO

TxG{F-\x))

i

= £ ai-Gi i=-oo

(F~\x)) .

(5.2.32) (5.2.32)

But from here it does not follow that ^G(F->(x))€L2(0,l). However, if the density function Eq. 5.2.32 is square integrable on the interval [0,1] for all sequences G,-€EF(a,6) (j = 0 , ± l , ± 2 . . . ) and arbitrary (a,) 6 S, then Ejr(a, b) is a totally convex metric space with respect to the scalar product Eq. 5.2.30. Since the set of absolutely continuous, strictly increasing distribution functions on [a, 6] is a subset of E(a, 6), the existence of a totally convex metric space is also assured here by Theorem 5.1.2. Next on the basis of Paragraph 3.3 we deal with the special decomposability problem where F E E(a, 6), G(z, x) e E F ( a , 6),

x6R,

(5.2.33)

and the set of weight functions is the set E 2 of distribution functions in E(0,1) that are absolutely continuous with respect to Lebesgue measure and have square integrable density functions. ! The two questions to be examined are the following: A/ What is the necessary and sufficient condition for the solvability of the integral equation / G(z,x)h(x)dx = F(z) Jo over E 2 if F e E ( a , 6 ) , and the family G(z,x).£ functions are given?

(5.2.34) Ep(a,6), of distribution

B / In the case of solvability of Eq. 5.2.34 what is the density function h ? By the definition of a family of the distribution functions Eq. 5.2.33 with pa­ rameter x G R the distribution functions G(F'1{z),x)

xeR

are absolutely continuous functions of z with respect to Lebesgue measure, and ^-G(F-1(z),x)eL2(0yl),

x£R.

67

5. DECOMPOSABILITY OF DISTRIB UTION FUNCTIONS Thus the discrepancy function of the present problem is given by $FG(h)=

/*<E E 2 ,

[ [ K(x,y)h(x)h(y)dxdy>l, Jo Jo

(5.2.35)

where we suppose that the symmetric positive definite kernel function K(x,y)

=

(G(z,x),G(z,y))F

+l =

=

J \TzG ( F - 1 ( 2 ) ' X )

G

TZ ( F _ 1 ( Z ) ' »)]

dz

5

*' V e [0.(42.36)

satisfies the Hilbert-Schmidt condition dx dy < oo . JO

(5.2.37)

JO

Functional Eq. 5.2.35 is convex on the convex set E 2 . But it is also convex on the set Ef of elements belonging to L 2 (0,1) and having integral one on the interval

[0,1]. According Chapter I inf QF,G(h) = m F , G (E 2 ) ,

/i€E 2

where

inf $F,G(h) = m F , G (E 2 ) ,

/i€Ej

m F , G (E 2 ) > m F>G (Ej) > 1 .

As it is usual, we apply the notation (h, k)=

f h(x)k(x) dxJo

L 2 (0,1) .

h,ke

Let the eigenvalues in increasing order of the symmetric positive definite kernel function Eq. 5.2.36 of Hilbert-Schmidt type be denoted by {A*}^ and let the sequence of the corresponding orthonormal eigenfunctions be denoted by

{?*(*)}?,

*€[0,1].

Here and in the following u means a positive integer or infinity according as the kernel function Eq. 5.2.36 is degenerate or non-degenerate. The following statement can be proved similarly to the proof of Theorem 3.3.1. T h e o r e m 5.2.3

The only solution of the equation $F,G(h) = ™ F , G ( E 2 )

with ™F,G(V1) =

V

c

*

m

> 1

68

CHAPTER II DECOMP OSABILITY OF DISTRIBUTION FUNCTIONS

is given by =

^W

^"

1 n

w

x2x J2(h
I6[0,l],

which is an element of the set Ef . From here the following statement can be obtained: Corollary 5.2.1

The inclusion

h*(x) £ E 2 holds if and only if

m F , G (E 2 ) =

1

E:=I(I,^) 2 A* •

The principal result of this paragraph is the following: T h e o r e m 5.2.4 Let the eigenvalues in increasing order of the symmetric positive definite kernel function Eq. 5.2.36 of Hilhert-Schmidt type be the elements of the se­ quence {\k}i , and let the sequence of the corresponding, orthonormal eigenfunctions be

*e[o,i].

{M*)K,

Then F G E(a,6) is decomposable by the family Eq. 5.2.33 of distribution functions over E 2 if and only if

In this case the representation F(x)=

I Jo

G(z,x)h(x)dx

holds, where h(x) = £ ( 1 ,
x G [0,1] .

(5.2.38)

We also have representation

( ) = Z ^ 1 ' ^*) A * /

F Z

1.-1 k=i

where the convergence is uniform for

G Z

( > X)

Jo

zGR.

Z e R

(5.2.39)

5. DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

69

Proof: Except for the statement Eq. 5.2.39 the assertions of the Theorem can be proved similarly to the proof of Theorem 3.3.2, and Corollary 5.2.1. To verify Eq. 5.2.39 let G(z)= where h(x)

f G(z,x)h(x)dx, Jo

.

is the weight function defined by Eq. 5.2.38. Thus

GW-E(1.^* / w

I f1 =

G(z,x)tpk(x)dx

/ G(z,x) \Jo

^T (l,
<^G2(z,x)dXy (jr(i,
f

< J2 (^^A; \k=n+l

The right hand side of this inequality converges to zero as n —► oo, independently of the points of z G [a, 6], because

J2(l^k)2Xl=

[\2(x)dx.

This is exactly what we wanted to prove. Finally we show that F G E(a, b) is not decomposable by the family G(z,x) = F1+x(z),

0
of distribution functions. Namely in this case G(F-\z),x)

= z1+x,

*e[0,l],

0<x
thus we get the following representation of the kernel function:

K{*,y)

=

j^G{F-\z),x)±G{F-\z),y)} dz = jf [lG
-|(rfi)'(i^)''

°<-'^

70

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

Now if h e E 2 , then $F,o(h)

=

K(x,y)h(x)h(y)dxdy

=

Jo Jo

= i + E f / Y r ^ ) K*)**) >!»

o<x
which gives us the statement. It is interesting to note that the kernel function

fctiKLtil,

„<,„<,

X + 2/ + 1 is a totally positive function. is a totally positive function.

5.3

Decomposability problems on the set of continuous dis­ tribution functions

Since degenerate distribution functions, and more generally, discontinuous distribu­ tion functions do not have Stieltjes integral with respect to themselves, the set E c of continuous distribution functions is the widest subset of E, the elements of which have Stieltjes integrals with respect to each other. It is obvious that E a C E c . It is well known ([32], 54., p. 110), that if F , G G E C , then f"

F(t) dG(t) + r

«/ — oo

G(t) dF(t) = F(x)G(x),

x e R .

«/—oo

It is easily to see that the set E c is totally convex. Namely the convexity conditions (I)-(IV) are satisfied trivially, condition (IV)* and (V) are also satisfied by the reasons which have been used in connection with the set E(a, b). In the following we introduce a metric which can only be defined on the totally convex set E c . T h e o r e m 5.3.1

Let F, G, H e E c . The formula

(G, H)F = I

[G(x) - F(x)] [H(x) - F(x)] dF(x) e R

(5.3.40)

J—oo

defines a scalar product on E c , and E c is a totally convex metric space with respect to the metric generated by this scalar product. Proof: First we show that the functional Eq. 5.3.40 is a scalar product defined on the totally convex space E c . Condition (a) is satisfied trivially.

5. DECOMPOSABILITY

OF DISTRIBUTION

71

FUNCTIONS

Condition (b) is also satisfied. Namely, if Gj e Ec ,

aj > 0

(j = 1,2) ,

a! + a 2 = 1 ,

(5.3.41)

then ( a 1 G 1 + a 2 G 2 , H)F = oo

/

[a x (
oo

/

[G x (x) - F(x)\ [H(x) - F(x)] dF(x) + ■oo

+ a 2 f0

[G2(x) - F(x)] [H(x) - F{x)} dF(x) =

J —OO

= <*!&!, H)F +

a2(G,,H)F.

The validity of condition (c) will be traced back to the following Lemma: L e m m a 5.3.1

Under the conditions Eq. 5.3.41 we have the relation oo

/

[G2(x) - G,(x))2 d(aiGi{x)

+ a2G2(x))

=

•oo

-£iftW-o,(.)f/**<■> ; * ' • » . This Lemma is a consequence of the following statement. Lemma 5.3.2 Let Gj £ E c (j = 1,2), a-|-6 = 0. Then oo

/

[G2(x) - dix)}2

and let a and b be real numbers with

d(aGi(x) + bG2{x)) = 0 .

■oo

Namely, for a = b = 0 the statement holds. If a ^ 0, then it can be supposed without loss of generality that a = 1. In this case OO

/

1

i[(G2(x)-G1(x))3]roo=0 [G (x) G^x)]2 22 d(G d(G222(x) (x) -- Gi(x)) Gx(x)) = \G G^x)} Gi(x)) [(G22(x) [G222{x) (x) -- G^z)] d(G (x) = -; [(G (z) - C O O O )) 33 ] ^ = 0

*■ in conformity with the statement. Returning to the proof of the Lemma 5.3.1, let a = | — a i , 6 = | — <*2 ; then a -h 6 = 0, i. e. Lemma 5.3.2 may be applied. Thus we get the statement of Lemma 5.3.1. If we show that

•OO

(G, G)F = f J — OO

[G(x) - F(x))2 d{F{x)

+ ^

G{X))

>0

(5.3.42)

72

CHAPTER II. DECOMPVSABILITY

OF DISTRIBUTION

FUNCTIONS

for F , ( ? 6 E C with equality if and only if G = F , then it will be proved that the functional Eq. 5.3.40 satisfies condition (c). The relation (F, F)F = 0 is obvious. We now prove, that if (G,G)F = 0 then G = F. By Lemma 5.3.1 and the Schwarz inequality

(J'[G(t)-F(t)]dF(t)j = (J^G(t)dF(t)-l-F2(t?) < <

['[G(t)-F(t)]'dF(t)< J —oo oo

/

[G(t)-F(t)]2dF(t)

=

(G,G)F.

■oo

If ( G , G ) F = 0, then G(t)dF(t)

= ]-F2(x),

xeR

J —c

and similarly, the relation

(j*JG(t)-F(t)]dG(t)j<(G;G)f implies that

f

F(t)dG(t)

= ^G2(x),

xeR.

Consequently

f

[G{t)dF{t) + F(t)dG{t)]

=

F{x)G(x)

=

J -c

\[F\x)

+ G*{x)] ,

therefore (F(x) - G{x)f = 0, i. e. G(x) = F(x), xeR. The distance concept given by Eq. 5.3.42 is a special case of the distance con­ cept, used in the convergence theory of functions square integrable with respect to a weight function. Moreover, this distance concept was introduced in the mathematical statistics by E. L. Lehman in 1950 in connection with nonparametric problems ([22], 3.24 to 3.26; [10], pp. 164-167; [31], § 2). A consequence of Lemma 5.3.1 is that the metric generated by the scalar product Eq. 5.3.40 is symmetric. The functional Eq. 5.3.40 satisfies also condition d). This is clear from the fact that IV (Gj (j = 1 , . . . , n ) ) is the Gram matrix of the distribution functions Gj E E c (j = 1 , . . . , n) with respect to the weight function F(x) .

5. DECOMPOSABILITY

73

OF DISTRIB UTION FUNCTIONS

It remains to show that E c is a totally convex metric space with respect to the scalar product defined by Eq. 5.3.40. For this purpose by the Theorem 1.2.5 it is sufficient to show that the functional Eq. 5.3.40 is uniformly bounded on E c . Let F,Ge E c . Then

°°

1

/

F2(x)dF(x)=-,

which gives us the statement. Here the upper bound cannot be improved. Really let F e E c ; then Fy G E c for y > 0. In this case CO

(F»,F»)F

jF»[F»(ar) - F(x)]2 ix)-Fix)fdF{x)

/

dF(x) ==

CO

1 - - { y + 2 )3yy 2 y + i y 3 (y + 2)(2y + 1) '

and therefore

(/",*>-|,

»->«>.

On the basis of property c) of the scalar product Eq. 5.3.40 we get a two sided inequality for continuous distribution functions. T h e o r e m 5.3.2

/ / F,G G E c then 2 3

oo

/

<

F2(x)^(x) +

OO /»oo

/

F2(x) dG(x) + / ■oo

G 2 (x)^F(x)< 1

■OO

G2(x) dF(x) < 1

J — oo

with equality on the left hand side if and only if F = G. Proof: 1/ Proof of the left hand inequality. Integration by part yields oo

/ Thus

F(x)G(x) dF(x) = 1 -

■CO

F2(x) dG(x) .

/

•oo

oo oo

/

y»oo

J—oo i

2

[F(x) - G{x)} dF{x)

/»oo

/«oo

± + J^G2(x)dF(x)- /

[F(z) - G(x)]2 dF(x) =

■CO

^ + / CO ^

/ /

J — CO

2

J—OO

G (x) dF(x) TOO - /

J—OO

CO

TOO

2

OO

F2 (x) dG(x) + /

F(z)G(z) dF(x) = F(x)G(x)

dF{x) = O C\

£*(*) dF(a?) - - > 0 .

F (x) dG(x) + J-OO / G2(x) dF(x) - -^ > 0 .

oo

^

J-oo

The case of equality can be treated with the aid of property (c) of the scalar product Eq. 5.3.40. 2/ Proof of the right hand inequality. This inequality follows from the relation OO

i»00 /»oo

oo

/

22

F (x) (x)dG(x)+ dG(x) + / •OO •oo

JJ—oo — OO

OO /*oo

22

G (x) (a;)dF(:r) G dF(x) <

/

[F(x) ^G(x) + G(x) dF(x)] =

■OO

/ [F(x) dG(x) + G(x) dF(x)] = [Fwcwir^i. = [F(x)G(x)\^ = l. J—oo

0

=

74

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

Equality is impossible here, since it could occur only if F2(x) = F(x) and G2(x) = G(x), x 6 R, i. e., if F(x) and G(x) were discrete distribution functions with a single jump. But such distribution functions cannot belong to the set E c . The right side inequality cannot be improved. In fact if n and m are positive integers, and

1

1

F(x) = I x

if

G(x) = { x

0
0

0 then

X> 1

m

n

x <0

£
+

^ - -. i

as m - 4 o o . 5.3.a In this paragraph we deal with the decomposability problem in the narrow sense for the case where the distribution function to be decomposed and the elements of the family of the distribution functions belong to E c , and the metric on E c is generated by the scalar product Eq. 5.3.40. Accordingly let C C Ej be the set of the discrete distribution functions, which have jumps only at the prescribed points Xi < ... < xn ,

(5.3.43)

n > 2.

Let F G E a , and let the family

G(z,x)eEc

xeR

of distribution functions be given. Let the distribution functions G(z,xk)

= Gk(z)

(5.3.44)

(t = l , . . . , n )

be linearly independent. This means — using the scalar product defined by Eq. 5.3.40 — that the Gram matrix

r = rF(Gj(j with

= i,...,n))

(5.3.45)

= (bjk)ik=1

oo

°jk

/

Gj(*)G*(*)dF(s)

(j,k =

l,...,n)

•oo

is positive definite. We shall use the distance oo

/»oo

/ [F(x) - G(x)}2 dF(x) = /J—oo [F(x) - G(x)}2 dF(x) = / oo

/»oo

c\

F2(x) dG(x) + / G2(x) dF(x) - - > 0 (5.3.46) oo 2 J—oo v F (x) 0 (5.3.46)

J—oo

J—oo

generated by the scalar product Eq. 5.3.40, where only if G = F.

F,GGEC

d

with equality if and

5. DECOMPOSABILITY

OF DISTRIB UTION FUNCTIONS

75

Thus the discrepancy function of the present decomposability problem is equal to the inhomogeneous quadratic function 9Fja(q) = q'Tq + 2q'a - | > 0 ,

(5.3.47)

where

q = fe) e Q n , with

i r°°

aj

= -j

a = (aj) e Rn (j = l , . . . , n ) .

F\x)dGi{x)

Since the set S n is closed in the Euclidean metric, and the convex functional Eq. 5.3.47 defined on this set is continuous, there is a vector p = j/°) 6 S n such that m F , G (S n ) = inf $FtG(p) = $F,G(P{0)) • p€S n

This quantity is the measure of decomposability of the distribution function the distribution functions Eq. 5.3.44 over the set S n . According to Theorem 2.2.2, the functional Eq. 5.3.47 is convex on the and Q n . Since this functional is continuously differentiable in each variable, use the Lagrange multiplier method to calculate the minimum Eq. 5.3.48 as the point of minimum over S n , and the minimum mFiG(Qn)

(5-3.48) F by set S n we can well as

inf $F,G{ ° »

=

96Q n

as well as the point of minimum over Q n under the auxiliary conditions n

]Tp f c = i , and

p=

{Pk)esn,

n

£ # = i,

? = te)eQn,

respectively. These auxiliary conditions are independent of the sets S n and Q n . Therefore, by the Lagrange multiplier procedure we actually get the minimum and the point of minimum over the set Q n only. In the following we deal with this calculation. So let n
fc

,

q = (qk)

e Qn .

3=1

The functional Eq. 5.3.47 has a point of absolute minimum on the convex set Q n , where T? + a - Xe = 0 .

76

CHAPTER

II. DECOMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

Here, as earlier, e 6 R n is the vector with components one. From here g(°) = r - 1 ( A e - a ) G Q n and so, by e*q^ = 1,

*-MS?.

(5.3.49)

<—>

Substituting the values Eq. 5.3.49 and Eq. 5.3.50 into the functional Eq. 5.3.47 we obtain that mF,a(Qn)

=

*F, G (<7 ( 0 ) ) =

=

(1+ fr a ) 2 -aT-*a-?>()■ CT^e 3 ~

The following decomposability Theorem follows from Theorems 3.1.1 and 3.1.2. T h e o r e m 5.3.3 Let F G E c , and let T be the Gram matrix Eq. 5.3.45 of the linearly independent distribution functions Eq. 5.3.44- Then the measure of decom­ posability of the distribution function F by the distribution functions Eq. 5.3.44 over the set Q n is given by (l + e T - i q ) 2 —— e*Y~le Equation

2

a L

a— — > u. 3 ~

• W ) = (1+ et e r- 1 ? )2 - aT"lfl - 1

<5-3-51)

is satisfied by the step function H0 that has jumps only at the points X\ < . . . < xn , the jumps are given by the components of the vector Eq. 5.3.49, and A is given by Eq. 5.3.50. The relation q^ = p^ £ S n is satisfied if and only if mFiG(Sn)

= m F ) G (Q n ) •

In this case, equation Eq. 5.3.51 is satisfied by the distribution function H = H0 € Ej , that has discontinuities only at the points x\ < ... < xn , with jumps defined by the components of p(°)=r-1(Ae-a)GSn> where A is given by Eq. 5.3.50. The principal result of this paragraph is contained in the following decompos­ ability Theorem. T h e o r e m 5.3.4 F G E C can be represented by the linearly independent functions Eq. 5.3.44 over tne set S n if and only if (l + e T - ' a ) 2 -— =al

x

2 a+ -,

distribution

5. DECOMPOSABILITY

77

OF DISTRIB UTION FUNCTIONS

where T is the Gram matrix Eq. 5.3.45 of the distribution functions Eq. 5.3.44Then the equation $F,G(P^) = 0 i 5 satisfied by the distribution function H0 G Ej , that has discontinuities only at the points x\ < ... < xn with jumps defined by the components of the vector p<°) = r - 1 ( A e - a ) 6 S B , where A is given by Eq. 5.3.50. 5.3.b In this paragraph it will be supposed that the following are given: F G E c ; a family G(z, x) G E c , x G R of distribution functions; a strictly increasing sequence {^fc}i° of real numbers; and a sequence of vectors a<»> = (af)

e Sn

(n = l , 2 , . . . ) .

(5-3.52)

Moreover, we suppose that the distribution functions of the sequence G(z,xk)

= Gk(z)

(4 = 1,2,...)

(5.3.53)

are linearly independent. The purpose is to give a necessary and sufficient condition in order that the sequence

Ln(z) = J2<*i;)GJ(z)

(n = l,2,...)

i=i

of distribution functions converges in metric to the distribution function F. other words, what is the necessary and sufficient condition in order that

Or, in

oo

/

[Ln(x) - F{x)]2 dF(x) -► 0 ,

n -► oo

(5.3.54)

•OO

be satisfied? We do not deal with this problem in full generality, but in the following two special cases only: First we suppose that a^ = p^ G S n is the vector, which satisfies the condition mFfG(»)(Sn) = inf $F,G(n)(p) = ^F,G(n)(p(n)) > 0 , (5.3.55) peSn

where G(n) denotes the n first elements of the sequence G = {Gj}^° . obvious that the sequence (Ln , Ln)F = mFtG(n)(Sn)

It is

(n = 1,2,...)

is non-increasing. Thus the limit ^F,G(n)(S n ) \

rnFiG > 0 ,

n -+ oo

exists. This number will be called the measure of decomposability of the distribution function F G E c by the distribution functions Eq. 5.3.53. To sum up, we have the following statement:

78

CHAPTER

II. DECOMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

T h e o r e m 5.3.5 Let F G E c , and let the sequence Eq. 5.3.53 of linearly indepen­ dent distribution functions be given. Let p<"> = (p< n) ) g S n be the solution of the equation Eq, 5.3.55. In this case the sequence n

Ln(x) = J2p(f)Gj(x) Ln(x) = J2p(f)Gj(x)

(n = l,2,...) (n = l,2,...)

converges in the metric Eq. 5.3.54 t° ^ne distribution function ne ™>F,G = 0 . in the metric Eq. 5.3.54 t° ^ converges distribution function

F F

if and only if if and only if

™>F,G = 0 .

In the second special case we suppose that the distribution functions Eq. 5.3.53 together with the distribution function F G E c form a linearly independent system. Moreover, let n

Ln+1(x)

= a^F(x)

+ X ) ^Gjix)

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

i=i

where

„<•+!) = ( ^

6

gn+i .

T h e o r e m 5.3.6 / / a £ ° -> 1 as n —► oo, then the sequence {Ln+^x)}™ converges to the distribution function

,

x GR

F G E c m £Ae metric defined by Eq. 5.3.54-

Proof: Applying Theorem 1.2.3 to the metric Eq. 5.3.54, we obtain n

< a< n ) (F,F) F + £<*<"> (Gj ,GS)F . < a< n) (F, F)F + £ <*<"> (Gj , GS)F .

0 < (Ln+1,Ln+l)F 0 < (Ln+1,Ln+l)F

Since ( J P,F) F = 0, and 0 < (Gj ,Gj)F < 1/3 (j = l , 2 , . . . ) , we get that Since (F, F)F = 0, and 0 < (Gj ,Gj)F < 1/3 (j = 1,2,...) , we get that 0 < (Ln+l,

Ln+1)F


«i n ) = | ( 1 - «o n) )

and, with the aid of the assumption, this gives us the statement of our theorem. 5.3.C In this paragraph we deal with two methods, different from the general methods of Chapter I. The tools for both methods form are described in Appendix J. Let n be a positive integer. Definition 5.3.1 Let F G E and FjeE

t/ = l , . . . , n ) .

(5.3.56)

5. DECOMPOSABILITY

OF DISTRIB UTION

If the identity 1

79

FUNCTIONS

n n

(5.3.57)

£ajFj(x) = F{x)

= 1 Ji=i

holds with an a = (aj) G S n , £/fcen, as earlier, Eq. 5.3.57 is said to be a decompo­ sition of F by the distribution functions Eq. 5.3.56. Definition 5.3.2

// Fj

F

=

(j = l , . . . , n ) ,

then Eq. 5.3.57 is said to be a trivial decomposition of Definition 5.3.3

F.

Let F G E and GjZE

(j = l , . . . , n ) .

(5.3.58)

If the identity n

F(x) = Y[Gj(x)

(5.3.59)

i=i

holds, then Eq. 5.3.59 is said to be a product representation of F by the distribution functions Eq. 5.3.58. Definition 5.3.4

/ / F G E and in Eq. 5.3.58 Gj(x) = Fa>(x)

(j = l , . . . , n )

with an a = (aj) G S n , then Eq. 5.3.59 is said to be a trivial product of F.

(5.3.60) representation

There are at least three decomposition problems for distribution functions. (a) The product decomposition of characteristic functions (Fourier transform of distribution functions) by characteristic functions. This problem has been in­ vestigated by several famous mathematicians; there is a rich literature on this topic ([25]). (b) The decomposition of distribution functions in the sense of Definition 5.3.1. As we have seen, such questions have been raised since long ago in connection with applications. The problem oriented procedures toward the solution of these problems have "ad hoc" features. The first two monographs on this subject — as mentioned in the Introduction — are due to Medgyessy ([28], [29]). To the best of my knowledge, present book is the first attempt to treat these questions in a systematic way. (c) The product representation of distribution functions in the sense of Definition 5.3.3. It seems that such investigations do not occur in the literature. On the other hand, by the results to be described later, this problem has importance in connection with the decomposability problem (b).

80

CHAPTER

II. DECOMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

An immediate consequence of Theorem J.l of Appendix J is the following state­ ment. T h e o r e m 5.3.7 Suppose that F G E c has the product representation Eq. 5.3.59, where Gj G E c (j' = 1 , . . . , n) . Then the decomposition n

Y^ a>jGj{x) = F(x),

(5.3.61)

xeR

3=1

holds, where [X

Gj(x) = a

Gl(t)...Gj.1(t)Gj+1(t)...Gn(t)dGj(t)

3 J-oo

with Go = Gn+i = I, and «i = f°

Gi(t)...

Gj-tftGi+iit)...

Gn{t) dGj(t) > 0

(j = 1 , . . . , n) ,

J — CO

Theorem 5.3.8 If F G E c has the trivial product representation Eq. 5.3.60, then the decomposition Eq. 5.3.61 that corresponds to Eq. 5.3.60 is also trivial. Proof: By Corollary J.2 of Appendix J in this case f

Gi(t)...Gi-MGi+tf)...

Gn(t)dGj(t)

F1-"'^)

= f

J— CO

dFa>(t) =

aiF{x)

J—CO

(j = l , . . . , n ) . T h e o r e m 5.3.9

Let n > 2 be an integer, and let a = (ctj) G S n be given. Let Gi(x) = exp ja,- J'

j & l J ,

x e

R

( 5 . 3 .62)

(j = l , . . . , n ) , where FGE T/ien

C

,

F,GEC

n

5 3 «i^i W = ^0*0 , /io/ds i/ and on/y i/

(j = l , . . . , n ) . x

eR

(5.3.63)

x GR .

(5.3.64)

n

J ] G ^ s ) = F(x) ,

5. DECOMPOSABILITY

81

OF DISTRIBUTION FUNCTIONS

Proof: First we suppose that the decomposition Eq. 5.3.63 is valid. Since

n<«.>-*{/:^}. by Theorem J.3 of Appendix J, we obtain that

f[Gj(x) = F(x), i=i

i. e. also Eq. 5.3.64 holds. Now we suppose that Eq. 5.3.64 is valid. Then by Theorem 5.3.7

E r n G*o d°w=F^ • 7=1 ^ - ° °

<5-3-65)

k=l k*j

Moreover, by Theorem J.2 of Appendix J,

J'^f[Gk(t)dGi(t) = ^M.dGi(t)

=

k*J

=

.

ajFj(x)

(5.3.66)

Comparing Eq. 5.3.65 and Eq. 5.3.66 we obtain Eq. 5.3.63. If conditions of Theorem 5.3.9 are satisfied, it is obvious that the functions Gj(x)

0 = l,...,n)

are non-negative, non-decreasing and continuous. But they are not distribution func­ tions in general. For example, let ai=a2 = i,

F&EC,

Fx = F2 ,

F2 = 2F - F2 .

Then FjeEc

(j = 1,2),

1

-(F1

+ F2) = F.

In this case

*■<*> = A\LdJm }-«* {/>"}=

expF(:r) > 1,

x GR ,

i. e. G\ is not a distribution function. This example is due to Gy. Pap. Consequently the representation n

WG^^Fix) i=i

in Theorem 5.3.9 is not a product representation of F in general. The following theorem is a converse of Theorem 5.3.8.

82

CHAPTER

II. DECOMP OSABILITY

OF DISTRIBUTION

FUNCTIONS

T h e o r e m 5.3.10 Let a = (a,) G S n . / / n

]TajFj(x)

= F(x) ,

i e R

i=i

is a trivial decomposition of F £ E c , i. e. Fj-Oc) = F(x)

and

(i = l , . . . , n ) ,

<«•>—K.»

then G,-(*) = *"">(*)

(i = l , . . . , n ) ,

x€R,

i. e. these functions are the components of a trivial product representation of

F.

Proof: Indeed from Theorem J.3 of Appendix J we get

°«->-HJCW-<->In the following we deal with the statistical estimate of the distance of two continuous distribution functions. Let Xi , X2 , . . . , Xm be a sample from F G E c , and >i , Y2 , . . . , Yn an independent sample from G € E c . We wish to find the uniformly minimum variance unbiased estimate ([33], 13.2) of

A(F,G) = JjG(x) - F(x)rdF{x) + G{x) . Following Lehmann's ideas ([22]), let P(F, min(Y1 ,YY22)}{] P(F, G) G) = P {[max(Xx, X * 2,)) < < min(y,, )\ ( J [max( [max(Y, Y,,,YF2)2) << min(X min(X11,, XX2)}} = 2 )]} = =

P {max(A: 1 , X2) < rma(Yx, Y2)} + P {max(r 1 , Y2) < min(X 1 , X2)}

Since P {max(X 1 , X2) < x) P {mmfr,Y2)>y}

,

=

F\x)

=

(1 - G(y))2 ,

therefore /*oo

oo

/

(1 - G(x)) 2 <*F2(x) + / •OO

J — OO

(1 - F(x)f

dG\x)

=

.

5. DECOMPOSABILITY

OF DISTRIBUTION

83

FUNCTIONS

/»oo

oo

/

2G(x)] dF\x) dF\x) + [1 + G22(x) (x) - 2G{x)]

F2(x) - 2F(x)] [1 + F\x) 2F(x)] dG dG22{x) {x) == + G{X) 2 2 F(x) 2 = 2 + l°° | [G (z) dF (x) + F\x) dG (x)] - 4 F = "x)~ ^ '{~ * - > ■oo

oo

3-2 /

i

= 3-8

J — oo 2

F( G x)r^

{[F(x) + G ( x ) ] 2 - [ F ( a ; ) - G ( x ) ] 2 } d

}} =

F(x) + G{x)

■OO

/

F(x) + G(x)

,F(x) + G(x)

+ 2A(F,G) =

J—c

- +

2A(F,G).

Let us define 1

if max(Xi , X2) < min(F 1 , Y2) (5.3.67)

or max(Fi, Y2) < m i n ( X i , X2)

ip{X1,X2,Y1,Y2)={ 0

otherwise.

Then Eq. 5.3.67 is an unbiased estimate of P(F, G), and actually a kernel of P(F,G) ([33], pp. 532-533). Therefore ([33], p. 534) the corresponding "Lehmann statistics" U(X,Y)

=

(2X2)]^ S^'^*15*'15'

is the uniformly minimum variance unbiased estimate of P(F, G), consequently

itf(x,y)-i = v(x,y) is the uniformly minimum variance unbiased estimate of A ( F , G) . Let rank X t = rt- ,

rank Yj = Sj

in the permutation Z\ < Z2 < . . . < Zn+m

(i = 1 , . . . , m; j = 1 , . . . , n) of the sample elements

X\ , A2 , . . . , Xm , Y\ , 12 , . . . , Vn •

In this case the statistics £/(X, K) can be expressed in the form ([36]) ruy

y , _ £ £ i ( r . ~ Q f o - i ~ 1)

m(2)

E L l ( ^ - * ) ( * * - * ~ 1)

n(2j

,

84

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION is denoted by D2(U),

and if the variance of U(X,Y)

FUNCTIONS

then ([36])

4 (m + n+ l)(ra + n - 2 ) D2(U) = 25 rn(m — \)n(n — 1) The Lehmann criterion is consistent with any alternative hypothesis G(x) ^ and V(X,Y)-A(F,G){F,G) D(V) ~^HMJ if n —» oo, m —► oo ([23]), where ^ ( 0 , 1 ) expectation zero and with variance one.

F(x),

denotes the normal distribution with

Decomposability problems on the set of discrete distri­ bution functions

5A

Denote by Zu

the set of the real numbers z1 < z2 < . . . < zu ,

(5.4.68)

where Zw = Zs if s is a positive integer, and Zw = Z, if u = oo. Let JZ(Zv) be the set of discrete distribution functions which have discontinuities at the points Eq. 5.4.68, and only at these points. Suppose that if F G ^{Z^) then F(Zj + 0) - F(ZJ) = fj > 0

(j = 1 , . . . , w ) ,

(5.4.69)

where £ / ; = !.

(5.4.70)

i=i

The decomposability problem can be expressed in these cases as follows: A/ Let F G E(Z W ) , and let the family G(z, x) G E(Z W ) of distribution functions with parameter x G R be given. What is the necessary and sufficient condition in order that the relation oo

F{z)

/

G(z,x)dH(x)(x) •oo

be satisfied by a distribution function C ^ 0 of distribution functions?

H(x)

which belongs to the given set

B / In the case of decomposability, what is the weight function H G C ? According to the conditions, the functions G(ZJ + 0,X)-G(ZJ,X) V^

gj(x)

(j = 1 , . . .

=

!>

X R

, x

l^9Ax) 3=1

=

^

,w) (5«4-71)

5. DECOMPOSABILITY

OF DISTRIB UTION FUNCTIONS

85

are measurable. Using this property and notation Eq. 5.4.69, question (A) can be expressed in the following equivalent form: A*/ What is the necessary and sufficient condition in order that the system of equation oo oo

/

•OO

g3(x)dH(x) g3(x)dH(x)

0 = 1 , . . . , a,) 0 = 1 , . . . , a,)

(5.4.72) (5.4.72)

•oo

be satisfied by a distribution function H E C, where / = (/j) € S w , and g(x) = (gj(x)) G S W , X E R , is a sequence of measurable functions, S w being one of the sets S s , S+ ? It is easy to see that S^ is totally convex. Indeed, conditions (I)-(IV) of convexity are satisfied evidently. Condition (IV*) and (V) are satisfied by the same reason which has been indicated in connection with the set E(a, b). In the following, a metric will be introduced on the set E(Zu,) . Theorem 5.4.1 Let the jumps of the distribution functions the points <2 < ..... . <
F,G,H

(E E(Z^) at

(5.4.73)

be given, respectively by the components of the vectors

// ==( /( /j j) )6eSs BB 1I

j
h= A = ((hi)es ^ ) 6 Su. „ .

(5.4.74) (5.4.74)

J2 {9fo~y-£> >-fj)^-M ==JT £ *g& -! = JT -^-l

(5.4.75)

Then the functional

(HH))FF

i=i

*>

i=i

ij

determines a scalar product on the totally convex set E(Z W ) . Moreover E(Z 5 ) is a totally convex metric space with respect to an arbitrary element of this set, and there is a totally convex subset of "£i{Zw) , which is a totally convex metric space with respect to a given element of this subset. Proof: Properties (a), (b) and (c) are satisfied by the functional Eq. 5.4.75 automatically. But condition (d) is satisfied, as well. In fact, if the jumps of the distribution functions Gk£E(Zu) (fc = l , . . . , n ) at the points Eq. 5.4.73 are given by the respective components of vectors

gW=(gW)"eSu

m

(k = l , . . . , n ) ,

then the matrix IV (Gk (k = 1 , . . . , n)) is equal to the Gram matrix of vectors

(* = l , . . . , n ) .

86

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

Now we return to the proof of the second assertion of the theorem. If u = s,

G,G) + l=±9f<±±g3

= -}-,

(G,G) + 1 = F F {

7=1

J

3

J m

,=1

J

m

where fm is the smallest among the components of the vector / € S 5 . Thus the assertion holds in this case. If u> = oo, then first we show that the set E(Z W ) has a pair of elements F, G satisfying (G,G)F = oo. Let 0 < p, q < 1, and let the components of the vectors / and g appearing in the formula Eq. 5.4.74 be given by fi = (1 - P)j-X

9i = (1 - iW'1

,

0 = 1,2,...).

Then

(G,o,+i =

{^f£;£,

which is divergent if q2 > p. Now let 0 < p < 1 be a fixed number, and let a > 1. Let Ga £ "Fi{Zj) be a distribution function the jumps of which at the points Eq. 5.4.73 (OJ = oo) are determined by the respective components of the vector

J'> = (<,]"») e s + , where gf

= (l-p")p{j-l)a

(i = l , 2 , . . . ) .

If F = G\ , and oc\ , a2 > 1 , then

(Gai,Ga2)F

+

l

=

V-^-^^i^-W-^ P i=i (l-p"i)(l-^) (l-p)(l-pQ(l+0(2-l) '

(5.4.76)

From here it follows that (GaiGa)F-h

1

(i-p°) (i-p)(i-pto-1)'

and after a short calculation we obtain ( r

r )

(Ga,Ga)F

p (i-p"- 1 ) 2 , P I - P " - 1 1 - p 1 - p2"-1 1 - p 1 + pa~x

r

P 1- p '

From Theorem 1.2.3 we conclude that the same bound holds for the distance with respect to F of an arbitrary finite or infinite linear combination of elements of the

5. DECOMPOSABILITY

87

OF DISTRIBUTION FUNCTIONS

set {Ga | a > 1} of distribution functions. This means that the totally convex set E a (Zu,) C ^{Zw) built from the elements of the set {Ga | a > 1} in this way, is a totally convex metric space. 5.4.a In the next paragraph we deal with the decomposability problem in the narrow sense for the distribution functions in *E(ZJ) . As before, we suppose that C C Ej is the set of discrete distribution functions, which have jumps at the points x\ < . . . < xn , and only at these points. Let the family G(z, x) € E(Z u; ) of distribution functions with parameter x £ R be given, and suppose that the jumps at the points Eq. 5.4.73 of the distribution functions F e E(Z W ),

G(z, xk) = Gk(z) e E(ZW)

(k = 1 , . . . , n)

are given by the respective components of the vectors / = (/i)GS„,

= (gf>)"

€ S„

(*=l,...,i»)

Suppose that

£-^V-<°°

(* = l,...,n),

h

;=1

and that the vectors
= l,...,n))

where

tf(n)

= TF(Gk(k

= l,...,n))

+ M =

H(n)Hm(n),

a w *w - ^ \ A/fr (o)\- 1 ' 9in) gin) -

^ ' 7 ,(0) 'JK)

be positive definite. If u> = 5 is a positive integer, the condition of the linearly independence means that n < s. The discrepancy function of the present decomposability problem is the quad­ ratic form *F,G(
=

i=i k=i

- u&tf-^


n

.

After this preparation, the results concerning the present decomposability prob­ lem are expressed word for word by Theorems 5.1.3 and 5.1.4.

88

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

5.4.b Next we deal with a special case making use of the results of Para­ graph 3.2, using the metric introduced in the present paragraph. In this special case the set CN of weight functions is the set of discrete distribution functions, which have discontinuities at the points x\ < x
(k = 1,2,...)

Gk(z) = G(z,xk)eE(ZN)

be distribution functions the jumps of which are given by the components of the vectors / = (/;) e S + , 0 = 1,2,...), gw = (gf) e s + respectively, where

^-^feV"

/, = ^ V ,

p><*+1>

(j,k =

1,2,...)

with a fixed number p, 0 < p < 1. This time our aim is to give an answer to the question: what can we say about the decomposability

F(z)

oo

/

G(z,x)dH(x)(x)

■oo

of the distribution function F over the set CAT of weight functions? In other words, what can we say about the representation

F(z) = J2ajGj(z) i=i

of F , where oy>0

(j = l , 2 , . . . ) ,

]>>j = l. j=i

To answer this question, let the scalar product of the distribution functions Gk (k = 1,2,...) with respect to the distribution function F be defined by the expression

{Gj,Gk)F

= ( - 1 ),;+*

[(l-p*')(l-p**)p ^ pj+ip*+i(l _ p )

Z ^

rr^-^-^TZj^^

{j+k+l)

_1

j+k

(i,fc = 1,2,...) in conformity with the scalar product introduced in this section. The following Theorem contains the answer to the previous question.

(5.4.77)

5. DECOMPOSABILITY

OF DISTRIBUTION

89

FUNCTIONS

T h e o r e m 5.4.2 The following statements hold: (1) The transsignation of the Gram matrix defined by the scalar product Eq. 5.4-77 is totally positive. (2) The distribution functions in the sequence {Gk}^°

are linearly

independent.

(3) The inequalities mF,G(Sn) > 0

(n = l , 2 , . . . )

hold, i. e.} the distribution function F cannot be decomposed by a finite number of elements of the sequence {Gk}™ . (4) mF,G (S n ) \

0,

n —► oo .

Proof: (1) Writing expression Eq. 5.4.77 in the form

(G3, Gk)F = (-ly+^ori - i)(i - / ) ^ 3 7

.

(5.4.78) ( 5 - 4 - 78 )

and using the notations

°i=G)

'

h =

> ~^ 0' = 1.2,...).

we obtain that ai < a2 < . . . ,

bi < b2 < . . . ,

and (j,fc = l , 2 , . . . ) .

aj + h>0

Thus by Corollary C.2 of Appendix C matrix

(—Y

(5.4.79)

is totally positive. Consequently, statement (1) holds. (2) Using the notation rF(G*(fc = l , . . . , n ) ) = r(n) again, and relaying on the scalar product Eq. 5.4.78, by the formula Eq. E.8 of Appendix E, we obtain

An

Detr(n) = ( _ l - ) n 2 +l)/6 p n(2n*+l)/6

n^

n

n (i^l! / n

D e t C

(„)

=

\ "

n+j+1) r ^ nr lfl d (p)cjn)(p)J "(1-P) \ -J) \J[ci =

90

CHAPTER II. DECOMP OSABILITY OF DISTRIBUTION

FUNCTIONS

i. e., the distribution functions Gj (j = 1,2,...) are linearly independent. (3) Using the scalar product Eq. 5.4.78 we get n

-

n

e* r'" 1 (n) e = (1 - p) £ E ^Bk——xm

aj + h

j=lk=1

XjVk ,,

where

Aj

a \ i+l\p)Cf{p)>Q, = p°>(l-pi)CLPn+j+1) '{\-f)C^

Bj

=

(j = l , . . . , n ) ,

/i(l-p')(7("+i+1)(p)C7Jn)(p)>0

and

^

=

^rri'

w = T37

0- = i , - , » ) .

Consequently, m f , G (S n ) = ( e T ' - 1 ( n ) e ) " 1 > 0

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

(4) Using the previous notations we have

2 ( n + x ) l 2 v ( l _ P P*)2 << p(„+i)/2 ]j]j^iij xiyj =~ pp("+D/2 ^ p { n + l ) / 2 '•

Thus Corollary E.l of Appendix E may be applied. Therefore 0 < ( e T - ( n ) e r < G ^ )

2

and this gives us the statement. 5.4.C Let us start from the sequence Gk(z) = G(s, xk) e E(ZN)

(k = 1,2,...)

(5.4.80)

of distribution functions as before. We have shown that the transsignation of the Gram matrix generated by the scalar product Eq. 5.4.77 is totally positive. Conse­ quently, we can treat the orthogonal sequence of distribution functions generated by the sequence Eq. 5.4.80 by the help of the results of Paragraph 3.2. So let

r(n) = ((
(Gj,Gk)F = ( - 1 ) ^ ^ ( 1

-P>)(I

0\*=1,2,...).

-p*)r^Wr

5. DECOMPOSABILITY

91

OF DISTRIBUTION FUNCTIONS

Then, as we have seen, r_1(n) =

T^

(i-^i-p*^r^),t=1

where the definition of the quantities Aj and Bk are given in Appendix E. From here it follows that ( & a4jr(n) = J — ^ A J P

(0)

" ••.

\

/ &

•..

\G(n)\

1(0) t^J

(0)

V(0)

| ,

(5.4.81)

£r

where by the formula Eq. E.8 of Appendix E

A„

= DetT(n) = (-?-) f[(l „n(2n 2 +7)/6

n

/

-p*)2DetG(n) =

n

\ *

(5.4.82)

Relying on Eq. 5.4.81 and on Appendix E we find o(n)

=

1

jn

A

J Bn \-j)3\-pnl_

~ P A

p

n

1 = pn+j+l

= J-^gAn/"Cl^)(p)P^C7("^)(P)CJ")(P)t _ ^ . + l . Using the identity

r r ^ r ^ + i + 1 ) W = TZp&Pto), we get

^ ^ = 1 '^ tnpynC(nn+1\p)Pai^\p)C\n){p)

(j = 1, • • • , n) .

Hence

Bln) = E < n ) = ^p ^1 —lpn ^ ^ t wE^^'w^V)o ^ . (5.4.83) Thus

&n'

E;=.pa'QT'(p)qn,(p)

0 = l,...,n) .

92

CHAPTER II. DECOMPOSABILITY

OF DISTRIBUTION

FUNCTIONS

Consequently, the orthogonal system of distribution functions, generated by the sequence Eq. 5.4.80 of distribution functions with respect to the scalar product Eq. 5.4.77 is given by

, .

Vn(z) =

EUPa'CtV\p)CJn)(p)G}(z) T,j=iPa,Cn-i

[P)C] '(p)

e E(Zn)

(5.4.84)

(n = l , 2 , . . . ) . If n ^ m then using the relation {Wn,Wm)F = 0 ,

n ^n^m m

derived from the expression Eq. 5.4.77, and the identity

(l-^)C7f ) (p) = ( l - P > ) ^ 1 ) ( p ) , we get the polynomial identity

E E ^^/"^"^^(p^w^^wcfcr 1 ^) so, P e R with

,„)=n(n_-3)+i(._1)

«r = = f c p + i a - i ) + i-

Q

+

1

In the following we calculate {
An_iAn

(*<">)' on the basis of paragraph 3.2 and of the results of Appendix E. By Eq. 5.4.83

(z#>)2 = (i _ p )v^-i)_^L_ w+%))2 fe^etJ)(p)^n)(p)) , where 2(/?n - 1) = - 2 ( n 2 + 1) . Thus (V?n,V?n)F

A„ 1 A

(1-

n n lV< n 2 + 1 >

- (I-P) 2 (C^ + 1 »(P)) (S£l (zUpa'ct\i]ip)c^(p))

(5.4.85)

5. DECOMPOSABILITY

93

OF DISTRIB UTION FUNCTIONS

Starting from the identities

1

c n)

i ^ =

n -p p

(i = l , . . . , n - l )

(5.4.86)

h&^lr)

and the definition

w "111 1 _ „"+j+l ,■ , Jrl-P" = c£?(p), 11

1 _ pn-j

we get n-l

I I Ct^\p)C(rl\p) ■

I I C
i=i

(5-4.87)

i=i

Moreover the relation n

/- 2 . «\

6 ^

n - 1

+ 7 )

6

HX9 . «n / .x 3 [2(n-l)2 + 7 ] = n ( n - l ) + -

Frt/

holds. By Eq. 5.4.87 and Eq. 5.4.82, we have A n _1x 1 - p 1 g = ^ - ^ b e

W

» W « ( p )

•

(5.4.88)

Using the relation _ 1_ 2 [n + (n + l) 2 ] , ~ 2

2(n 2 + 1) - n ( n - l ) + -

and the first identity of Eq. 5.4.86 in the case j = n, then substituting the value Eq. 5.4.88 into the expression Eq. 5.4.85, we obtain that {¥n,Vn)F =

(l-^)V n 2 + ( n + 1 ) 2 ] / 2

(EUpajcnn-v\p)^n)(P)y

(i - P )(i -p^)

(n = l , 2 , . . . ) . By Eq. 3.2.49,

™>Fyip ( s n )

>0

V

I

JT{ (
(U=l,2,...),

(5.4.89)

94

CHAPTER

II. DECOMPVSABILITY

OF DISTRIBUTION

FUNCTIONS

because now

_J__ _ O - g y (^.-[—.•I/.^W^W)' 'tocfoo) •. (5.4.M) So the distribution function F is not decomposable by a finite number of elements of the orthogonal sequence Eq. 5.4.84. Moreover, the sequence Eq. 5.4.89 is decreasing. We now show that the limit of the sequence Eq. 5.4.89 is equal to zero if 0 < p < 1. In fact, 1 + p + . . . + p2n

(l-p)(l-p2n+l) (1-Pn)2

>

(1+P+...+P11-1)2

>I.

1i

>

(5.4.91)

Further by Eq. D.5 of Appendix D,

c J r - t o > c**->(o) = Nn<m (n^ni+I)) = Thus

^p0j_Hn2+(n+1)2]c(„r)(p)C7,„)(p)

>

i=i

(5.4.92)

>

Dn+*

L

J

j=2

n»+i

> 1.

Making use of Eq. 5.4.81 and Eq. 5.4.92, from Eq. 5.4.90 we obtain that 11 (
1_

>I

(n = l,2,...).

n

Taking this inequality into account, from Eq. 5.4.89 we get our statement since the harmonic series is divergent. Theorems 3.2.3 and 3.2.4 imply the following two results: Theorems 3.2.3 and 3.2.4 imply the following two results: Theorem 5.4.3 Let F € E(Z#). Theorem 5.4.3 Let F e E(Z^).

Then the function Then the function

M*) = £ $ B W*),

(/*jn))e s*.

(5-4-93)

5. DECOMPOSABILITY

OF DISTRIB UTION

where

FUNCTIONS

95

1

/f>=

nn

(y

»' y » ) F 1

(t = l f - . f n ) ,

es £/ie 6es£ linear estimation of the distribution function F by the first n distribu­ tion functions of the orthogonal sequence Eq. 5.4-84 generated by the scalar product Eq. 5.4-77. Furthermore the error of this linear estimation is given by Eq. 5.4-89 and Eq. 5.4.90. Theorem 5.4.4 If F £ E(Z;v), then the sequence defined by Eq. 5.4-93 converges to the distribution function F in the metric generated by the scalar product Eq. 5.4-77. It may be interesting to note that the coefficients in the expression Eq. 5.4.84 and Theorem 5.4.4, as well as the measure of approximability, depend only on the parameter p which determines the distribution function F.

Appendices Appendix A Let (X,<S,/i) be a measure space, where S is a Boolean cr-algebra of subsets of X , and \i is a finite measure defined on <S, i. e. \i is a non-negative and countably additive set function defined on S such that fJ>(X) < oo. If a real-valude function / is integrable on i c X , A G S with respect to //, the value of the integral is denoted by JA f dfi(t). The following theorem of Lebesgue is well-known. Theorem Let {/ n }i° be a sequence of measurable functions defined on A G S such that \fn\ < g a. s., where g is an integrable function on A with respect to fj,. If fn —► / a. s. on A, then fn and f are integrable on A G S, and

J f»M*)-+J fdfi(t)

,

n —► oo .

In the following let the measure space (R, 5 , H) be given, where S is the Borel cr-algebra of the set R of the real numbers, and H is a probability distribution function. The following Theorem has a fundamental role in the mixture theory of proba­ bility theory. T h e o r e m A . l Let G(z,x) be a family of distribution functions x G R, and let H be an arbitrary distribution function. Then F(z)=

I

(A.l.)

G(z,x)dH(x)

J—oo

is a distribution

function.

Proof: Since

0
< 1 ,

x e R,

the integral Eq. A.l exists for all z G R. 97

oo

/

l-dH(x) oo

with parameter

= l,

98

APPENDICES (a) F(z)

is non-decreasing. Indeed, if z\ < z2 , then F(z2) - F(Zl) = f°

[G(z2, x) - G(zu x)] dH(x) > 0 ,

J —oo

since G(z, x) is a distribution function in z. (b) F ( - o o ) = 0, F(oo) = 1. Really from G ( - o o , z ) = 0, G(oo,a?) = 1 we obtain

=

F(-oo)

r

G(-oo,x)dH(x)

= 0,

J—oo /•oo

F(oo)

=

/

G(oo,x)dH{x)

= l.

J OO

(c) F(z)

is right continuous. Let z be a fixed point of R, and let hn > 0

(n = l , 2 , . . . ) ,

hn —> 0 ,

n -* oo .

Let /„(x) = G(^ + A n ,x)

(n = l , 2 , . . . ) ,

/ ( x ) = G(2f,a?),

xGR.

These functions are integrable on R with respect to H and bounded with bound one. Since G(z, x) is right continuous in z> we obtain: fn(x) -> f(x) ,

n -> oo ,

x € R .

Using the theorem of Lebesgue, it follows that oo

/ oo OO

i.e.,

/*oo

fnndH{x)= dH(x)= /

J -J-00 oo

lim F(z + hn) =

/<*#(*), fdH{x),

F{z),

hn-*0

so the distribution function Eq. A.l is right continuous. T h e o r e m A.2 Let G(z,x) eter x G R . Then j

G E c be a family of distribution functions with param­

G(z, x) dH(x) e E c ,

^ R ,

(A.2)

J—oo

where H 6 E is arbitrary. Proof: By Theorem A.l it is sufficient to show that Eq. A.2 is continuous, i. e., Eq. A.2 is left continuous. This statement follows similarly as the right continuity of Eq. A.l proved under (c).

APPENDIX

99

A

T h e o r e m A . 3 Let the family of the distribution functions G(z,x) with parameter i 6 R , and an arbitrary distribution function H(x) be given. Let S be the Borel cr-algebra of R . Let fi(A,x) , AeS be the a-finite measure generated on S by the distribution function

G(z,x).

Then

oo

/

//(A,x)dH(x) = fi(A) ,

AeS

■oo

is a a-finite measure generated on S by the distribution F(z)=

I

G(z,x)dH(x)

,

function z£R.

J—oo

Proof: It is obvious that fi(A) > 0, / J ( R ) = 1. Further, the set function ^(A), A E S is a-additive. Indeed if Aj G S (j = 1,2,...) are disjoint, then by the cr-additive property of the measure ^(A,:r), A G 5 , and the Theorem of Lebesgue,

^ (E ^) = £/ (E ^' *) ^w= = /^E^'*)) dHw = OO

OO

-oo

i=l - 7 - 0 0

i=l

The second statement follows from the fact that if a\ < h < a2 < b2 < . . . , I(ak ,h) = {xeR\ak<x
(k = 1,2,...) ,

and CO

A = ^2l(ak,bk), k-\

then oo

n(A) = J2lF(h)-F(ak)}. k=l

Suppose again that S is the Borel a-algebra of R. Let v and // be measures on S generated by the distribution functions G(z) and F(z), respectively. The measure v is said to be absolutely continuous with respect to the measure //, if v(A) = 0 whenever fj,(A) = 0.

100

APPENDICES

T h e o r e m A . 4 Let //* be a a-finite measure generated on S by the distribution function Fk, where fc = l , 2 , . . . . Let oo

aj>0

(j = 1,2,...) ,

and let

£ " ; = !,

oo oo

6e £/ie a-finite measure generated by the distribution

function

oo

i=i

77ten £/ie measures measure fi.

fij with aj > 0 are absolutely continuous with respect to the

Proof: The relation oo

//(A) = ^ a j 7 x i ( A ) = 0 i=i

holds if and only if ajN(A)

=0

(i = 1,2,...) ,

which implies our statement. Compare with [5], Ch. V. Section 5.6. The following statement is a consequence of Theorem A.3. Corollary A . l Let F G E, and the family of the distribution functions with parameter x G R be given. Then the integral equation

(i(A,x)dH(x) = fi(A) ,

I

G(z,x)

AeS

J — oo

has a solution

H G E if and only if the integral equation CO

/ has the solution

G(z,x)dH{x)

= F(z),

zeR

•oo

H G E.

Appendix B A finite or infinite matrix is said to be totally positive (non-negative) if its subdeterminants of finite order are positive (non-negative).

101

APPENDIX C

The matrix A' = ( a ^ ) is said to be the transsignation of the matrix A = (cijk), if (-l)j+kajk.

a'jk =

A matrix A is said to be sign regular, if A' is totally non-negative. If A' is totally positive, then A is said to be the sign regular in the strong sense. It can easily be seen, that in the case of finite square matrices the following statements hold ([11], II. §. .2.2): (a) DetA' = D e t A (b) If C = A ± £ , then C = A' ± B'. (c) If C = AB, then C = A'B'. (d) If C = A~\

(A')-1.

then C =

(e) The product of two totally non-negative matrices is a totally non-negative ma­ trix. (f) The product of a totally positive matrix by a regular totally non-negative matrix is a totally positive matrix. (g) If A is a regular, totally non-negative matrix, then (A')"1 is totally nonnegative. If A is totally positive, then also A'~l is totally positive.

Appendix C Let the complex numbers aj, b3, (j = 1 , . . . ,n) satisfy the conditions aj + h^iO

(C.l)

(j,k=l,...,n).

For calculating the determinant of the matrix

/

C= \ C=\

1

1 ..

|I

V an + 6i

•"

\

::

I

(C.2)

an + bn 7

the method of Cauchy ([7], 151-159) is used in the following way: After subtracting the last row from the other n — 1 rows, the factors 1 an + &i

1 an -|- b2

1 an + bn

can be taken out from the columns, and the factors an-

ai , an-

a2 ,...,

an-

an-i , 1

102

APPENDICES

from the rows. Next we substract the last column of the remaining determinant from the first n — 1 columns, and take out the factors h - bi , bn - b2 , . . . , bn - 6n_a , 1 from the rows, and the factors 1

1

Q>1 + bn '

1

1 ' On-l + &n '

a2 + &n '

from the columns. Then the left upper corner determinant of order n — 1 of the matrix C remains unchanged. Thus we conclude by induction that

A^et^n^-;^;-^.

(C.3)

Uj,*=i( a j + 6*)

Consequently A ^ 0 if and only if «j^"k,

bj^bk,

j^k

(j,fc = l , . . . , n ) .

(C.4)

Now we suppose that condition Eq. C.4 is satisfied, and we calculate ( C ) " 1 . Let Cjk denote the matrix which arises from C by omission of the j t h row and the kth column. For calculating (C')~l it is necessary to calculate the determinants Ajk = Det Cjk

(j,& = l , . . . , n ) .

Comparing the determinants of Cjk and C we see that those and only those elements of C are missing from Cjk which have first index j and second index k. Thus from Eq. C.3 we get the representation (j,* = l,...,rc),

^L = Aj-^—Bk A a j + bk

(C.5)

where

riLK +

A. =

o

Dk

(an - a,j)... (aj+i - aj)(aj -
=

nr=i(«.-+M

fr)

(j =

l,...,n),

(cij - di)

(bn-bk)...(bk+1-bk)(h-bk.1)...(bk-bl)

rt=1 {K i

„x

'---'n^-

Therefore, by Eq. C.5 (A, ((cr C "1) ■■ -1^

( 0 ) \ (Bx ( 0 ) \ ■.. )C[ •.. .

V(0)

(C.6)

A , / \ ( 0 ) fin/

Since Det C = Det C, from Eq. C.6 we obtain another representation of the determinant A

(Delc)

'-IEW

(C7)

APPENDIX

103

C

Corollary C . l / / aj = bj (j = 1 , . . . ,n) , and conditions Eq. C.l and Eq. C4 are also satisfied, then

D< c

* =rEi;-

Corollary C.2

If the real numbers aj, bj (j = 1 , . . . ,n) satisfy

<*!<... < a n , £/&en £/ie matrix

&!<...<&„,

condition

(j,fc = l , . . . , n )

aj + bk>0

(C.8)

C is totally positive, and 4>0,

fl,->0

(i = l , . . . , n ) .

(C.9)

Let us now consider the special case, where the numberws appearing in Eq. C.2 are «j=j + l,

(i = l , . . . , n ) .

h=3

Further, let /

i

r=

c

(o)

V

/ I

\

(0)

n

\

(0)

(0)

n

)

)

In view of Appendix B, /

i

(r) V

V1

(0)

/

(cr

(o)

i

(0)

(0)

n J

n J

where, by Eq. C.6, where, by Eq. C.6,

(C)- 1

whith

* - ( "

+

:

A,

(0)\

(A,

(0)

<(0)

A./

\(0)

An

+

% " >

0 = l,...,n).

So

(!")where

V(o) x A j„ / V(o)

V(o) AAJn y \(o)

* - M " + ; ! + , ) G ) «-• •»•

104

APPENDICES

Consequently,

fxrxy

DetC

DetT:

n("+i+I)C.

Let us start now from the conditions of Corollary C.2. Taking the representation Eq. C.6 into account, we obtain n

0 < x*C'-\n)y

n

= J2H

1

A

>Bk^TTxM<

where x = (XJ) € R n , V = (yj) € R n are arbitrary vectors with positive compo­ nents. Since all quantities appearing in this quadratic form are positive numbers, the well-known inequality between the arithmetic and geometric means can be applied. Consequently,

^<x*{C')-1{n)y
\

(n^-)"(n^)"n^-

Using representation Eq. C.7,

0 < {X*C>-\n)yr < ^ D e t 2 C ( " ) M ^ II K + h) ■ Since C(n)

is totally positive, n

DetC(n)
1

-.

Therefore

0 < (x* C'-\n) j,)"1 < 1

;

ff J _ ( J f] _L_ )

N

Y[(aj

+

bk). (C.10)

3,k=l

Let now ai=j

+ l,

bj=j,

Xj

= yj = J

(j = l,...,n)

.

APPENDIX

105

D

Then n

i

n

(C.ll) j=i

\ j,k=i

l=lai

+ fe.

< 112j

n n

(C.12)

2»n!'

n n

(C.13) Substituting expressions Eq. C.ll, Eq. C.12 and Eq. C.13 into the inequality Eq. C.10, we obtain the following result: Corollary C.3

If the matrix C(n) aj=j

+ l,

arises from formula Eq. C.2 by choosing (j = l , . . . , r c ) ,

bj=j,

and if i

(

/

\

(0)

1

\

(0)

c

T(n) = (0) then

n

/

(0)

n

)

n+ 2 4n 2 ' are equal to one.

0<(eT'-1(n)e)"1< where all components of e € R n

Appendix D In this appendix we deal with the representation of two polynomials. Lemma D . l

/ / x g R, and

Pv{x)

=

PuO)

=

(1 - x 2 ' 2 - 1 ) ) (1 - s2<2"-2>) . . . (1 - x2")

*1,

(l-x)(l-x3)...(l-x2"-1)

(D.l)

22

for v = 1,2,... , then P„(x) = (l+x)(l+x2)...(l

+ x2"-1)

.

(D.2)

106

APPENDICES

Proof: If v — 1, then

1 -x2 Pi(x) = = l + s. 1—x Suppose that the representation Eq. D.l holds for all indices less then or equal to v. Since applying the induction hypothesis we obtain that Pu+l+1{x) P„ (x)

=

(l + x ) ( l + x 2x)(l+x*)...(l+x*+ ) . . . ( l + x 2 " + 11))xx (l + (1 _ sJpH-i)) (i _ ^a-to) (1 - x ) (1 - s 2 "+i) (1 + x2") (1 + x 2 "+i) 2v

(l + z ) ( l + * 2 ) . . . ( l + z2"+1) , which proves the statement of the Lemma. Let C^n\x)

c^-Hx)

=

1,

iGR,

-

(i-*n+m)(i-*"+m-1)---(i-*"+1)

x € R

x 4 l

(D.3)

n

m

cfir^D = ( + ) (n = 0 , l , 2 , . . . ; m = l , 2 , . . . ) . L e m m a D.2 Let m, n, m + n > 0 6e non-negative integers. Let Nm,n(k) the number of representations of the positive integer k in the form k = n + . . . + rm , tuyere r i , . . . , r m n + m. Then

are positive integers satisfying condition

be (D.4)

1 < r\ < . . . < rm <

C%+m\x) = PiT +m '(x),

(D.5)

*—>(«) = g ivm,n (I2(ll±i) + t y .

(D.6)

where

Proof: It can be seen easily that N

N

m=0

r=l

(D.7)

APPENDIX E

107

Indeed, the coefficient of ymxk on the right side of Eq. D.7 is evidently equal to the number of representations of k in the form k = n + . . . + rm ,

1 < n < . . . < rm < N .

On the other hand, the following generalization of the binomial theorem was already known to Gauss:

I](l + yxr) = £ ,v(- +1 W J C(f>(*); rr = = ll

(D.8)

m m= = 00

perhaps the easiest proof is by induction. From definition Eq. D.3 it is obvious that the symmetry m m+ m ) C%++m \x) C£" >(s) = C^ C< n \x) (*)

(D.9)

holds. Moreover, by Eq. D.8 N

N

m=0 m=0

r=l

Y, *m(m+1)/2C
(D.10)

Appendix E Let 0 < p < 1, and let C(n)

ai = / l Y

be the Cauchy matrix generated by

+1

(U -'

«i = ( - )

bi =

h

~pi U(>=' =1.•••,«) i, •••,«)••

( R(E.l)1 )

0 = 1l ,,.. . -. , nn)) .. 0'

(E.2)

i = ~V?

Further let ^ • = PP, i++ 1 ,. *;

tfi % = l1

Taking the conditions ai fli < < .. .. .. < < aann ,,

bi 6i < < ... ... < < b6nn ,,

and

aj 0',fc= aj + + 66*f c>> 00 (j,fc = ll ,, .. .. .. ,, nn )) into account, we obtain that C(n) is a totally positive matrix. Since aj + bk < dj (j, k — 1 , . . . , n) , thus n

n

dn^

1

=

p(n+l)/2 •

(E.3)

Moreover, 1 = JTT I Jfr_L 11 \}i*iyi \JLTP*1 n

11

p(n+1)/2

(E.4)

108

APPENDICES

holds. Finally, using the identity l-p2j+1

= (l-p)(l+p

(j = l , . . . , r a ) ,

...+p2j)>l-p

+

we obtain that

P7 +1

,1Sr ( { n ^ )
.1n«,+»,

D 2j+1

(E.5)

n

« +!

Now substituting the inequalities Eq. E.3, Eq. E.4 and Eq. E.5 into the relation Eq. C.10 of Appendix C, we get the following result: Corollary E . l If the elements of the matrix C(n) of Cauchy type is generated by Eq. E.l, where 0 < p < 1, while the components of vectors x = (XJ) £ R n and y = (yj) £ 'Rn are determined by Eq. E.2, then

2

0<{x.c>-i{n)yr<(^±-j)

.

p)J

We now deal with the determinant of the matrix C(n). From relation Eq. C.7 of Appendix C, and from Eq. E.l above, after a short calculation we get

(DetC(n))"2 =

n

JjAjBj, i=i

where

4- = p ^ ( i - ^ ) c ^ + 1 ) ( p ) c | n ) ( p ) , Bj = ^■(l- P ')Ci" +i+1) (p)CJ B) (p)

0' = l,...,n)

with

"i = [0' + 2) + ... + 0' + n)] + 0 ' - l ) 0 ' + l)-»>(i + l ) . ft = -[(l + ... + n ) + j ( n - i ) + (l + ... + 0'-l))] Cn (p) being defined by Eq. D.3 of Appendix D. Since <*i + Pi = i 2 - (2n + l ) i - 1 we have

(;' = 1, •.. ,n) ,

n

n

^ ( a j + / ? i ) = -^(2n 2 + 3n + 4).

109

APPENDIX E Using this, we get the representation | n(2n

2

+3n+4)/6

(n^+j+i)(p)^n)(p)

Detc{n)=

fi^o-go

(E.6)

which has been our aim. Hence, by the well-known identity I-pi 1-p

= l+p+...+pJ-1

(j = l , . . . , n ) ,

0
it is easy to obtain that

+i i ,. e ,i(nC^ w D tcwtc ,»,=i(n(" + r)0)"'. i™,P1 -p)-De )G))"'

(E.7) (E 7)

-

The next step is to deal with the representation of the matrix

*

I l_

p

l

1 3

1 - p

G(p)

4

\l-p +

\ I

0
i

1 !_pn+3

n 2

n+2 «J-9

l-p2n+d

/

It can be seen easily that

/

7

(o)

1

p

7

G(p)

V

<7(n),

(0)

7^ I

thus (7(p) is a totally positive matrix, and

Det(

?(j>)

=

I ,n(n+l)/2 ^77Det^(") = P' „»<2n*+l)/6

/

«

« ' ) n^ ni=1(i-p ^

+i+i)

n)

(^i (p)

The analogue of Eq. E.7 holds for G(p) since

lim|(l -p)G(p)\ p-fl

/1 = , 1 \ nT2

1 1 ra+lf

HT"2 ^ •••

2HT3 /

(E.8)

110

APPENDICES

Appendix F First of all, we prove the following statement: T h e o r e m F . l Let n > 2 be an integer. Let L be a n x n matrix with real or complex entries. Let Ljk be the (n — 1) x (n — 1) matrix derived from L by omission of the j t h row and the kth column. Let £jk = e* adj Ljk e where the components of e £ R n - i

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

are equal to one. Then

j = l k=l

Proof: The following Lemma is needed: Lemma F.l

The sum A(L)

is invariant against an exchange of rows or columns

of L. Proof: The n x n matrix P is called a permutation matrix, if it arises by interchanging the rows (columns) of the unit matrix. If L is a matrix derived from L by interchanging rows and interchanging columns, then there are permutation matrices Pi and P2 such that L = P\LP2 . Let now T = L-M. Then Det r = Det L - e* adj L e , where all components of e £ Tin are equal to one. From here we get that ( - i y + * D e t I > = (-iy+k

[Det V

- /,-*]

and e*(adj T)e = e*(adj L)e - A(L) .

(F.l)

If P is a permutation matrix, then Pe = e obviously. For the sake of brevity we introduce the notation A(L) = adj A . known that A(L1L2) = A(I 1 )A(L 2 ) .

(F.2) It is well

Since permutation matrices are orthogonal matrices, A(P\) = P? and A(P2) = P2* are permutation matrices, whenever Pi and P2 are so. Thus taking Eq. F.2 into account e*A(PiLP3)e = emA(P1)A(L)A(P2)e = e*A{L)e . (F.3)

APPENDIX

111

F

This property says that e*A(L)e is invariant against an interchange of the rows or columns of L. The proof of Lemma F.l is complete. It is easy to see, that if e = — 1, then £

e'

A ( i ) = e*

(0)

(0)

flu

■■■

e / i » \

en /

\

e'

/

V

\

(0)

e.

(0) /

Lemma F.2 A(L) is invariant against the left hand side, and the right hand side permutations of the elements sk (k = 1 , . . . , n) . Proof: Really, if P is a permutation matrix, then e = PP*e = Pe by Eq. F.2. Thus if Pi and P 2 are permutation matrices, then (

\

(0)

(

f In

e2

A ( i ) = e*Pl (0)

e

\

(0)

^nn J

\e„i

n

Pte, n

e

\

/

(0)

1

i. e. the statement of Lemma F.2 holds. Proof of Theorem F . l : Let L^ be the (n — 2) x (n — 2) matrix, which can be obtained from L by omission of the rows with indices z, j , and of the columns with indices &, L The elements of the (n — 1) x (n — 1) matrices MLJk)

(F.4)

(j,fc = l , . . . , n )

are the determinants Det Lf-

(l
with suitable signs. Let us consider the value Det/^_}[J matrices Eq. F.4 the matrices adj Ln-\n-\

5

adjLn_in ,

adjX n n _i ,

from them. Among the adjLnn

and only these matrices contain this determinant. Thus the coefficient of Det L"Z\" in the sum A(L) is (_l)2(n-l)

+

(_!)2n-l

+

(_!)2n-l

+

(

_

l ) 2

0.

The procedure is similar if we consider the coefficient of Det L^j in the sum A(L). Namely, by suitable changes of rows and columns, the matrix L will be transformed into a matrix which has L^ for the principal minor matrix in the left upper corner. By Lemmata F.l and F.2 A(L) is invariant against such permutations. So, by the statement above, the coefficient of DetL^ is equal to zero. Thus the proof of the theorem is complete.

112

APPENDICES

Appendix G As it is well-known, the formula of Sherman-Morrison has importance in calculating the inverse of certain matrices. In this appendix we deal with the solution of a matrix equation. A special case of the results obtained is an extension of the ShermanMorrison formula ([15]). Let C(m, n) denote the set of m x n matrices with complex entries. A* G C(n,ra) is the transpose of A 6 C(ra,n), and adj B G C(n,ra) the adjoint of B G C(n,rc). Let E and 0 denote the unit and the zero matrix, respectively. The main goal of this appendix is the proof of the following theorem: T h e o r e m G . l Let A G C(n,n) . Then the solutions excepting the trivial solution X = 0 of the matrix equation adj AVet{A are given by the following

+ X) - adj (A + X)Vet A = adj A X adj A

(G.l)

statements:

(a) If A is singular and adj A = 0, elements X G C(n,n).

then equation Eq. G.l is satisfied by all

(b) Let A be singular and adj >1 ^ 0. Then X G C(n,rc) Eq. G.l if and only if it satisfies the equation

satisfies equation

Det(i4 + X) = b*Xa ,

(G.2)

where adj A = ab* with a, b G C(n, 1) . (c) Let the matrices the following: (I) X = AN, 2.

A and A + X

be regular. Then all solutions of Eq. G.l are

where N G C(n, n) is an arbitrary nilpotent matrix of index

(II) X= (expl-^il-ljAPk

(j = l , . . . , * - 2 ; * = 3,...,n)

where Pk G C(n,n) is an idempotent matrix of rank (Ill)

X = (y — l)APi from 0 and 1.

, where y

k.

is an arbitrary complex number

different

(d) If A is regular and A+X is singular, then all solutions of equation Eq. G.l are given by X = uv*, where u,v G C(n, 1) satisfy the condition v*A~1u + l = 0.

APPENDIX

113

G

Proof: Since equation Eq. G.l is trivially satisfied by X = 0, in the following we suppose that X ^ 0. (a). The statement is trivial. (b). In this case, Eq. G.l can be reduced to the form adj A Det(A + X) = adj A X adj A . Since A is singular, adj A = ab*, where a,be

(G.3)

C(n, 1). By Eq. G.3

ab*Vet(A + X) = ab*Xab* . From here in view of ab* / 0 we obtain that condition Eq. G.2 is necessary for the solvability of Eq. G.3. If we start from condition Eq. G.2, we obtain Eq. G.3, i. e. condition Eq. G.2 is sufficient for the solvability of Eq. G.l. (c) and (d). Using the notation Y = A +X equation Eq. G.l can be expressed in the form adj A Det Y - adj Y Det A = adj A(Y - A)adj A . (G.4) Multiplying the equation Eq. G.4 from the left by the matrix A, taking into account that Aadj A = £Det A , and, finally, dividing by the determinant of A, we obtain Aadj Y + Fadj A = E (Det A + Det Y) .

(G.5)

In the case of the regularity of A equations Eq. G.l and Eq. G.5 are equivalent. In order to find a form more suitable than expression Eq. G.5, we write equation Eq. G.5 in the form (Y - X)adj Y + (A + X)adj A = E (Det A + Det Y) . Using the relations FadjF = £ D e t F ,

Aa,d)A =

EDetA

we obtain that X[adj(A + X) - adj A] = 0 which, by the identity adj(A + X) = adj [A (E + 4 " 1 * ) ] = adj ,4adj [E + A'lX]

,

gives X a d j [E + A-XX]

=X

(G.6)

.

Since A is regular, equation Eq. G.6 is equivalent to Eq. G.5, thus also to Eq. G.l. Using the relation (E + A~lX)

adj (E + A~*X) = £ D e t (E + A~lX)

,

114

APPENDICES

by Eq. G.6 we get X (E + A~lX)

= X Det (E + A'lX)

,

or, what is the same, A~lX

[Det (E + y T 1 * ) - l] = ( A " 1 * ) 2 .

(G.7)

On the other hand, Eq. G.7 is equivalent to Eq. G.6, and thus also to Eq. G.l if and only if A + X is regular. If A + X is singular, then condition Eq. G.7 is necessary for X to be a solution of equation Eq. G.l. (c). If y = Det (E + A~lX) = 1 then by Eq. G.7 A~XX = N is a nilpotent matrix of index 2. On the other hand, if A~XX = N is a nilpotent matrix of index 2 then in view of the relation E + N = exp N we have Det(E + TV) = exp{tr TV} = 1. Thus the statement (I) is proved. If 2 / ^ 0 , 1 then, by Eq. G.7, A~lX = (y - 1)P, where P G C ( n , n ) is an idempotent matrix. From here E + A-1X

= (y-\)P

+ E.

Since P is idempotent, its eigenvalues are the numbers 1 and 0. Moreover, it can be transformed to a diagonal form. Let P = Pk . Then, by our last equation, y = yh Thus in the case k > 3 conditions y ^ 0,1 imply that y is a (A: — l ) t h root of unity different from 1. If k = 1 then yk~x = 1 is satisfied by arbitrary number y ^ 0. Thus X = (y — 1)AP\ , where y is an arbitrary complex number different from 1 and 0. Taking into account that in the case y ^ 0 the relation

Dek[E + (y-l)Pk]

=

yk?l

holds for y = exp< —-i

\

(j = 1 , . . . , k - 2; k > 3) ,

while in the case y ^ 1 it holds for k = 1 we get the proof of the statements (II) and (III), respectively. (d) In this case from Eq. G.7 it follows that X can be a solution of equation Eq. G.l only if X = —AP, where P is an idempotent matrix. In order to determine those X which really satisfy Eq. G.l, we substitute them into Eq. G.l. After a short calculation we conclude that the matrix P must satisfy the condition P = adj(i? — P). Since the matrix A + P is singular, E — P is also singular. Thus P = c6*, b,c € C(n, 1), which has the eigenvalue 1; therefore b*c = 1. Let now C = -A~la. Then X = ab*, where b*A~xa + 1 = 0 . Now the proof of Theorem G.l is complete. The following result is an easy consequence of Theorem G.l.

APPENDIX

115

G

T h e o r e m G.2 If A £ C(ra,n), further u,v G C(n, 1), whereas either A is singular and adj A = 0, or A is singular and adj A ^ 0, or A and A + uv* are regular, then adj ADet(A + m;*) - adj(A + m;*) Det A = adj Au v* adj A .

(G.8)

If A is regular and A -f wi>* is singular, then Eq. G.8 holds if and only if the additional condition v*A~lu + 1 is satisfied. Proof: It is sufficient to show, that X = uv* satisfies the conditions of statements (a), (b) and (c) of Theorem G.l. The case of uv* = 0, as well as the case where A is singular and adj A = 0 are trivial. If A is singular and adj A = ab* ^ 0, then Det(A + uv*) = v*ab*u = b(uv*)a i. e., X = uv* satisfies condition Eq. G.2. Let A and A + uv* be regular. Then Det (E + A - W ) = 1 + v*A~lu . Hence A " W [Det (E + A " W ) - 1] =

v*A-luA~xuv*,

i. e., condition Eq. G.7 is satisfied by X = uv*. The proof of the Theorem G.2 is complete. If A and A -f uv* are regular then Eq. G.8 can be brought to the form (A + uv*)-1 = A" 1 v

;

1

——A-1uv*A-1, l + v*A~lu

(G.9)

v

;

and this is the formula of Sherman-Morrison. Theorem G.l yields the following generalization of Theorem G.2. Corollary G . l

The representation (A

+

Xr

1

= A'

A--Det{ElA_lx)A-XA^ XA

Vet{E + A~lX)A

holds for regular matrices A,A + X G C(n, n) if and only if X belonges to one of the types (I), (II) and (III) described under (c) in Theorem G.l.

116

APPENDICES

Appendix H In this part we deal with the moment problem of Hamburger, and with the full moment theorem of Stieltjes. Definition H . l

The sequence {Mn}g° of real numbers is said to be positive, if $(P) = a0M0 + axMx + . . . + anMn > 0

(H.l)

is satisfied for all polynomials P(x) = a0 + axx + . . . + anxn > 0 ,

P(x) =£ 0 ,

x GR .

If $(P) > 0 for all such polynomials, the sequence is said to be strictly positive. Definition H.2 The sequence {Afn}g° of real numbers is said to be positive in the Hankelian sense, if

An =

Mo Afi

Mi M2

M2 M3

Mn

Mn+1

Mn+2

... ...

Mn Mn

>0

(n = 0 , l , . . . )

••• M2n

The equivalence of the previous two definitions is expressed by the following Lemma. L e m m a H . l The sequence {Mn}g° of real numbers is strictly positive if and only if it is positive in the Hankelian sense. Proof: The functional defined by Eq. H.l is obviously linear. Thus $ (xki>n(x)) = 6knAn

(fc = 0 , l , . . . , n ) ,

(H.2)

where ^o(^) = 1, MQ

Mi

Mx

M2

Mn

Mn+1

Mn_x Mn

1 x

1>n(x) = . . . M2n-i

(n = l , 2 , . . . )

xn

and 6kn is the Kronecker symbol. If the sequence {Mn}g° is strictly positive then by Eq. H.2 *(#(*)) =A„_1An, and from here it follows by induction that that A n > 0, i.e. the sequence is positive in the Hankelian sense.

APPENDIX

117

H

Now suppose that {Mn}g° is a positive sequence in the Hankelian sense. Let the polynomial P(x) be not identically equal to zero, and let the degree of P be equal to n. Then the representation P(x) = A0il>o(x) + Aiip^x)

+ . . . + Ani/>n(x)

holds. From here by Eq. H.2

$(P 2 (x)) = E 4 A * - i A , .

(H.3)

k=0

Now let P(x) representation

be an arbitrary non-negative polynomial. As we shall see, the P ( s ) = / * ( * ) + i*(*)

holds, where P\(x)

and i^fa) are real polynomials. Thus by Eq. H.3 *(P(X))

= *(P1(X))

+

*(P!(X)),

i. e. {Mn}o° is a strictly positive sequence. It remains to show that if P(x) is a polynomial with real coefficient and P(x) > 0, x G R, then P(x) is representable as the square sum of two polynomials with real coefficients. Since the real roots of P(x) have even multiplicity, the representation Since the real roots of P(x) have even multiplicity, the representation P(x) = Q(x)R2(x) P(x) = Q(x)R2(x) holds, where the roots of the polynomial R(x) are real numbers, and holds, where the roots of the polynomial R(x) are real numbers, and

Q(x) = A*f[[(x-aky + blY> , where a* -f M (k = l , . . . , s ) are complex roots of P(x) with multiplicity cr* . Thus Qk is the product of factors which are square sums of two polynomials. Since the identity (a2 + b2)(c2 + d2) = (ac - bd)2 + (ad + be)2 holds, each such product is the square sum of two polynomials. T h e o r e m H . l (Theorem of Hamburger) For the existence of a distribution function F(x) which has infinitely many points of increase which satisfying the condition oo

/

xndF(x) = Mn

(n = 0,l,...)

■oo

it is necessary and sufficient that sense.

{Mn}g°

be a positive sequence in the Hankelian

118

APPENDICES

Proof: In order to show that the condition is necessary we prove the following statement: If F(x) is a distribution function with infinitely many points of increase and the integrals

xndF{x)

Mn= I

(n = 0,l,...)

J — CO

exist, then {Mn}™ is a strictly positive sequence. Really, if the polynomial P(x) is not identically zero, while A and B with A < B are finite numbers, then $ (P(x))=

f

P(x)dF(x)>

J-oo

j

P(x)dF(x)

JA

Let A and B be chosen so that the number of the points of increase of F(x) in [A, B] is larger than the degree of P(x). Then there exists a point of increase x0 in (A, B) which is not a root of the polynomial P(x). Moreover there exists a number h > 0 such that the interval [x0 — h, x0 + h] lies inside the interval [A, B] and contains no root of P(x) lying in this interval. Let m be the minimum of P(x) in this interval. Then rB

/

rxo+h

P(x) dF(x) > /

JA

P(x) dF(x) > [F(x0 + h)-

F(x0 - h)] > 0 ,

J XQ—h

and this has been our statement. In order to show that the condition of the Theorem is sufficient, suppose that the sequence {Mn}£° is strictly positive. Then there exists a distribution function F(x) with infinitely many points of increase such that CO

Mk

/

xkdF(x)

(A; = 0 , 1 , . . . ) .

(H.4)

CO

In the proof, the following Lemmata will be used. L e m m a H.2 / / {Mn}g° is a strictly positive sequence, then all roots of (n = 1,2,...) are real and have multiplicity one.

ipn(x)

Proof: i/>n(x) has a real root with odd multiplicity. Namely, by Eq. H.2, in the opposite case i>n(x) would not be a non-negative functional satisfying the identity $ (ipn(x)) = 0, contradicting the condition, that the sequence is strictly positive. Let xi < . . . < xs be the real roots of ipn(x) with odd multiplicity, and suppose that s < n. If R(x) = (x - xi)...

(x - xs) ,

then Eq. H.2 gives $ (R(x)^n(x)) = 0. Since R(x)i^n(x) strict positivity of the sequence {Afn}g° . Thus s = n.

> 0, this contradicts the

119

APPENDIX H L e m m a H . 3 Under the conditions of Lemma H.2 let P(x) nomial of degree less then 2n. Then

be an arbitrary poly­

$(p(x))=f:4 n) J p(4 n) ), k=l

M where x^'

(k — 1 , . . . , n) denote the roots of \j)n(x), and

4-> = * Proof:

^n(a:)

n)

^(4 )(*-4 n) )J

Since P{x) = R{x)1>n(x) + Q{x),

where the degrees of the polynomials R(x) and g(x) are less than n. By Eq. H.2 it follows that 9(P(x)) = *(e(x)) . But we have

) = e (4">)

(* = i , . . : , « ) .

Lemma H.4 Under the conditions of Lemma H.2, we have

4n)>0

(* = l,...,n).

Proof: Substitute the polynomial \ 2

P(x) =

\x-xPJ

of degree 2n — 2 into the expression Eq. H.3. If i ^ k, then

p(4*")=o,

*(«i"9=fa(*i"})r

Consequently * ( P ( X ) ) = A|"»[V.;(X|"»)] 2 >O, i.e. A< n ) >0.

120

APPENDICES

We now return to the proof of the sufficiency of the condition of Theorem H.l. Let Xi < . . . < Xn be the roots of the polynomial V'n(^), and let the discrete distribution function Fn(x) be formed as follows. I 0,

n)

if —oo < x < Xi

n

if

Fn(x)= I Ai + ... + 4 \ l4n)

+ ... + 4 n ) ,

,

n)

4 <^<4+i (* = l,.-.,n-l),

if z l n ) < z < o o

(n = l , 2 , . . . ) . By Lemma H.3 Mi = *(*•") = ] T 4 n ) (^i n ) ) = /

xl dFn(x)

^-°°

k=i

(t = 0 , l , . . . , 2 n - l ) . Since functions Fn(x) are distribution functions, they are uniformly bounded. By the selection theorem of Helly, there exists a sequence {rik}™ of positive integers such that the limit lim Fnk(x) = F(x), xeR k—HX>

exists and F(x) is a distribution function. We show that the distribution function F(x) has infinitely many points of increase and satisfies condition Eq. H.4. To do this, let i be a fixed positive integer and nm > i. Then

Mi= f°° x'dFnJx). J—oo

Let A and B be finite numbers satisfying the condition A < 0 < B. that \Mi\MiI I

I

i \x\ \x\idF dFnm (x)+ nm(x)+

x'dF^x)]^ x'dF^x)]^ I I

JA JA

\x\ \x\iidF dFnm (x)+ nm(x)+ oo ■oo

JB JB

we obtain that

/»oo /»oo

A A

/

x'dF^x). x'dF^x).

J-oo ./-oo

For 2r > i and K = min(|A|,£?)

fA A

/

x'dF^x)^ x'dF^x)^

JB

0

0

x2rdFnM f 1 C°° ^w^Lx2rdFnJx)+ x2rdFnJx)+B^Lx2rdFnM1

1

^w^L if nm > 2r.

I f

B^L

-

Consequently

\Mi- J

^Fnm(z)

M2T - AT21-1

Then we get

nm > 2r .

APPENDIX

121

H

By the theorem of Helly, which gives the limit of a sequence of Stieltjes integrals of functions defined on the same finite interval [A, B], we get tB f

li-

x{dF{x)

<

JA JA

:

M2r

K2^

'

If A —¥ —oo and B —> oo, then also K —► oo, i. e. Eq. H.4 holds. Finally we show that function F(x) has infinitely many points of increase. Really, in the opposite case let P(x) be the non-negative polynomial, the roots of which are the points of increase of F(x). Then oo

#(P(*))

P(x)dF(x)=0,

/

•OO

contradicting the assumption that {Mn}™ is a strictly positive sequence. Definition H.3 The sequence {Mjfe}£° totally positive, if the infinite matrix

of real numbers is said to be Hankelian

H = (Mj+k)~=0

M0 = 1

,

is totally positive. T h e o r e m H.2 The sequence {Mk}™ is Hankelian totally positive if and only if there is a distribution function F(x) which has infinitely many points of increase, and satisfies the condition F(x) = 0, x < 0, and roo

xkdF(x)

M, =

(jfc = 0 , l , . . . ) ,

(H.5)

M0 = 1.

Jo Proof: Suppose that Eq. H.5 holds with a distribution function the conditions of the theorem. Let 0 < ii < ... < j n ,

F(x)

satisfying

0 < &! < . . . < kn .

Using the theorem of Landsberg ([20]) we obtain that

Det(AW: j/3=1 = / . . . / D e t v ( ^ -

£■)

0<Xi<...<Xn

(H.6)

xDet V (

kl

\ ^1

'"'

kn

) dF(n)...

• • • Xn J

dF{xn)

where

/xf

( ]l

■■■ Jn \ _ j

\Xi

...

Xn

.

... s» \ .

.

)

O'l < • • • < Jn , 0 < Xx < . . . < Xn)

(H.7)

122

APPENDICES

is the so called generalized Vandermonde matrix. It is well-known that the determinant of the matrix Eq. H.7 is positive ([11]). Therefore and since F(x) has infinitly many of points of increase, we find that the determinant Eq. H.5 is positive. Thus the sequence {Mk}™ defined by Eq. H.5 is Hankelian totally positive. Let us now suppose that {Mjt}g° , M0 = 1 is a Hankelian totally positive sequence. Applying the procedure we used in the proof of the Theorem of Hamburger, it can be shown that the roots of the polynomials are positive numbers. Really, in this case the transsignation of

^i(M = ( A(A^ jk) ;)lf ck=0 adj(M );,t=0 = + tk=0 =0 j+ki )l is a totally positive matrix. Therefore i ++ n Ain A i n = ( - li )y "|| A ii n | ,

| A ijnn| | > 00

(j = 00, ,1l,,..... ., , n) 0' n).

Thus if x > 0, then ll>n(~x) =

> (-xYAjn = (-1) LrV|Ain|*VO,

iz=0 j=0

tfn(0) M0)

= =

i=0 3=0 j=0

A oo n„ ^ O 0,

in conformity with our statement. A consequence of this statement is that Fn(x) = 0,

xx<0 <0

(n = ll,,22,,......)) -.

Thus the solution F(x) of equation Eq. H.5 is a distribution function having infinitely many points of increase, and F(x) = 0 for x < 0. This completes the proof of Theorem H.2. T h e o r e m H.3 the sequences

The sequence {Mk}™ \M {M k}% t }g° ,

is Hankelian totally positive if and only if {M{M }%> }£ k+1k+1

are strictly positive. A similar result is due to Gantmacher and Krein in the case where F(x) is a distribution function having finitely many points of increase ([11]). Proof: The full moment theorem of Stieltjes ([1], p.76) says the following. The system Eq. H.5 is satisfied by a distribution function having infinitely many points of increase and F(x) = 0, x < 0 if and only if the matrices (Mi+t)~=0 ,

(Mj+k+l)^0

are positive definite. Comparing this result of Stieltjes with Lemma H.l and Theorem H.2, we get the statement of our Theorem.

APPENDIX

123

J

Theorem H.4 If {Mk}™ and {Nk}™ are Hankelian totally positive sequences, then {MkNk}™ is also a Hankelian totally positive sequence. an(

Proof: Under the assumption, the matrices (Mj+k)Tk=o positive definite. By Theorem H.3 the matrices (AO+AO^-O also positive definite. So the matrices (Mj+kNj+k)^0

,

an

^ (^i+*+i)£fc=o d (^i+fc+i)Ti=o

are are

(Mj+k+1Nj+k+1)~k=0

are positive definite by the following theorem of Schur ([34]): When matrices

A = (ajk)lk=1 ,

B = (6 >4 );, tel

are positive definite, then their Hadamard product

A*B =

(ajkbik)lk=l

is also positive definite. Using Theorem H.3 again, we get the statement of the theorem. The following theorem can be proved in a similar way. T h e o r e m H.5 / / {Mk}™ and {A^}g° are Hankelian positive sequences, then {MkNk}™ is also a Hankelian positive sequence.

Appendix J As earlier, let E c be the set of continuous distribution functions. It is known that if F,G eEc , then /

F(t) dG(t) + / " G(t) dF(t) = F{x)G(x)

J— oo

,

x € R

(J.l)

«/ — oo

([32], 54., p.110). We now prove the following generalization of this identity. Theorem J . l

Let Gj£Ec

(j = l , . . . , n ) .

Then

Y,

G1(t)...Gj-1(t)Gj+l(t)...Gn(t)dGj(t)

3 = 1 J~°°

where G0 = G n +i = 1.

= Y[Gj(x), j=l

xeR

124

APPENDICES In order to prove this theorem, the following Lemma is needed.

Lemma J.l

G Ec .

Let F,G,H

I

G(y) dH(y) = /

F(t) d I

J —oo

Then

J—oo

F(t)G(t) dH(t) ,

xG R.

J—oo

Proof: This identity is triviality by ft dj

df' ^G(y)dH(y)

=

J

"

0 0

G(y)dH(y) ^) "'dH{t) =

G{t)dH(t),

which holds by the assumption. Proof of Theorem J . l : Using the identity Eq. J.l, we obtain that = r

f[Gj(x) •

^

Gl(t)...Gn-l(t)dGn(t)+

f

J—oo

Gn(t)d(G1(t)...Gn-i(t))

(J-2)

J—oo

for x G R. By the induction hypothesis f

Gn(t)d(Gi(t)...Gn-t(t))

=

J—oo

= f

Gn(t)J2d f

J — OO

_ j

G1(y)...Gj-1(y)Gj+1(y)...Gn-1(y)dGj(y),

J—OO

where now Go = Gn = I is understood. Applying Lemma J.l we obtain that

f

Gn(t)d(G1(t)...Gn-l(t)) =

J — OO

= fX J-oo

Gn(t)^G1(t)...Gj-1(t)Gj+1(t)...Gn.1(t)dGj(t). j=1

If this expression is substituted into Eq. J.2, we get the statement of the theorem. Corollary J . l

If GeEc

, then

I Gn-l{t)dG(t) J-oo

= -Gn(x), n

xeR

(n = l , 2 , . . . ) .

Proof: Applying Theorem J.l to G\ = . . . = Gn = G G E c , we get the statement of the Corollary.

APPENDIX

125

J

T h e o r e m J.2

LetO
Let

G,W=(ny)exp{jT^})°, with F,H G E c .

x>y

Then

f W) dGy(t) = alH{X) " H{V)] ' * " " " Proof: Let <£

:

XQ = x > X\ > . . . > XJV-I

> a;//

=

y

be a partition of the interval y < t < x. Using the notation

(* = 0,l,...,tf-l),

P* = -£PK the sum

s(z) = E ^ 4 iG»(**) - G»(**+i)i = E n^+iXw -1)

(J.4) (J-4)

is a Riemann-Stieltjes sum for the integral

lwJdGy{th

x > y

'

Since, according to Eq. J.3

f/

x [/'* > dH(t)\ dH(t)\

and since by the continuity and increasing property of the functions F and H there is a number {* with £fc+i < & < re* such that

Cd-m'm[HM-HMi-

ff(**+o],

we get the relations

F(xk+1)(p) k( ? * ■- 1) =

F(xk+1)

l a j

k

y ^ i + o(xk - xk+1) I (J.5)

=

F(xk+1)lajr--[H(xk)-H(xk+i)]

+ o(xk-xk+1)\

,

126

APPENDICES

where o(xk - Xk+l) £fc -

ft

>0,

.-

n

if

, , - v

z* - x^ + i - » 0 .

(J.6)

Xk+l

Substituting Eq. J.5 into Eq. J.4 we obtain that S(Z) = E

^ g ^ [H(xk) - H(xk+1)] + £

k=o

^k'

F(Xk+1) o(xk - xk+l)

.

k=o

Now for arbitrary number e > 0 a (5(e) > 0 can be determined in the following way. If xk-xk+i <S (A: = 0 , 1 , . . . , T V - 1 ) , then < | |jT|$*w.)-5(*)|
(J.7)

by integrability,

^1-W<5WPP*(Si

I*-"-1--*-1)

<J-8'

by the uniform continuity of F(^) in the interval y < t < x, and \o(zk - xM)\

< j-f

(J.9)

(k = 0,l,...,N-l)

by Eq. J.6. Using the inequality

l<

\a[H(x)-H(y)]-S(Z)\<

^ a £ (l - ^%T) [#(**) - * (**+>)] + £ F (^+i) i°(** - **+»)i, fc=o ^ from the relation

*

w

'

k=o

y^dGv(t)-a[H(x)-H(y))

A[-mdGAt)-s{z)

+

m»)i < . \a[H(x)-H(y)]-S(Z)\

by the help of Eq. J.7, Eq. J.8 and Eq. J.9 we obtain

f

F(t) dGy(t) = a[H(x) - H(y)] , Gy(t)

x>y

in conformity with (J.3). T h e o r e m J . 3 Let F 6 E c .

Then

F(y)exPyX^}=F(x),

x,y£R.

(3.10)

127

APPENDIX J

Proof: Under the assumption the statement is triviality by the following known identity. If g(x), x G R is a continuous function, and F G E c , then

f

g(F(t))dF(t)=

f

X

g(t)dt,

(J.ll)

JO

J-oo

where the right hand side is a Riemann integral. Indeed if g(t) = l/t we get

™->{«}-'HCTH by ( J . l l ) . Let ( ? G E C Corollary J . l ,

and Gn(x) = F(x), /

F1-a(t)dFa(t)

where n is a positive integer. Then, by = aF(x),

xeR

(J.12)

J — oo

with a = 1/n . As a generalization of Eq. J.12, we prove the following statement. Corollary J.2 Let F G E c . satisfying 0 < a < 1 .

Then Eq. J.12 holds for an arbitrary number

Proof: Let F(t) = y in (J.12). Then

T F1-"(t)dFQ(t) = a j J-oo

X

Jo

by (J.12) under the condition of the corollary.

yl-°ya-1 dy = aF(x)

a

This page is intentionally left blank

References and Bibliography [1] N. J. Akhiezer, 1965).

The classical moment problems (Hafner Publ. Co., New York,

[2] M. H. Alamatsaz, Completness and self-decomposability of mixtures. Ann. Inst. Statist Math. 35 (1983) 355-363. [3] Zhi Dong Bai and Chon Su, On the Lebesgue decomposition higher-dimensional infinitely divisible distributions. (Chinese) J. China Univ. Sci. Techn. 10 (1980) 76-95. [4] O. Barndorff-Nielsen, John T. Kent and M. S0rensen, Normal variance mean mixtures and ^-distributions. (French summary) Internat. Statist. Rev. 50 (1982) 145-159. [5] N. Bourbaki, Elements de mathematique. Livre VI. Integration. (Paris, Hermann k C** Editeur.) [6] C. Bruni and G. Koch, Identificability of continuous mixtures of unknown Gaus­ sian distributions. Ann. Probab. 13 (1985) 1341-1357. [7] A. Cauchy, Exercices d'analyse et de phys. math. (Deuxieme edition. Paris, 1981, Bachelier). [8] Satish Chandra, On the mixture of probability distributions. Scand. J. Statist. 4 (1977) 105-112. [9] Bradlex Efron and Ricard A. Olshen, How broad is the class of several mixtures? Ann. Stat. 6 (1978) 1159-1164. [10] D. A. S. Fraser, in Nonparametric methods in statistics. (John Wiley, New York, 1965) 164-167. [11] F. R. Gantmacher and M. G. Krein, Oszillationsmatrizen, Oszillationskerne und Heine Schwingungen mechanischer Systems. (Akademie Verlag, Berlin, 1960). [12] M. Girault, Les fonctions characteristiques et leurs transformations. In Publ. Inst. Stat. Univ. Paris (1945) 223-229. [13] B. Gyires, Contribution to the theory of linear combinations of probability distribution functions. Studia Math. Hung. 16 (1968) 297-324. 129

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[14] B. Gyires, On the superbonality of the strictly monotone increasing continuous probability distribution functions. In Proc. of the third Pannonian Symposium of Math. Stat. (Akad. Kiado, Budapest, 1982) 89-104. [15] B. Gyires, Egy matrixegyenlet megoldasa es ennek alkalmazasa valoszmusegi eloszlasfiiggvenyek linearis kombinacioinak elmeleteben. Alk. Mat. Lapok 9 (1983) 134-141. [16] B. Gyires, The mixture of probability distributions by absolutely continuous weight functions. Ada Sci. Math., Szeged 48 (1985) 173-186. [17] B. Gyires, An application of the mixture theory to the decomposition prob­ lem of characteristic functions. In Contribution to Stochastic. (Physica Verlag, Heidelberg, 1987) 137-144. [18] B. Gyires, Valosziniisegi eloszlasfuggvenyek felbontasarol. Alk. Mat. Lapok 14 (1989) 1-25. [19] V. M. Kruglov and A. M. Ulanovski, Mixtures probability distributions, which are determined uniquely by their behaviour on a half line. (Russian) Teor. Veroyatnost. Primenen. 32 (1987) 670-678. [20] G. Landsberg, Theorie der Element art eiler linearer Integralgleichungen. Math. Ann. 69 (1910) 131. [21] Austin F. S. Lee and John Gurland, On sample £-test when sampling from a mixture of normal distribution. Ann. Stat. 5 (1977) 803-807. [22] E. L. Lehmann, Theory of estimation (Mimeographed notes. University of Cal­ ifornia, 1950). [23] E. L. Lehmann, Consistency and unbiasedness of certain nonparametric tests. Ann. Math. Statist. 22 (1951) 165-180. [24] G. S. Lingappaiah, (1975) 403-411.

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Kozlemenyei

Index Absolutely continuous measure 36, 99 Associative-distributive property 6 Asymptotic decomposition 24, 40, 41, 45, 46, 47, 49 Commutative property 6 of scalar product 8 Convergence of a sequence in a convex metric space 11 Convex metric space 8 Convex space 6 Decomposability problems by a sequence of linearly indepen­ dent distribution functions 40, 45, 47, 49 by an orthogonal sequence of prob­ ability distribution functions 60, 62 if the weight functions are absolutely continuous distribution functions with square integrable density functions 29, 33 if the weight functions are discrete distributions with a finite num­ ber of discontinuity points 20, 22,23 if the weight functions are discrete distributions with infinitely many increasing points 23, 24, 55 in a special case of a family of char­ acteristic functions 39 in a totally convex metric space 16 in case of a family of distribution functions 36 in case of convolution 38 in the narrow sense 38, 52-54, 74, 87 132

of probability distribution functions 35 on the set of continuous distribu­ tion functions 70, 76 on the set of discrete distribution functions 84, 89, 94, 95 on the set of distribution functions, which are concentrated on a fi­ nite or infinite interval 63, 68 on the whole set of probability dis­ tribution functions 50 Decomposition of a distribution function 36, 79 theory of Lebesgue 36 trivial 79 Density function 37 Determinant theorem of Cauchy 101— 105, 107-109 Discrepancy function 16 Distance 8 Distribution function 6, 35 generalized 6 Distributive property of scalar product 8 Family of characteristic functions 36 Family of distribution functions 35, 97 Generalization of the Sherman-Morrison formula 112-115 Generalized binomial coefficients 106107 Generating function without zero state 42 Gram property 8 Hankelian totally positive sequence 121— 123 Hilbert-Schmidt kernel 29

133

INDEX Infinitely additive property 7 Integration in a totally convex metric space 14 Kernel function of the discrepancy func­ tion 17 Lebesgue decompositon 36 Lebesgue's convergence theorem 97 Lehmann statistics 83 Linear approximation by an orthogonal system of distribution functions 27,29 Linear approximation of distribution func­ tions 1 Linearly independent distribution func­ tions 39 Matrix identity 110-111 Measure of the decomposability 19 if the weight functions are discrete distribution functions 22, 24 in case of an orthogonal sequence of distribution functions 29 Metric on the set of distribution func­ tions 50 concentrated on a finite or infinite interval 64, 65 of the continuous distribution func­ tions 70 of the discrete distribution functions 85 Mixture of probability distribution func­ tions 1 of a family of characteristic func­ tions 36 of a family of distribution functions 35 Moment 38 Moment problem of Hamburger 117— 121 of Stieltjes (full) 122 Orthogonal system with respect to an element of a convex metric space 24, 26, 28 Positive (strictly positive) sequences 116

Positive sequences in Hankelian sense 116 Product representation of characteristic functions 79 of distribution functions 79 Scalar product defined on a convex space 8 defined on the set of probability dis­ tribution functions 50 Schwarz inequality 11 Superposition of probability distribution functions 1 Theorem of Landsberg 121 Totally associative-distributive property 7 Totally convex metric space 12, 70 Totally convex space 7 Totally positive (non-negative) matri­ ces 100 Totally positive sequence defined on a convex metric space 8 Transsignation of a matrix 101 Uniformly bounded scalar product de­ fined on a convex metric space 12 Uniformly minimum variance unbiased estimate 82, 83 Weight functions 1, 35