Selected Topics in Characteristic Functions 9783110935981

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M O D E R N PROBABILITY A N D STATISTICS SELECTED TOPICS IN CHARACTERISTIC FUNCTIONS

ALSO AVAILABLE IN MODERN PROBABILITY AND STATISTICS: Chance and Stability: stable distributions and their applications Vladimir V. Uchaikin and Vladimir M. Zolotarev Normal Approximation: New Results, Methods and Problems Vladimir V. Senatov Modern Theory of Summation of Random Variables Vladimir M. Zolotarev

MODERN PROBABILITY AND STATISTICS

Selected Topics in Characteristic Functions Nikolai G. USHAKOV

Russian Academy of Sciences

MYSPIN UTRECHT, THE NETHERLANDS, 1 9 9 9

Tel: +31 30 692 5790 Fax: +31 30 693 2081 E-mail: [email protected] Home Page: http://www.vsppub.com

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

© V S P BV 1999 First published in 1999 ISBN 90-6764-307-6

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands by Ridderprint bv,

Ridderkerk.

Contents Preface

vii

Notation

ix

1 Basic properties 1.1 Definition and elementary properties 1.2 Continuity theorems and inversion formulas 1.3 Criteria 1.4 Inequalities 1.5 Characteristic functions and moments, expansions of characteristic functions, asymptotic behavior 1.6 Unimodality 1.7 Analyticity of characteristic functions 1.8 Multivariate characteristic functions 1.9 Notes

1 1 4 8 23

2 Inequalities 2.1 Auxiliary results 2.2 Inequalities for characteristic functions of distributions with bounded support 2.3 Moment inequalities 2.4 Estimates for the characteristic functions of unimodal distributions 2.5 Estimates for the characteristic functions of absolutely continuous distributions 2.6 Estimates for the characteristic functions of discrete distributions 2.7 Inequalities for multivariate characteristic functions 2.8 Inequalities involving integrals of characteristic functions . . . . 2.9 Inequalities involving differences of characteristic functions . . . 2.10 Estimates for the first positive zero of a characteristic function . 2.11 Notes ν

39 43 52 54 64 67 67 81 88 95 100 Ill 114 133 142 154 158

3 Empirical characteristic functions 3.1 Definition and basic properties 3.2 Asymptotic properties of empirical characteristic functions . . . 3.3 The first positive zero 3.4 Parameter estimation 3.5 Non-parametric density estimation I 3.6 Non-parametric density estimation II 3.7 Tests for independence 3.8 Tests for symmetry 3.9 Testing for normality 3.10 Goodness-of-fit tests based on empirical characteristic functions

159 159 164 182 187 198 210 226 232 242 251

3.11 Notes

256

A Examples

259

Β Some characteristic functions

281

C Unsolved problems

331

Bibliography

335

Subject Index

351

Author Index

353

vi

Preface Characteristic functions play an outstanding role in the theory of probability and mathematical statistics. Due to this reason, there are a number of excellent books concerning the mathematical theory of characteristic functions and their applications. But, at the same time (and due to the same reason), the abundance of theoretical results on characteristic functions and variety of their applications is so great that one book (of reasonable volume) can hardly hold all of them. Moreover, this field is intensively developing, and many results which have recently been obtained are out of existing monographs. Inequalities for characteristic functions and the theory of empirical characteristic function are among those parts of the theory of characteristic functions that are inadequately slightly reflected in monographs. This book is supposed to fill this gap. The monograph consists of three chapters. Chapter 1 contains basic general results concerning characteristic functions. Usually, proofs are given only for those (mainly recent) results which are not included in existing textbooks and monographs. The main goal of this chapter is to be a handbook on characteristic functions. At the same time, it contains some new results. Various inequalities for characteristic functions are presented in Chapter 2 (both in univariate and multivariate cases). Lower and upper bounds for characteristic functions are given both in the neighborhood of the origin and for large values of the argument. The estimates involve such characteristics as moments, the maximal value or the total variation of the probability density function, etc. Some inequalities involving integrals of the characteristic function are given in Section 2.8. In Section 2.9, recent results are presented concerning the problem of estimating the closeness of distribution functions in terms of the closeness of the corresponding characteristic functions. The last section is devoted to the problem of estimating the position of the first zero of a characteristic function. In Chapter 3, we study empirical characteristic functions and their applications in statistics. We begin with the statistical properties of empirical characteristic function as estimators of the underlying characteristic function. The results of this part of the chapter are the theoretical foundation for the statistical inference based on empirical characteristic functions which is previi

sented in the remainder of the chapter. Some results are presented on statistical estimation (both parametric sind nonparametric) and on testing statistical hypotheses. A collection of various examples, counterexamples and assertions, demonstrating interesting (sometimes unexpected) properties of characteristic function, is presented in Appendix A. Constructing counterexamples permanently accompanies the investigation of characteristic function and their applications and sometimes leads to beautiful and intriguing results. Such results are often not only of theoretical interest but helpful in solving applied problems as well. Many examples have already been included in monographs concerning counterexamples in probability theory and statistics and in those concerning characteristic functions. However, recently quite a number of new interesting examples have appeared in various publications and the goal of the appendix is to bring together and systematize all known counterexamples. In Appendix B, we give formulas and graphs of some frequently used characteristic functions as well as characteristic functions demonstrating interesting properties. In Appendix C, several unsolved problems are presented. I am very grateful to Yu.V. Prokhorov, D.V. Belomestnii, A.V. Kolchin, V.Yu. Korolev, and A.P. Ushakova for many useful comments. N. G. Ushakov Moscow, May 1999.

viii

Notation Throughout the book, a triple numeration is used: the first digit denotes the number of a chapter, the second one denotes the number of the section in the chapter and the last one denotes the number of an item (theorem, lemma, formula) in the section. For a complex number ζ, ζ denotes the complex conjugate. R m is the mdimensional Euclidean space. 91 and 3 denote the real and imaginary parts of a function, respectively; f(x+0) and f(x — 0) are respectively the right-hand and a.s.

Ρ

D

left-hand limits; —>, and —» denote, respectively, the convergence almost surely, convergence in probability and convergence in distribution; a=' means that an equality holds with probability one. The weak convergence is denoted LimF„Gc) = F(x)

71-» OO

or Fn-+F.

If A is an event, then I a is the indicator of A; the symbol * denotes convolution. Where the multivariate case is considered, m always denotes the dimension of the space. The sign • means the end of a proof.

ix

1

Basic properties of the characteristic functions In this chapter, we summarize basic results concerning characteristic functions. Most of these results are given without proof because the proofs can be found in many textbooks and monographs: see (Cramer, 1962; Cuppens, 1975; Feller, 1971; Galambos, 1988; Gnedenko & Kolmogorov, 1954; Ibragimov & Linnik, 1971; Kawata, 1972; Linnik & Ostrovskii, 1977; Loeve, 1977; Lukacs, 1970; Lukacs, 1983; Lukacs & Laha, 1964; Petrov, 1975; Prokhorov & Rozanov, 1969; Ramachandran, 1967). Usually, proofs are presented for recent results which are not contained in the mentioned books.

1.1. Definition and elementary properties Let X be a real-valued random variable with distribution function F(x). The characteristic function f(t) of the distribution function F(x) (or of the random variable X) is defined as oo

/

ltx itX -oo e dF(x) = Ee ,

- o o < t < oo.

If Fix) is absolutely continuous with density p(x), then oo

/

-oo

eltxp(x)dx.

I f X is discrete with P(X = x n ) = p n , η = 1,2,...,

- 1, then

oo

Μ =

Σρηβ**. n=l

Characteristic functions of the most frequently arising distributions can be found in Appendix B. 1

1. Basic properties

2 THEOREM 1 . 1 . 1 .

Any characteristic function f{t) satisfies the following condi-

tions: (a) fit) is uniformly continuous; (b) /(0) = 1; (c) 1/(01 < 1 for all real t; (d) f(—t) = f(t) where the horizontal bar denotes the complex conjugate. Two distribution functions are identical if and only if their characteristic functions are identical. THEOREM 1 . 1 . 2 ( u n i q u e n e s s t h e o r e m ) .

According to this theorem, there is one-to-one correspondence between characteristic functions and distribution functions. A distribution is said to be a lattice distribution if it is concentrated on a set of the form {a + nh, η = 0, ±1, ±2,...} where a and h are some real numbers, h > 0.

A characteristic function f(t) corresponds to a lattice distribution if and only if |/(io)| = 1 for some to Φ 0.

THEOREM 1.1.3.

Note that, due to Theorem 1.1.3, any periodic characteristic function is the characteristic function of a lattice distribution. The converse is false (see Appendix A, Example 3). In fact, the following theorem holds true. Let fit) be a characteristic function. Then the following statements are equivalent: THEOREM 1.1.4.

(a) there exists ίο > 0 such that /(ίο) = 1/ (b) fit) is periodic with period ίο; (c) the distribution function Fix), corresponding to fit), is purely discrete, and all its points of discontinuity belong to the set {nh, η = 0, ±1,...} where h = 2π/ίο· COROLLARY 1 . 1 . 1 . A characteristic function fit) corresponds to a lattice distribution if and only if there exists a real number a such that the function emi/(i) is periodic.

A characteristic function fit) corresponds to a discrete distribution if and only if it is almost periodic. This immediately follows from the definition of an almost periodic function.

1.1. Definition and elementary properties

3

THEOREM 1.1.5. If fit) is the characteristic function of a discrete distribution, then lim sup |/(ί)| = 1.

(1.1.1)

|ί|-»οο

1.1.6. If fit) is the characteristic function of an absolutely continuous distribution, then THEOREM

lim |/(f)| = 0.

(1.1.2)

In Theorems 1.1.5 and 1.1.6, the converse is not true: there exist singular distributions whose characteristic functions satisfy (1.1.1) or (1.1.2) (see Appendix A, Examples 6 and 7). THEOREM

1.1.7. A characteristic function is real if and only if it is even.

A distribution function Fix) is called symmetric if Fix) = 1 — Fi—χ — 0). 1.1.8. A distribution function is symmetric if and only if its characteristic function is real and even. THEOREM

1.1.9. Let fit) be the characteristic function of a random variable X. Then the characteristic function of the random variable aX + b, where a and b are real numbers, is fiat)eibt. THEOREM

THEOREM 1.1.10. Let X and Y be independent random variables with characteristic functions fit) and git). Then the characteristic function of Χ + Y is fit)git). Note that the independence is not a necessary condition in this theorem: there exist dependent random variables such that the characteristic function of their sum is the product of characteristic functions of summands. In terms of distribution functions, Theorem 1.1.10 means that the characteristic function of the convolution of distribution functions is the product of the corresponding characteristic functions. The converse is also true. 1.1.11 (convolution theorem). A distribution function Fix) is the convolution of two distribution functions F\ix) and F2ix):

THEOREM

Fix) =

Fxix - y)dF2iy) =

if and only if fit) = f\it)f2it), where fit), fiit) functions of Fix), F\(x) and F2ix) respectively.

F2ix - y)dF1(y), and f2it) are characteristic

4

1. Basic

properties

The convolution theorem implies, in particular, that the absolute value of the characteristic function of the convolution of two distributions decreases as \t\ —> oo faster than the characteristic functions of the components. This explains why the convolution is always 'smoother' (or at least no less smooth) than each of the components: if distribution functions F\(x) and F^ix) are absolutely continuous, then their convolution F(x) is also absolutely continuous; if F\{x) and F2(x) have continuous densities, then the density of F(x) is continuous; if the densities of Fi(x) and /^Ot) are η times differentiable, then the density of F(x) is no less than η times differentiable, etc. In addition, the maximum value, maximum of the derivative modulus, and the total variation of the density of F(x) do not exceed those of the densities of F\(x) and F2(x). On the other hand, it follows from the convolution theorem that the characteristic function of the convolution tends to 1 as |i| —» 0 more slowly than the characteristic functions of the components. This explains why the tail of the convolution is always 'heavier' than those of the components.

1.2. Continuity theorems and inversion formulas Let F\(X),F2(X), ... be a sequence of distribution functions. By definition, this sequence converges weakly to a distribution function F(x) if lim Fn(x) = F(x) n—>00

for all continuity points χ of F(x). The weak convergence is denoted Lim Fn(x) = F(x)

n-too

or

w

FN^F.

THEOREM 1.2.1 (continuity theorem 1). Let Fi(x), i ^ t o , ...be

a sequence

of dis-

tribution functions and f\(t),f2(t),... be the corresponding sequence of characteristic functions. The sequence F\(x),F2(x),... converges weakly to some distribution function F{x) if and only if the sequence f\(t), /2W,... converges at all points to some function f(t) which is continuous at zero. In this case, fit) is the characteristic function ofF(x). COROLLARY 1.2.1. Λ sequence FI(X),F2(.X),... of distribution functions converges weakly to a distribution function F(x) if and only if the corresponding sequence of characteristic functions f\(t), /2(f),... converges to the characteristic function f(t) ofF(x) at all points t. COROLLARY 1.2.2. If a sequence of characteristic functions fi(t),f2(t),... converges to some function f(t) at all points, and f(t) is continuous at zero, then f(t) is a characteristic function.

1.2. Continuity theorems and inversion formulas

5

Note that a sequence of characteristic functions may converge at all points to a function which is not a characteristic function (see Example 28 of Appendix A). THEOREM 1.2.2 (continuity theorem 2). Let F\ix), F^ix), ...be a sequence of distribution functions and fiit),f2it),... be the corresponding sequence of characteristic functions. The sequence Fi(x),F%(.x),... converges weakly to some distribution function Fix) if and only if the sequence /ι(ί),/2M,... converges uniformly on each bounded interval to some function fit). In this case, fit) is the characteristic function of Fix).

Theorem 1.2.2 means that the one-to-one correspondence between characteristic functions and distribution functions is continuous if the topology of weak convergence and the topology of uniform convergence on each bounded interval are taken in spaces of distribution functions and characteristic functions respectively. Theorems 1.2.1 and 1.2.2 imply, in particular, that if a sequence of characteristic functions /i(£), fiit),... converges to a characteristic function fit) at all points, then it converges uniformly on each bounded interval. COROLLARY 1.2.3. Let Fix),F\ix),F2ix),... be a sequence of distribution functions and fit), fiit), fzit),... be the corresponding sequence of characteristic functions. The sequence F\ix), F2ix),... converges weakly to Fix) if and only if

for any a limoo lim Λ—>0 THEOREM 1.2.5. Let F(x) be a distribution function with characteristic function fit). Then

for any χ being a continuity point

ofF(x).

We can use the inversion formula of Theorem 1.2.5 in all points if redefine F(x) at discontinuity points as F(x) =

F(x + 0) + F(x - 0)

2 For absolutely continuous distributions, inversion formulas for the distribution functions are supplemented by the inversion formula for the density function. THEOREM 1.2.6. Let fit) be an absolutely integrable characteristic function. Then the corresponding distribution function F(x) is absolutely continuous, its density function p(x) is bounded and continuous, and (1.2.1)

Note that this theorem implies that if f(t) is an absolutely integrable complex-valued continuous function such that the function

is a probability density function, then f(t) is a characteristic function, namely, the characteristic function of the distribution with density p(x). COROLLARY 1.2.4. Let f(t),fi(t),f2(t),... be a sequence of integrable characteristic functions, p(x),pi(x),p2(x), ...be the corresponding density functions. If

then the densities pn(x) uniformly converge to p{x): s u p \pnhc) — p(*)| —» 0

χ

as

η —> oo.

1.2. Continuity

theorems and inversion

7

formulas

If the characteristic function of an absolutely continuous distribution is not integrable, then Theorem 1.2.6 cannot be applied for the calculation of the density function. In this case, some modifications of (1.2.1) can be used, for example, pix) = lim [ e~itx (l - ξΠ fit) dt. oo 2π J-τ \ ΤJ Let f(t) be the characteristic function of a lattice concentrated on a set {a+nh, η = 0, ±1, ± 2 , . . . } . Denote jumps of the function in points a + nh by pn. Then THEOREM 1 . 2 . 7 .

h

distribution distribution

Γπfa

pn = ^ e~l(a+nh)t fit) dt. Απ J-nth THEOREM 1 . 2 . 8 .

LetF{x) be a distribution function with characteristic

function

f(t), then Fix) - Fix - 0) = lim - ί f e~itxfit)dt Γ—>oo 2,1 J-T for any t. To conclude the section, we present several identities being useful in many situations. THEOREM 1.2.9 (Parseval). Let Fix) and Gix) be distribution functions characteristic functions fit) and git) respectively. Then oo roo

/ -oo e~itxgix)dFix)=

/ fiy J—oo

with

— t) dGiy).

The theorem implies, in particular, that OO fOO / -oo fit)dGit)=

J/ — oo

git)dFit).

THEOREM 1.2.10 (Parseval-Plancherel). Let fit) be the characteristic function of an absolutely continuous distribution with density function pix). Then \ fit)\2 is integrable if and only ifp2ix) is integrable, and, in this case, oo

1

/ -oo p\x)dx=—

r oo J—oo

\fit)\2dt.

A discrete analog of this theorem can be formulated as follows. THEOREM 1 . 2 . 1 1 .

Let Fix) be a distribution

Then oo

n=1

,

-T

function with jumps p\, Ρ2, ••••

1. Basic properties

8

If F(x) has only a finite number of jumps, then the series at the left-hand side should be replaced by the finite sum. COROLLARY 1.2.5. A distribution function F(x) is continuous if and only if the corresponding characteristic function f{t) satisfies the condition

Another probabilistic variant of the Parseval-Plancherel theorem concerns the difference of characteristic functions. 1.2.12 (Parseval-Plancherel). Let p(x) and q(x) be two probability density functions with characteristic functions f(t) and g(t) respectively. Then THEOREM

OO /

-OO

1 roo

\p(x) - q(x)]2dx = ^— / OG |/(f) - g(t)\2dt, ^ J—

provided that the integrals exist.

1.3. Criteria We start with some general necessary and sufficient conditions, and then present various sufficient conditions for a function to be the characteristic function of a probability distribution. We also give some methods of constructing characteristic functions satisfying given properties. At the end of the section, some necessary conditions are presented which supplement necessary conditions given by Theorem 1.1.1. Other necessary conditions (in the form of inequalities) are given in Section 1.4. Criteria related to unimodal distributions will be given separately in Section 1.6. A complex-valued function f(t) of the real variable t is called non-negative definite if it is continuous, and the sum NN

_

j=l k=l is real and non-negative for any positive integer Ν and any real t\,t2, •••Jn and complex ξι, ξ2, •••, ζ,Ν1.3.1 (Bochner-Khintchine). A complex-valued function f(t) of the real variable t is a characteristic function if and only if THEOREM

(a) fit) is non-negative definite; (b) f(0) = 1.

1.3. Criteria

9

THEOREM 1.3.2 (Cramer). A bounded and continuous function f(t) is a characteristic function if and only if (a) /(0) = 1, fA r A (b) / fitJo Jo

s)e"(i~s) dtds > 0 for all real χ and all A > 0.

THEOREM 1.3.3 (Khintchine). A complex-valued function fit) of the real variable tisa characteristic function if and only if there exists a sequence ofcomplexvalued functionsg\it),g2it),..., satisfying the condition oo

/ -oo \gnit)\2dt = 1,

and such that

oo

/ -oo gnit + s)gn(s)ds uniformly in each bounded interval.

For absolutely continuous distributions, this criterion has a simpler form. THEOREM 1.3.4. A complex-valued function fit) of the real variable t is the characteristic function of an absolutely continuous distribution if and only if it can be represented in the form oo

/ -oo g(t + s)g(s) ds

where git) is a complex-valued function satisfying the condition oo

/ -oo \git)\2dt = 1.

The Khintchine criterion is very useful in constructing examples of characteristic functions satisfying given conditions, for instance, those vanishing on some sets, say, outside a neighbourhood of the origin. An advantage of the Khintchine criterion over the Polya one (see Theorem 1.3.8 below), which is also often used in constructing such functions, consists in the possibility to obtain smooth characteristic functions, i.e., distributions having as many moments as it is required, whereas all distributions satisfying the Polya criterion do not have expectation. A special case of the Khintchine criterion (Zolotarev, 1967) gives characteristic functions vanishing outside the interval [—1,1]: letg(i) be a real-valued even function such that git) = 0 for |i| > 1/2 and f^g2^) = 1; then the convolution oo / -oo git + s)gis) ds

1. Basic properties

10

is the characteristic function of the absolutly continuous distribution with the density 1/2 2 ( rr 1/2 ν p(x) = " ( / g(t)cos(tx)dt) . A criterion similar to Theorem 1.3.4 (Berman, 1975) is formulated in terms of the covariance function. A complex-valued Borel functioni?(s, t), —oo < s,t < oo is a covariance function if for any positive integer Ν and any real Si, ...,SN and u\,..., UN, Ν Ν THEOREM 1 . 3 . 5 .

i=1 j=1 Let R(s, t) be a covariance function satisfying the condition

/

oo

oo.

R(s,s)ds< —oo

(1.3.1)

Then the function f(t)=J-~

(1.3.2)

m

J_00R(s,s)ds

is the characteristic function of an absolutely continuous distribution. Conversely, if fit) is the characteristic function of an absolutely continuous distribution, then there exists a covariance function R(s,t) satisfying (1.3.1), such that fit) is representable as (1.3.2). PROOF. Let Ris, t) be a covariance function satisfying (1.3.1). Consider a complex Gaussian process Xit), —oo < t < oo, associated with Ris, t), i.e., such that EX(t) = 0 for all t and EX"(s)X(i) = Ris, t) for all s, t (such a process exists due to the Kolmogorov existence theorem). It follows from (1.3.1) thatX(f) belongs to L2 almost surely, and so there is a measurable version of the Fourier transform process oo /

-00

elusXis)ds,

-oo 0;

dt tf(f)-/(-*)]t

12

1. Basic

(e)

t h e r e

e x i s t s

k

o s u c h

t h a t

f o r

a n y

properties

k > k o

a n d .

r°° (sgnx)',Α+1

a n y

χ

f ( m t ) d t

J-oo (x + iii t

Φ Ο ,

(1.3.4)

>0. +

f

1

To prove the theorem, we need two lemmas. LEMMA 1.3.1. N > 14

(a)

1 N — /r

πι J-ν (b)

F o r

a n y

F o r

a n y

Ι — ζ

r e a l

χ

ϊ

r e a l

N

a n y

u

e

Ν

PROOF.

i

e

N

e

=

J - N

|θχ | < c a n d

c o m p l e x

ζ

s u c h

t + i x

16*21 < c ,

3z * 0 a

t h a t

n d

a n y

Ν

> 1 ,

d t

/ w h e r e

Φ 0 , a n y

_ . dJ t = e (sgnx + sgnSz) + Θχ

0 a n d

f

χ

i

~

N

+ Ν

+ i xx

c i s a

+ 02 Ν

— ii xx

n a b s o l u t e

1 +

χ

2

'

c o n s t a n t

Due to the Cauchy theorem, for Ν > \z\, 1 2 π ι

f

N

J - ν

e

i x i

d t

Ζt —

z

1

1

= ö2 e

f

(sgnx + s g n S z ) - — / 2πι J r

»

e

ν

ζ

C —

z

Here Γ^τ is a half-circle of radius Ν centered at zero (the upper half-circle if χ > 0, and the lower one if χ < 0). In the first case (x > 0), for instance, π 1

f N rζ — Ζ^ Jr

-

eixNe*Νί(,ίφ

Ν

-άφ

f Jo

N e

i (

P

-

Ν

ζ

,—Νχ sinνάφ.

z Jo

-

Obviously, Γ , —Nx sinψάφ < π, Jo

and, since sin φ >

π

-φ

for 0 < φ < π/2, ρπ

/ β~Νχ"ϊηφάφ Jo

Hence

ρη!2

rπ/2

e~Nxsin(pd(p
0, that g\t&) —> g(x) as Μ —» oo uniformly in each bounded interval (we use the boundedness of f(t) in the first integral and the Abel test in the second one). Therefore rN /

J-N

. elxyg(x)dx=

fN lim

/

M-kjo J-N

. elxygM(x)dx=

foo /

J-oo

fNy f(t)dt

eiudu

—.

J—Ny " + ltJ

14

1. Basic

properties

Let us take the limit of the right hand side as Ν —» oo. By virtue of of Lemma 1.3.1 (N > l/|y|), •Ny

iu e

eiNy

du

e~iNy +

-Ny u + ity

UN + it)y

eN(y,t)' 1+

UN - it)y

t2y2'

In view of the boundedness of fit) and the Lebesgue theorem, d t

lim / fit)QNiy,t) N^oo y_oo

2 2

1+ty

f°° ,· dN(y,t) = / f i t ) lim dt J-oo N->oo 1 + t2y2

(the existence of the limit in the right-hand side follows from the previous equality). Thus, it remains to verify that roo

(

lim /

f(t)

eiNy ——

N—>oo oo

e-iNy

+ lt

\

roo

r \dt - /

+ Ν -It

J

(

J-oo

iN

giNy

f i t ) lim

e~

y

\

- + ——r

N->oo yN + lt

dt.

Ν - it J

The right-hand side of this relation is equal to zero, and the left-hand side is l i m \giN)eiNy N—>oο

-gi~N)e~iNy}

= 0.

Thus, by virtue of Lemma 1.3.1, oo

/

-oo

roο elxygix)dx

/·οο coo

= / f(t)dt J — oo

oo

pixy pixy

/ rdx J — oo X + it

/ -oo fit)etyisgny

- sgn

t)dt.

•

PROOF OF THEOREM 1.3.6. The necessity of conditions (aHd) is obvious (see

(Lukacs, 1970). Let us prove the necessity of condition (e). Let oo / -oo eixtdFix)

for some distribution function Fix). For any realy * 0, by virtue of the Fubini theorem gM(y)

fM fit)dt f°° rM eltx = / ^ = / dFix) / -dt. J-M y + it J-oo J-M y + it

By Lemma 1.3.1 iz = iy), the inner integral is bounded by a constant depending only on y and has a limit as Μ —> oo; therefore, for y * 0, giy) = v.p. f

J-ooy

f^dt

+ it

_

π

f

J—oo

e -^( S gna:

- sgny)dF(x),

15

1.3. Criteria

which implies (for y > 0 and y < 0, respectively) roo

g(y) = 2π / Jo

(1.3.5)

e~yxdF(x)

and Γ0 g(y) = —2π / e~yxdF(x). J—oo

(1.3.6)

Therefore, for any k > 0 and y ϊ 0, ( - l ) V % 0 s g n y >0 (see (1.3.4)). The necessity is thus proved. Let us turn to the proof of sufficiency. First note that if g(x) is bounded on the positive half-line and there exists η > 2 such that for any χ > then g (n_1) (:r) < 0 for χ > 0. Indeed, for any > 0 and x>xq,

V

v=0

'

which implies that o) < 0. Therefore, in view of the conditions of the theorem (the boundedness of g(x) was proved in the proof of Lemma 1.3.2), for any k > 0 and any χ * 0, (-l)y»(*)sgnx>0. From the theorem on the completely monotone functions (Feller, 1971), taking into account that one-side limits of g(x) exist at zero, we conclude that there exist bounded increasing functions Fi(x) and i^Oe) such that roo

g(t)=

/ Jo

e'^dF^x), r0

g{t) = - / e~txdF2(x), J—oo

t> 0,

(1.3.7)

t < 0.

(1.3.8)

Without loss of generality, we may assume that Fi(0) = ^ ( 0 ) . Set Fix) =

^-F\(x) 2π

Fix) =

J-F2ix) 2π

f o r x > 0 and

for χ < 0. Then the function oo

/

-oo

eUxdFix)

1. Basic properties

16

is a characteristic function. Indeed, fo(0)=

lim - ί ( Γ e-txdF1(x)+ ί->ο+ο 2π \Jo

f° etxdF2{x)\ J-oo J

1 ,· r , s , s, 1 ,· f°° ^ 2tdx = — lim \g(t) -g(-t)] = — lim / fix)-, 2π *->o+o 2π t~>o+o j^QO ^ + ^ 1 , r°° ft\ du - - lim / / 2= =1

In view of (1.3.5H1.3.8), for t Φ 0, ν·Ρ· /

J-OO

f0(t)dt r- = g(x). X + lt

Now it suffices to prove that fit) = fait), i.e., thatg(i) uniquely determines fit). Assuming git) = 0, we, due to Lemma 1.3.2, obtain Γ fi-tsgny)e-tMdt Jo

=0

for anyy Φ 0, or after the substitution t = ln(l/u), f1 / /(lnusgny)u w_1 du = 0. Jo Considering this relation for all complex y Φ 0 and taking into account that fit) is continuous, we obtain fit) = 0. • A number of sufficient conditions for characteristic functions can be obtained if we study how characteristic functions change under some transformations of the corresponding random variables and distribution functions. Let fit) be a characteristic function. Then the following functions are also characteristic functions: THEOREM 1.3.7.

(a) fit) (or, which is the same, fi—t)); (b) fnit) for any non-negative integer n; (c) */(*); (d) \fit)\2; (e) elbtfiat) for any real a and b. These characteristic functions correspond to the following transformations:

1.3. Criteria

17

(a) ifX is a random variable with characteristic function fit), then fit) is the characteristic function of —X; (b) if Xi, ...,Xn are independent identically distributed random variables with the common characteristic function fit), then fnit) is the characteristic function of the sum X\ + ... + Xn\ (c) if Fix) is a distribution function with characteristic function fit), then 9ί/(ί) is the characteristic function of the distribution function + Fix)-Fi-x0)); (d) if X\ and X% are independent identically distributed random variables with characteristic function fit), then X\ —X 3. Without loss of generality, we can assume n - 2 < 7i[9t/(f)]B. Since η > 3 we have 1 < η[91/(ί)] η_1 and 0 < Slf(t). Consider t h e function γ(ζ) = zn — an — (ζ — ά)ηαη~ι for \z\ < 1 a n d 0 < α < 1 < nan~l

a n d 1 < η — 2 < nan.

(1.4.3) An elementary

computation shows t h a t y(z) > 0 for \z\ < 1. L e t u s s e t a = 3if(t), ζ = c o s ( t x ) i n (1.4.3). T h e n cos n (fce) > Wf(t)]n

+ [cos(tx) -

XfmnWm]"-1.

After integration we obtain

Γ J —I

c o s n ( t x ) d F ( x ) > [9l/(i)] n .

25

1.4. Inequalities

Then from (1.4.1) it follows that oo

/

or

-oo [η

- 1 + cos(ntx)]dF(x) > n[3if(t)]n

1 - ö 2 [l-9l/(i 0 )]· Indeed, consider the characteristic function f(t) = cos t and set ίο = 2π. Then 0 = b2[l - cos 2π] =b2[ 1 - SH/(i0)] < 1 - cos2rc6 = 1 - |/(ίι)||/(ί2)| - (1 - N l W l

- l/M 2 ) 1/2 ·

(1-4.4)

PROOF. Set ίο = 0- By virtue of the Bochner-Khintchine theorem, / 1 f(~t 1) det /(ίι) 1 \fit2) f(t2-h)

/(-i2) \ fit 1 - i 2 ) > 0, 1 J

i.e., 1 + /(-il)/(i2>/(il - t2) + f(h)f(-t2)f(t2

- ίι) - ΐ/(ίι)ΐ2 - \m\2

- i/cii - w i2 > 0,

which yields 1 + 2|/(ίι)||/(ί2)||/(ίι - ί2)| - |/(ίι)|2 - |/(ί2)|2 - |/(ίι - ί2)|2 > 0. (1.4.5)

1. Basic properties

26 Let us demonstrate that

|/(*1 - 1 2 ) ι > |/(ίι)||/(ί2)| - (1 - |/(ίι)| 2 ) 1/2 (ΐ - |/(ί2)|2)1/2.

(1.4.6)

Indeed, if |/(ίχ - t2)\ > |/(fi)||/(f2)|, then (1.4.6) is obvious. If |/(ix - t2)\ < |/(fl)||/(f2)|, then (1.4.6) is equivalent to (1.4.5) (this can be verified directly) and therefore is again true. Now replacing t2 by —t2 in (1.4.6) and taking into account that |/(£)| = |/(-f)|, we finally obtain (1.4.4). • COROLLARY 1.4.2. Let f(t) be a characteristic function. If |/(fi)| > cos q>\ and \f(t2)I > cos ψ2, where φι>0,ψ2>0, and φι + q>2< π/2, then |/(ίι +1 2 )I > cos( cos φ, then \f(nt)\ > cos(nq>). From Theorem 1.4.2, we obtain the following interesting assertion. LEMMA

1.4.1. Let f{t) be a characteristic function. Suppose that |/(*i)| > 1 - ct\

(1.4.7)

|/(f2)| > ι - Ct\

(1.4.8)

and

where c is a positive constant, and t\ > 0 , t 2 > 0. Then |/(ii +12)I > 1 - citι + t2f.

(1.4.9)

The sign V can be replaced by ">' simultaneously in all three inequalities (1.4.7M1.4.9). PROOF. Without loss of generality we may assume that t\ < 1/c and t\ < 1/c. Otherwise the right-hand side of (1.4.9) is negative, and the assertion of the lemma is obvious. Suppose the contrary: let (1.4.7) and (1.4.8) hold but (1.4.9) do not hold, i.e., |/(fi +12)I < 1 - ciH +1 2 ) 2 .

(1.4.10)

27

1.4. Inequalities

From (1.4.7) and (1.4.8), taking into account that 1 - ct2 > 0 and 1 - ct\ > 0, we obtain l/(ii)ll/(i 2 )i - ( i - \f(h)\ 2 ) V 2 a

-1m\2)V2

> (1 - ct\){ 1 - ct2) - [1 - (1 - ct2)2]y2[l - (1 - ct2)2\v2 = 1 - ct2 - ct2 + c 2 i f i i - (2c*2 - c 2 it) 1/2 (2cii - c 2 ^) 172 , while (1.4.10) means that |/(*i + t2)I < 1 - ct\ - ctj ~ 20*1*2> therefore, making use of Theorem 1.4.2, we obtain c 2 i 2 i 2 + 2c*ii 2 < (2ci2 - c2i^)1/2(2ci| -

c2t$)y2,

which is equivalent to (fι + t2)2 < 0, which is impossible. The contradiction proves (1.4.9). The case '>' is proved in a similar way. COROLLARY 1.4.3. Let f(t) be a characteristic function. implies |/(rafo)| > 1 — c(nto)2 for any η = 1,2,...

• Then |/(ίο)| > 1 - ci^

THEOREM 1.4.3. Let f(t) be a characteristic function. Then the following assertions take place. (1) If\f(t) I > 1 — ct2 (c is a constant) in some neighborhood of the origin, t Φ 0, then |/(ί)| > 1 — ct2 for all t. The assertion remains true if'>' is replaced by ">' in both inequalities. (2) I f R f ( t ) > 1 — ct2 in some neighborhood of the origin, t * 0, then 91fit) > 1 — ct2 for all t. The assertion remains true if'>' is replaced by ">' in both inequalities. PROOF. Assertion 1 immediately follows from Lemma 1.4.1. Let us prove assertion 2. From the left inequality of Theorem 1.4.1, setting η = 2, we obtain m

t

)

f

< l ^ l Ζ

(1.4.11)

for all t. Suppose that > 1 — ct2 in some neighborhood of the origin but not everywhere. Let i—b, b) be the widest interval such that 3\fit) > 1 — ct2 for

28

1. Basic

properties

t Φ 0. Then 0. Then |/(i)| < 1 - ^ t for |i| < b. PROOF. Suppose the contrary: let (1.4.13) do not hold. Then there exists ίο e (0, b] such that l/M > 1- ^ t l

(1.4.14)

Let η be the nonnegative integer satisfying the relations b b —i< i 0 ^ —• 2 n+1 2n Since 2 n+1 io > b, we obtain |/(2 n + 1 i 0 )| 1 - ^ ( r t which contradicts (1.4.15).

0

)

2

>1- V

( 2 - ^ )

=c •

In connection with Theorem 1.4.4, the question arises whether the coefficient 1/4 at the term — ^ r t 2 can be improved (made larger) or not. It is clear that this constant cannot be larger than 1.

1.4. Inequalities

29

THEOREM 1.4.5. For any characteristic function f(t), -

|3/(f)|
0, k = 1,2,...) such that tk —> 0 and —> 0 as k —> 00, we have to construct a sequence of characteristic functions /i(£),/2W, ···, whose distributions have zero expectations and the same variances a 2 , and such that >1— for k large enough. Let us set fkit) =Pk + i 1 - Pk) cos ( \y/l

g

-pk

tk

30

1. Basic

properties

where 1 > pk > 1 — . Then, as is easy to see, the variances of the distributions corresponding to the characteristic functions fkit) are equal to a 2 , and, for k large enough, fk(tk) > Pk > 1 · Now we present several so-called truncation inequalities connecting the behavior of the tail of a distribution function with the behavior of the corresponding characteristic function in the neighborhood of the origin. Let Fix) be a distribution fit). Then for any 0 < α < 2π, THEOREM 1 . 4 . 7 .

ι

x2dFix)
2c n (l — cosy) -— 11

71

therefore Γχ c fx Ψ„(χ) = / ψη(ιι) 0, 2ϋι(π - y ) ( l - cosy)"

1.5. Moments, expansions, asymptotics i.e., (1.4.28) is proved. Relation (1.4.28) means that the degenerate distribution in π. This implies

39

weakly converges to

2π / ψη{χ)Κ{χ)άχ -> 2πΛ(π) = π. Jo Combining this relation with (1.4.27), we obtain (1.4.24).

•

1.5. Characteristic functions and moments, expansions of characteristic functions, asymptotic behavior Let Fix) be a distribution function and let λ > 0 be a positive real number (not necessarily integer). The absolute moment of order λ of F(x) is defined as

provided that the integral on the right-hand side exists and is finite. In this case the moment of order λ

exists also. In this section, we present some results concerning the relationship between the existence of moments of a distribution function and the behavior of its characteristic function in the neighbourhood of the origin. THEOREM 1.5.1. If a characteristic function fit) is η times differentiable at zero, then (a) all moments of order k < η exist if η is even, (b) all moments of order k < η — 1 exist if η is odd. For odd n, the condition k < η — 1 cannot be weakened (see Example 24 of Appendix A). COROLLARY 1.5.1. If a characteristic function fit) is infinitely differentiable at zero, then all moments of the corresponding distribution exist. THEOREM 1.5.2. Let Fix) be a distribution function with characteristic function fit). If the n-th moment of Fix) exists, then fit) is η times differentiable for all t, and

40

1. Basic properties

If the moment of order η exists, then it related to the rath derivative of the characteristic function by the formula an =

(-i)n/n}(0).

The existence of some moments of a distribution function guarantees the existence of the corresponding derivatives of the characteristic function, therefore, for the characteristic function an expansion can be derived. THEOREM 1.5.3. Let Fix) be a distribution function with characteristic function fit). If the first η moments of Fix) exist, then f(t) admits an expansion of the form fei

kl

as |i| —» 0. Conversely, if fit) admits an expansion of the form /(i) = l + £ a * i * + o( |*D, k=l

|f|-»o,

then each moment of Fix) of order k exists for k < η if η is even and k < η — 1 if η is odd. In this case iiak)k If a distribution function Fix) has a moment of order η + δ, where 0 < δ < 1, then an expansion for fit) can be derived in the following form, which differs from that given by Theorem 1.5.3 in the remainder term. THEOREM 1.5.4. Let Fix) be a distribution function with characteristic function fit). If its moment of order η + δ exists, 0 < δ < 1, then fit) admits an expansion of the form fit) = 1 + £ Μ.

+ 0(|ί|η+5) k]

as |ί| —» 0. Conversely, if fit) admits an expansion of the form η k=l then Fix) has all moments of orders inferior to η + δ. In this case iiakf

1.5. Moments, expansions, asymptotics

41

Since the existence of the absolute moment βχ (λ > 0) of a distribution function Fix) implies 1 - Fix) + F(-x)

χ —» oo,

= ο(χ~λ),

(1.5.1)

and, conversely, (1.5.1) implies that βμ exists for all μ < λ, the following theorem holds. THEOREM 1.5.5. Let Fix) be a distribution function with characteristic function fit). Then fit) admits an expansion of the form

A=1

k

·

as |i| —» 0, where η is a positive integer, and 0 < δ < 1, if and only if 1 - Fix) + Fi-x)

= oix-(n+5)),

χ

oo.

The accuracy of the approximation of a characteristic function by the first terms of its Taylor expansion is given by the estimate n+i fit+s)

- fit) - ^fa) 1!

-...

-

nl

^/^ω ±ßn+1

in + 1)!

which is true for all t and s. THEOREM 1.5.6. Let Fix) be a distribution function with characteristic function fit). Suppose that f(2n~1)(t), where η is positive integer, exists for all t. If there exists δ > 0 such that the function — t

(1.5.2)

is bounded for 0 < |f| < δ, then the moment of order In of Fix) exists. Conversely, if the moment of order 2 η of Fix) exists, then function (1.5.2) is bounded on the set i—δ, δ) \ { 0 } for some positive δ. THEOREM 1.5.7. Let Fix) be a distribution function with characteristic function fit). Suppose that there exists a sequence t\, t2,... and a constant λ, 0 < λ < 2, such that (a) lim„^oo tn = 0; (b) the series Σ ^ χ |ίη|ε converges for any ε > 0; (c) the sequence {tn-iltn}

is bounded;

1. Basic properties

42 (d) \og\fitn)\l\tn\x

is bounded.

Then Fix) has moments of any order inferior to λ. Condition (d) can be replaced with the condition that (1 — |/(ira)|V|in|A is bounded or with the condition that (1 — 91/(ί/ι))/|ίη|λ is bounded. Let Fix) be a distribution function with characteristic function fit). Suppose that a moment of order 2n, where η is a non-negative integer, exists. Then a moment of order 2n + λ, 0 < λ < 2, exists if and only if the integral THEOREM 1 . 5 . 8 .

exists and is finite for some c > 0. Let Fix) be a distribution function with characteristic function fit). Suppose that a moment of order 2n, where η is a non-negative integer, exists. Denote uit) = 3\fit). Then THEOREM 1 . 5 . 9 .

ft.

1

= "TV1 ΠΙ

71,5

+

^ sin · δ) T

^Γ

" ( 2 " ) (^0 ) ~

dt.

Some other results, concerning fractional moments and fractional derivatives of the characteristic function, can be found in (Hsu, 1951; Zolotarev, 1957; Ramachandran & Rao, 1968; Ramachandran, 1969; Brown, 1972; Kagan et al., 1973; Lukacs, 1983). Let Fix) be a distribution function with characteristic function fit) and 0 < a < 2. Then THEOREM 1 . 5 . 1 0 .

1 - Fix) + Fi-:c) = Ο ( j ^ ) -

|x|

1 - 91f(t) = 0(|ί|α),

0.

oo

if and only if \t\

This theorem can be formulated in the following equivalent form. Let Fix) be a distribution function with characteristic function fit) and 0 < a < 2. Then THEOREM 1 . 5 . 1 1 .

1 - Fix) + Fi-x) = Ο ( - j ^ ) ,

|x| -» oo

if and only if 1 - |/(ί)| = 0(|ί| α ),

\t\

0.

1.6. Unimodality

43

In Theorem 1.5.10, as well as in Theorem 1.5.11, Ο can be replaced with ο simultaneously in both relations (for the distribution function and for the characteristic function).

1.6.

Unimodality

The concept of unimodality was introduced in (Khintchine, 1938). A random variable X and its distribution function F(x) are called unimodal with the mode at a if F(x) is convex on (—oo, a) and concave on (a, oo). Unimodal distributions play an important role in many branches of probability theory and statistics. This section brings together some of the basic results concerning characteristic functions of unimodal distributions. For simplicity, we will consider below only unimodal distributions with the zero mode. Possible extentions to the general case are evident. In (Khintchine, 1938), the following criterion for characteristic functions of unimodal distributions was proved. THEOREM 1.6.1. Let cp(t) be an arbitrary

characteristic function, then the func-

tion (1.6.1)

is a characteristic function of a unimodal (with mode at zero) distribution. Conversely, any characteristic function of a unimodal (with mode at zero) distribution can be represented in the form (1.6.1), where (p{u) is a characteristic function. This criterion can be reformulated as follows. THEOREM 1.6.2. Let G(x) be an arbitrary

distribution function. Then the func-

tion (1.6.2) is the characteristic function of a unimodal (with mode at zero) distribution. Conversely, any characteristic function of unimodal (with mode at zero) distribution can be represented in the form (1.6.2), where G(u) is a distribution function. PROOF. Let a function f(t) be represented in the form (1.6.2) where G(u) is a

distribution function. Taking into account that Ψί(") =

sin(uf) e ut

44

1. Basic

properties

is the characteristic function of the uniform on [0,21] distribution, and using Theorem 1.2.9, we derive roo

I

f2t

ι

ft

fit) = / Wt(u)dG(u)= — / g(u)du = — / 0. Then fit) is the characteristic

function of a unimodal

distribution.

PROOF. We prove the theorem under additional conditions that fit) is three times differentiable and — f i t ) is non-negative for t > 0, /

/•OO

Jo

t\fit)\dt < oo,

lim t2fit) = lim t3fit) ί-»οο f->oo

= lim t4f"(t) = 0. t—too

A complete (without any additional restrictions) proof is contained in (Askey, 1975). It suffices to prove that the function r oo

oo

/

is unimodal. This will be

fit)eitxdt = 2 1 fit) cositx) dt if we showJothat p'ix) < 0 for χ > 0. But

-00 proved

p'ix) = -2 so we need to show that Iix)=

POO

Jo

tfit)sinitx)dt

roc / tfit) sin(fct) dt >0

Jo

for t > 0. Integrate by parts, differentiating fit) and integrating the other factor. Then 00 rt r oo pt Hx) = fit) / u sin(ux) du — fit) / u sin(ujc) du dt, Jo

ο

Jo

Jo

46

1. Basic

properties

and the integrated terms vanish by the assumptions we made. Integrate by parts twice more; again the integrated terms vanish to obtain poo

I{x) =

pt

fit)

Jo 2

pOO

= - -

u)sm{ux)dudt

pt

fit)

2 Jo

Since —fit)

uit -

Jo

Jo

u)2sin{ux)dudt.

u{t -

> 0 for t > 0, it is sufficient to prove that git) =

Jo

u(t — u)2 s i n ( u x ) d u

for t > 0 and χ > 0. Setting ux = v, we obtain 1 rtx

g(t) = —7

X4 Jo

{fx — v)2v sin ν dv,

so it is sufficient to prove that h(t) = [ it - v)2v sin vdv > 0 Jo

for t > 0. Denote by H{z) the Laplace transform of hit). We have poo

Hiz)=

Jo

zd

pt

e~tz

Jo

poo

it-v)2vsinvdvdt=

) Jo

ο

e~iz-i)vvdv

=

s2z3 iz-i)2

But 1

=_ /

ziz2 + 1)

1

Jo

fc

Jo

e~ztil

poo

e~tzt2 dt

z2{z2

e~z"vsinvdv

Jo

+1)2'

- cos t) dt;

therefore Jo

it — v)2v sin vdv = 4 / (1 — cosy)(l — cos(f — v))dv, Jo

and the right-hand side is obviously positive when t > 0.

•

The notion of unimodality has proved to be very fruitful, therefore its analog has been introduced for discrete distributions. A random variable X taking values a+nh, η = 0, ±1, ±2,..., with probabilities pn and the corresponding distribution function F{x) are called discrete unimodal with the mode at ao = a + noh i f p n > pn-\

for nno

+ l.

For simplicity, we will consider below only integer-valued discrete unimodal random variables (and corresponding distributions) with the zero mode and, respectively, use the term 'discrete unimodal' only for these distributions. For characteristic functions of discrete unimodal distributions, the following analog of the Khintchine criterion was obtained in (Medgyessy, 1972).

1.6. Unimodality

47

THEOREM 1.6.6. The characteristic function fit) of an integer valued random variable is the characteristic function of a discrete unimodal distribution (with mode at zero) if and only if it can be represented in the form jet rt Μ =^ T T T / ί(1 - elt) Jo

(1.6.4)

where θ (0 < θ < 1) is arbitrary and cp(u) is the characteristic function of some discrete distribution. There is a difference between the Khintchine criterion and the Medgyessy one. The first of them gives necessary and sufficient conditions for φ{ί) under which it generates the characteristic function of a unimodal distribution. The Medgyessy criterion gives only a necessary condition for φ(ί) (it must be the characteristic function of a discrete distribution). This condition is not sufficient: there exist discrete distributions such that the right hand side of (1.6.4) is not a characteristic function at all (but, as it follows from the Medguessy criterion, if the right-hand side of (1.6.4) is a characteristic function, then it is the characteristic function of a discrete unimodal distribution). Thus, the Medguessy criterion gives conditions for fit) to be the characteristic function of a discrete unimodal distribution under the condition that it is a characteristic function. Below we present two criteria of the Khintchine type. THEOREM 1.6.7. Let (pit) be a characteristic function of a random variable Y taking values k + 1/2, k = 0, ±1, ± 2 , . . . , and satisfying the following condition Σ Σ "(y ' J L m ι ^ J=k 7 +J 1/2 k=—oo

)

- »·

(1.6.5)

Then the function 1 f* f(t) = ——r (p(u)du, 2 sin § Jo

f(2nn) = 1,

η = 0,±1,±2,...,

(1.6.6)

is the characteristic function of a discrete unimodal (with mode at zero) distribution. Conversely, the characteristic function of any discrete unimodal (with mode at zero) distribution can be represented in the form (1.6.6), where q>(u) is the characteristic function of a random variable Y taking values k + 1/2, k = 0,±1,±2,..., and satisfying condition (1.6.5). PROOF. Let Y be a random variable taking values k + 1/2, k = 0, ±1 and satisfying condition (1.6.5). Denote its distribution function and characteristic

48

1. Basic properties

function by Φ(χ) and φ(ί) respectively. We set

j=k

j=-oo

J

Consider the distribution function G(x) such that Gix) = pkx + bk

(1.6.7)

for k — 1/2 < χ < k + 1/2, k = 0, ±1, ±2,... Denote its characteristic function by git):

oo

/ -oo eitxdG(x). Then, as we easily see, G(x) is a continuous distribution function such that GGc) - xG'(x) = Φ(χ)

(the forward derivative is taken at the points χ - k + 1/2). Then, due (Khintchine, 1938), G(x) is a unimodal distribution function (with mode at zero), and g(t) = -f\(u)du. t Jo

(1.6.8)

Further, (1.6.7) implies that oo / -oo H(x-y)dF(y),

where H(x)

=

χ+1/2,

- 1 / 2 < x 1/2,

(1.6.9)

and F(x) is the distribution function of an integer-valued random variable. Therefore , λ sin(i/2) x t/2

=

where

oo / -oo eitxdFix).

Taking (1.6.8) into account, we obtain f®

= ο ·

1

/

[*

2 sin(i/2) Jo

0. In other words, on each interval [k—1/2, k+1/2), k = 0±1, ±2,..., the distribution function G(x) can be represented in the form G(x) = akx + bk

where α* > 0, > 0, α* increases for k < 0 and decreases for k>0 ,bk increases for all k. Therefore (see (Khintchine, 1938)) G(x) - xG'ix) = Φ(χ)

(1.6.10)

and 1 Γ* g(t) = q>{u)du, t Jo where 0c) is some distribution function and cp(t) is the corresponding characteristic function. From (1.6.10) we obtain Φ(*) = bk for * e (k - 1/2,k + 1/2), k = 0,±1,±2,... This means t h a t Φ is a discrete distribution function having jumps bk+1 — bk at points k + 1/2, k = 0, ±1,±2,..., i.e., P(.Y = k + l/2) =

bk+1-bk.

From the continuity of G(x) we obtain ak(k + 1/2) + bk = ak+i(k

+ 1/2) + bk+1

or 2(bk+1

-

=

- bk)

2^ + 1

2 P ( Y = k + 1/2) =

2* + l

But ak = P(k - 1/2 0, and such that 0, then f(z) is an analytic characteristic function in the circle \z\ < R. Conversely, if fiz) is an analytic characteristic function in a circle \z\ < R, then F(x) satisfies condition (1.7.1) for all 0 < r < R. This condition can be also formulated as follows. THEOREM 1.7.2.

Let F{x) be a distribution function with characteristic function

fit). If 1 - F{x) + F(-x) = 0(e - r i ),

χ —» oo,

(1.7.2)

for all 0 < r < R, R > 0, then f(z) is an analytic characteristic function in the circle \z\ 0, b > 0, if and only if

1 - Fix) = Oie'™),

χ -> oo,

for allO 0, then fiz) is analytic in a strip containing the strip |3z| < R.

Note that the converse follows from Theorem 1.7.2. Assume that 0, b > 0. It is said to be a ridge function if |φ(ί + iy)\ < | 0, then it is a ridge function in this strip.

The maximal strip of the form —a < 3z < b, a > 0, b > 0, where a characteristic function fiz) is analytic is said to be the strip of analyticity of fiz). The strip of analyticity of an analytic characteristic function can be the whole complex plane or a half-plane or can have one or two horizontal boundary lines. If the strip has a boundary, then the points of intersection of the boundary with the imaginary axis are singular points of the analytic characteristic function. If the strip of analyticity is the whole plane, then the characteristic function is said to be an entire characteristic function. Let fit), fiit), and fiit) be characteristic functions such that fit) = fiit)f2it). If fit) is an analytic characteristic function, then both f\(t) and fiit) are analytic characteristic functions. If fiz) is analytic in a strip — α < 3z < b, a > 0, b > 0, then /i(z) and f?,iz) are analytic in this strip. If fiz) is an entire characteristic function, then /i(i) and fiit) are entire characteristic functions. The methods of the theory of analytic functions have proved to be a very powerful mean in the investigation of characteristic functions. One of the most celebrated examples is the following Cramer theorem on the decomposition of the normal law.

1. Basic properties

54

The characteristic function of the normal distribution has only normal factors. In other words, let fit), fi(t) and fzit) be characteristic functions such that fit) = fi(t)f2(t)· If fit) = βχρ{ίμί — σ 2 ^ 2 ^} is the characteristic function of the normal distribution, then each fjit) is the characteristic function of the normal distribution: fjit) = βχρ{ϊμβ — aft212}, j = 1,2, and μι + μ2 = μ, of + of = σ2. THEOREM 1.7.6 (Cramer).

The next theorem is among those results of this section which will be used below. THEOREM 1.7.7. If a characteristic function fit) coincides with an analytic characteristic function git) in some (real) neighborhood of the origin, then they coincide for all real t: fit) = git).

In particular, this implies that if f(t) = e~ct2 in some neighborhood of the origin, then fit) = e~ct . Note that this is not true if the interval where fit) and e~~ct coincide does not contain the origin (see Appendix A, Example 33).

1.8. Multivariate characteristic functions We make use of the following notational conventions: vectors in R m , m > 1, are denoted by bold lowercase letters, random vectors are denoted by bold capital letters, the scalar product of two vectors χ and y is denoted by (x, y), the Euclidean norm of χ is denoted by ||x||: ||x|| = y/(x, x). Let X = (Χι Xm) be an m-dimensional random vector with distribution function Fin), χ = (xi xm) e R m . The characteristic function fit) of the distribution function Fix.) (or the random vector X) is defined as /(t)= [

e i ^ d F i x ) = Eei j

^

where a = (αϊ, ...,am) is the expectation of X, C =

is the covariance

matrix of X, and Λ = ι is the inverse matrix to C: Λ = C _ 1 . characteristic function of X is

(

τη

^

τη

m

The

I

ι >1 Σ ajtj - zr j=Σ 1 Σk=l aJktJtk Jf · THEOREM 1.8.18. An m-variate probability distribution is normal if and only if all its univariate projections are normal. In other words, an m-variate characteristic function fit) is the characteristic function of the normal distribution if and only if fe(t) are univariate normal characteristic functions for all unit vectors e. A distribution F in R m is said to be spherically symmetric with respect to a point a e R m if it is invariant under rotations about a. Here we present some results concerning the characteristic functions of spherically symmetric distributions. A systematic analysis of this class of distributions is given in (Fang et al., 1989). Without loss of generality it is assumed that a = 0 in the remainder of the section. Let X be a random vector in R m whose distribution F is spherically symmetric (we will call such vectors spherically symmetric). The distribution F can be represented as (Schoenberg, 1938) - Jo f

V^dHir),

(1.8.2)

where U(D is the uniform distribution on the surface of the m-dimensional sphere of radius r with center at zero and H(r) is some distribution function.

62

1. Basic

properties

Each one-dimensional projection of U*·^ (these projections are identical) for m > 2 is an absolutely continuous distribution with the density

(r

u \x) =

iv io\ r(m/2) rv^r((m - 1 ) / 2 ) ^

ι r2 ^

(m 3)/2

-

0,

'

F * r,

(1.8.3)

Ixl > r.

It is easy to see that one-dimensional projections of £/*r) are symmetric and unimodal when m > 3. Thus, by virtue of (1.8.2), we obtain the following useful property of spherically symmetric distributions. THEOREM 1.8.19. For m > 3, the one-dimensional projections of an m-dimensional spherically symmetric distribution are unimodal. This theorem implies (in view of Theorem 1.6.3) the following important necessary condition for characteristic functions of spherically symmetric distributions. THEOREM 1.8.20. Let /(t) be the characteristic function of an m-dimensional spherically symmetric distribution, m > 3. Then for any unit vector e e Rm, the function fe(t) = f(te) is differentiate for at least all t * 0. This theorem shows, in particular, that the Polya criterion (Theorem 1.3.9) cannot be extended to multivariate case straightforwardly (see Appendix A, Example 37) at least for m > 3. Actually, as shown in (Velikoivanenko, 1987), the same situation takes place in the case of m = 2. Another consequence of Theorem 1.8.19 is (due to Theorem 1.6.1) the following representation of characteristic functions of spherically symmetric distributions for m > 3: each of these functions can be represented as 1 Μ Z(t)=üTTr/ g(u)du, Fll Jo where g(u) is a univariate characteristic function. A spherically symmetric distribution F in R m is said to be unimodal if it can be represented as follows (recall that we consider only the distributions which are spherically symmetric about the origin) F(x) = pE(x) + (1 — p)G(x), where 0 < ρ < 1, E(x) is the degenerate, concentrated at zero distribution, and G(x) is an absolutely continuous distribution whose density ς(χ) satisfies the condition g(x x ) > g(x 2 ) if ||xi|| < ||χ 2 ||.

63

1.8. Multivariate characteristic functions

Let F be a unimodal spherically symmetric distribution in R m . It can be represented in the form poo

F=

/ Jo

V* r) dH(r),

(1.8.4)

where is the uniform distribution in the m-dimensional sphere (in the ball) of radius r with center at zero, and H(r) is some distribution function. Each one-dimensional projection of m > 1, is an absolutely continuous distribution with the density r((m + 2)/2) ( *2Ym"1)/2 .. 1 {r "9 . 2 1 1m^r* v \x) = < ry/nT({m +1)/2) ^ r J ' " ' 0,

(1.8.5)

bcl > r.

Taking into account that any spherically symmetric distribution is completely determined by its univariate marginal distribution, and comparing (1.8.3) and (1.8.5), we come to conclusion that the uniform distribution in the m-dimensional sphere (ball) of radius r is the m-dimensional marginal distribution of the uniform distribution on the surface of the (m + 2)-dimensional sphere of the same radius. This implies the following theorem, which is a generalization of Theorem 1.8.19. THEOREM 1.8.21. A distribution in R m+2 , m > 1, is spherically symmetric if and only if its m-dimensional marginal distributions are unimodal spherically symmetric in Km. The characteristic function of the uniform distribution on the m-dimensional sphere of radius r (centered at zero) is fl>w(t) = Γ ( I ) ( ^ )

m / 2

'«WiirlltH).

(1.8.6)

The characteristic function of the uniform distribution in the m-dimensional sphere (ball) of radius r is vW(t)

=Γ

+ l) (-^)m/2e7m/2(it||).

(1.8.7)

Therefore, by (1.8.2) and (1.8.4) we obtain the following (quite similar) characterizations for spherically symmetric and unimodal spherically symmetric distributions respectively. THEOREM 1.8.22. A function fit), t e R m , m > 2, is the characteristic function of a spherically symmetric distribution if and only if it can be represented as /m\ m

=

T

\ i ) L

f°° f 2

\m/2_1

Ui)

:r||,||) 0)

Thus it suffices to prove t h a t the function

decreases in the interval (0, y/2). But this is obvious because both functions 2/*2 and ln(l - * 2 /2) decrease. To prove (2.1.4), let u s use the elementary inequality ln(l +z) — 1. Then we obtain (for 0 < * < 1) t

, χ

2

,

2

2 / 2

\

,

*2

ο

69

2.1. Auxiliary results

To prove (2.1.5), we rewrite it in the form x2 x 4 + —. 2 24

cosx < 1

(2.1.10)

Using an expansion for cos χ we obtain X2

cosx =1 —— +

COS(iV)(0x)

—

2

X2

= 1

+

24

X

4

COs(öx) Λ

χ

2 24 χ 2 χ4 2+24'

where |θ| < 1, which coincides with (2.1.10). Let us prove (2.1.6). Since both sides of the inequality are even functions, it suffices to prove that .x3 v(x) = χ — h,2(c) 6

sinx > 0

for 0 < χ < c. We have x2 ν (χ) = 1 — hzic)—— cosx > 0 Li

due to (2.1.2). Hence v(x) is non-decreasing for 0 < χ < c. Taking into account that v(0) = 0, we see that v(x) > 0 for 0 < χ < c. Let us prove (2.1.7). As in the previous case, assume without loss of generality that χ > 0. Set v(x) = 1

s-x2 — cosx. πΔ

We have v(0) = 0, ν(π/2) = 0. Consider the derivative of v(x): / ν (x) = sinx

8

öX.

Since the function sinx/x decreases in the interval [0, π/2], ν7(χ) has only one root in this interval, therefore either v(x) > 0 for all χ e [0, π/2] or v(x) < 0 for all χ e [0, π/2]. Demonstrate that the first relation holds. Indeed, we have .

(sinx

8\

provided thatbx is sufficiently small, i.e., for some α > 0, v(x) > 0 for χ e [0,a].

70

2. Inequalities To prove (2.1.8), consider the function x2 v(x) = cos* — 1 + —. Lt

We have v(0) = 0 and v'(x) = χ — sinx > 0, χ > 0, which implies (2.1.8). Finally, let us prove (2.1.9). Without loss of generality, we may assume that χ > 0. Consider the function v(x) = sinx — χ cosx. We have v(0) = 0, and V(x) = χ sinx > 0 for 0 < χ < π. These obviously imply that v(x) is positive on the interval (0, π), i.e., (2.1.9) is valid. A set of distributions and the set of the corresponding characteristic functions (denote the latter by are said to be closed with respect to translation if the inclusion fit) e & implies the inclusion f(t)eltb e & for any real b. It is easy to see that a class of characteristic functions is closed with respect to translation if and only if it can be represented in the form

& = U

ael

where I is some set of indices and for each & a the corresponding set of distributions is an additive type (a set of distributions is called the additive type of distribution F if it consists of all distribution functions of the form F(x — a), —oo < a < oo; see (Prokhorov, 1965; Kagan et al., 1973). The following lemma is frequently used in this chapter. LEMMA 2.1.1. Let & be a class of characteristic functions closed with respect to translation, Β be an arbitrary set of the real line, and let g(t) be a real-valued function defined on B. If for any f e & and any t e B, |9V(i)| * git), then \m\ 0,

because φ(χ) is non-increasing for χ > 0 and hence POO

/ Jo

D(x)dq>(x) < 0.

• LEMMA 2.1.3. Let fi(x), fi(x) and g(x) be integrable functions defined on the interval [a,b]. Suppose that g(x) is non-increasing on [a,b], pb

/ fi(x)dx= Ja

pb

/ f2(x)dx, Ja

and there exists c ε (a, b) such that fl(x) > f2(x) forxe

[a,c), and /!(*) f f2(x)g(x) dx. Ja Ja PROOF. It is clear that

fb h(x)g(x) dx - f f2(x)g(x) dx Ja Ja = fb Iflix) - f2(x)]g(x)dx Ja

= [C\fi(x)-f2(x)]g(x)dxJa

Je

fb[f2(x)-fl(x)]g(x)dx

Zgic) Γ If i(x) - f2(x)] dx - g(c) fb [f2(x) - flix)] dx Ja Je = g(c) [b[f1(x)-f2(x)]dx Ja which proves the lemma.

= 0, •

2.1. Auxiliary results

73

Let fix) be a non-negative continuous function defined in the interval [a, &]. Let us introduce the functional sup [ BeMu) ΒεΜμ)·J Β

f(x)dx,

where is the class of all Borel subsets of the interval [a, 6] whose Lebesgue measure is equal to μ (it is assumed that μ < b — a). We need the following properties of the functional 3Fba. LEMMA 2.1.4. Let a be a real number, b, c and d be positive numbers π b>—,

such that

, 1 cd > -.

2c

2

Then

^c(dcoSbx,-)< —

sm—.

PROOF. Without loss of generality, one can suppose that c = π/2 (the general case can be reduced to this one by the change of variables). Let 0 < λ < π. Then, for any real a and b > 1, Sraa-iUcosbx,X)

< ^l2(cosbx,X)

= l^llzicosxM) b

< videos

χ, λ);

hence, for d > 1/π, K-Ll

(dcosbx,




2. Then Γ (ϊϋ±2) 1 I 2 )

m

PROOF. If m is even, then .r(f r

(™±i)

l)

+

r(f)

fr(f) r(f)

m 2"

Let m be odd. Set η = (m — 1)12. Then

Γ (N+I)

+

13s(2n + 1)2" ν/π ~ 13s(2ra - l ) 2 n + 1 φ ι In +1 m =

2

=

2*'

• LEMMA 2.1.7. Lei X and Y be independent and identically distributed random variables with zero expectation and finite absolute moment of order 2 + δ, 0 < δ < 1. Then E\X - Υ\2+δ < 2(Ε|Χ|2+δ + ΕΧ 2 Ε^| δ ). PROOF. Denote the distribution function of X by F(x). Making use of the elementary inequality |α — 6|δ < |a|s + |ö|5, which is true for all real a and b,

2.1. Auxiliary results

75

we obtain oo

rOO

-oo J—oo

/

OO

/

roo

/ •oo

(Λ: - Y ) 2 ( | X | 5 +

\y\s)dF(x)dF(y)

J—oo

oo roo / -OO J —oo oo roo e e / -OO / J —ooay(|x| + |y| )dfXx)dFXy).

The second term on the right-hand side is equal to zero: oo /

roo

/

-ooJ—oo

roo

xy(\x\s+ \y\s)dF(x)dF(y) = 2

J—oo

roo

xdF{x)

J—oo

ybl^W

= 2ΕΧΈ(Χ|Χ|δ) = 0, therefore Epe - Υ\2+δ < Ε [(Χ2 + y2)(|Z|5 + \Υ\δ)] = 2(Ε|Ζ|2+δ + ΕΧ2Ε\Χ\δ).

LEMMA 2 . 1 . 8 .

Then

Let Χ and Υ be independent

random variables, and

EY" =

• 0.

E\X + Y\>E\X\. PROOF.

Denote the distribution function of X by F(x). For any real x, E|x + Y| > |E(x + 7)| = |*|,

therefore

oo /

-oo

roo

Ejx + YjdF(x)>

/ J—oo |x| dF(x) = E|X|.

• LEMMA 2 . 1 . 9 .

For any

0

π, in the interval (0,2π). In addition, φρ(χ) is positive for 0 < x < xq and negative for xq < x < 2π.

2. Inequalities

76

PROOF. First let us establish the following elementary inequality: sin* χ

>

1 + cos* ζ 2

(2.1.11)

for 0 < χ < π. Indeed, for 0 < χ < π we have 2 sin* — x(l + cos Λ;) = 4 sin — cos - — 2x cos2 — 2 2 2 χ ( .

χ

= 4 cos - sin 2 V 2

χ

x\

cos - > 0, 2 2/

therefore, making use of inequality (2.1.9), we obtain (2.1.11). Now consider the function φρ(χ). If 0 < χ < π, then, in view of (2.1.11), we obtain . 1 + cos* sinx ρ + (1 — ρ)cosx < < , 2 χ i.e., for these χ the function φρ(χ) is positive. Let χχ denote the root of the equation sinx = χ cos* in the interval (π, 2π). For χχ < χ < 2π, sin* χ

< cos*

0. Let us fix an arbitrary δ e (0,1] and consider the function x2 1/(2 + δ)), φζ(χ) > 0 for all χ > 0. At the same time, for small ζ (say, for ζ = 0), φζ(χ) can take negative values. Thus there exists the infimum zm of all ζ satisfying the condition t h a t φζ(χ) is non-negative for a l l * > 0, i.e., zm = min{z > 0: φζ(χ) > 0, χ > 0} (we write min instead of inf because this infimum is, obviously, attained). Let us prove t h a t zm = c(0,

χ > 0,

which contradicts the definition of zm. Denote the minimal positive root of φΖπι(χ) by xm. Then relations (2.1.13) and (2.1.14) mean t h a t l-^+zmx%s

-cosx

= 0

and -Xm + (2 + S)zmx!L+s + sinJCm = 0.

78

2. Inequalities

Finding zm from the second equation: Zm

xm - sinx„ —

(2 + 5)xl+s

'

and substituting this expression to the first one, we obtain the following equation for xm: 1 -

2(27S)*" ~ ä h )

X m SinXm

-

C°SXm =

It remains to show that the equation 1 -

δ , 1 xsinx — cosx = 0 r*22(2 + 5) (2 + δ)

(2.1.15)

has a unique solution in the interval (0,2π) for any 0 < δ < 1. Consider the left-hand side of equation (2.1.15): Ψ(Χ) = 1 -

-x2-

2(2 + δ)

-xsinx — cos λ:.

(2 + δ )

We have ψ(0) = 0,

and ψ'(χ) =

ψ(2π)

(1 + δ) sin χ (2 + δ)χ

=

2δπ*

η

(2.1.16)

-(ϊ7δ)

be satisfied.

Then

i

and

£·

The corollary follows from the theorem and inequality 2.1.6. One j u s t should take into account t h a t 2ac > 1. R E M A R K . Both inequalities ( 2 . 2 . 7 ) and ( 2 . 2 . 8 ) are sharp. In ( 2 . 2 . 7 ) , the equality is attained when p(x) is the uniform density in the interval of length 1 la. In

2.2. Distributions with bounded support

85

(2.2.8), the equality is attained for arbitrary large t under an appropriate choice of a distribution. In other words, although the characteristic function of any absolutely continuous distribution converges to zero as t —» oo, this convergence is not uniform even in each set of uniformly bounded densities having uniformly bounded supports (see Example 35 of Appendix A). The situation changes if distribution under consideration is unimodal. In this case, uniform estimates can be obtained even without the condition of boundedness of the support (see Theorem 2.4.3). Let h\(,a) and h^ia) be functions defined by (2.1.1). THEOREM 2.2.3. Let X be a bounded random variable, < c, fit) and σ 2 be its characteristic function and variance respectively. For any a e [0, π/4], |/(ί)| < 1 - Λ 2 (α)-— Li

for |ί| < ale. Fix an arbitrary α from the interval [0, π/4]. Denote the additive type of the distribution of the random variable Ζ by Since srf is closed with respect to translation, it suffices to prove (Lemma 2.1.1) that PROOF.

σ2*2 |9ty(f)| £ 1 - A 2 ( a ) — -

(2.2.11)

for |f I < ale for any characteristic function ψ(ί) whose distribution belongs to fi/. Let us fix an arbitrary ψ(ί) whose distribution belongs to si and fix an arbitrary t from the interval [0, α/c]. We will use the same notation as that introduced in the proof of Theorem 2.2.1. In particular, Lk (k = 0,±1,±2,...) is determined by (2.2.1). As in the proofs of Theorems 2.2.1 and 2.2.2, we should separately consider the cases > 0 and SRi^(i) < 0, and, because the reasoning is similar, we will consider only the first case. As in the proof of Theorem 2.2.1, we come to the conclusion that there exists an integer ko such that |9ty(f)| =^¥(t)
c.

\o,

We have g(c) =

Eg(\X\)

and hence, using the Chebyshov inequality, we obtain

e

(2.5.3)

g(c)

Further, q(x) is a probability density function, whose support belongs to the interval [—c,c], and such that sup q(x)
i/u rnu j-^ Jo and

I

x||>2/u

J _ V /Γ dF{x) < — mu ^

J-mu

[1- /(V1.

(2.7.7)

Vu Vv

+

...+x2m

> - ) < Σ p (\Xj\ > —). "/ \ rnu J

Taking into account that for each j = 1, ...,m, the characteristic function of Xj is fiSyti, ...,8mjtm), and applying Theorem 1.4.8 to each summand of the right-hand side, we arrive at (2.7.7). • Another multivariate analog of Theorem 1.4.8 is given by the following theorem. Consider the cube I m = { x = (*!, ...,x m ): - 1 < X! < 1 , . . . , - 1 < xm < 1}, and let

= (ε^,

= 1,..., 2 m , be the vertices of F 1 numbered so that

THEOREM 2.7.3. Let Fix.) =

Fixi,...,

xm) be a distribution function with characteristic function fit) = f(t\, ...,tm). Then for u > 0 [ dFix\,...,xm) Jut-11**1^"

< - y ; Γ[1 - Kfieft,.$>t)\ U Jo

PROOF. L e t £ i ,

dt.

denote the '(l/2 m )th spaces' ofR m labeled by ε (1) , ...,e ( 2 m \ respectively, i.e., ε^ e Ej,j = 1, ...,2 m . This means that if χ = (x\, ...,xm) e Ej, j = 1 2 m _ 1 , then efxk = k = 1, ...,m, and if χ = (λγι, e EJ+2m-i, j = 1 2 m - \ then efxk = -\xk\,k = 1, ...,m. Denote Cj = Ej u Ej+2m-i, Au = j x = (xi, ...,* m ):

2. Inequalities

118 and

s(x) = 5 > * i · k=l Then dt

•=1 u Jo L

+

- V

f

[l

-

"(£l)xl

s i n (

+

···

+

£mxrn))

$xm)) dt dF(x)

dF(x)

2m — 1

J ^ _ sin(ttS(x))j dF(x) JcjrAyu ί 1 ~ uS(x) J

^ " ^ 4

Σ

/

=

dF(x),

•

because 1 — sin jc/ac > 1/7 for χ > 1.

One more multivariate analog of Theorem 1.4.8 is given by the following theorem. THEOREM 2.7.4. LetF{x), Χ € function f(t). Then for u > 0 f

be a distribution function with characteristic

diXx) < ^

f

[1 - 9t/(t)]dt

(2.7.8)

/or any a > m/y/π, where c(a) =

r ( f + 1) rrm/2 1 - - J 2 νπα

PROOF. First let us prove that |y/r)(t)|