Asymptotic Behaviour of Linearly Transformed Sums of Random Variables

Mathematics and Its Applications
Volume 416
by
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6346-3 ISBN 978-94-011-5568-7 (eBook) DOI 10.1007/978-94-011-5568-7
This is a completely revised, updated and expanded translation of the original Russian work Functional Methods in the Problems of Summation of Random Variables, @Naukova Dumka, Kiev, 1989 Translated by Vladimir Zaiats.
Printed on acid-free paper
AII Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 Softcover reprint of the hardcover Ist edition 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner
Contents
Preface
Part I Random series and linear transformations sequences of independent random elements
Chapter 0 Random elements and their convergence (preliminary notions) 0.1 F -spaces and separating setb ·of functionals . . . . . .
0.1.1 Topological vector spaces ........... . 0.1.2 Separating sets of functionals and weak topologies . 0.1.3 F -spaces . . . . . . . . . . . . . . 0.1.4 Classes of topologies on F-spaces 0.1.5 The space Rtl .......... . 0.1.6 Linear operators and matrices ..
0.2 u-algebras and measures. Convergence of measures 0.2.1 u-algebras .................. . 0.2.2 Pre-measures, measures and characteristic functionals . 0.2.3 Weak convergence of measures . . . . . . 0.2.4 T-weak and essentially weak convergence
0.3 Random elements and their characteristics 0.3.1 Random elements ............ . 0.3.2 Distributions of random elements . . . . 0.3.3 Mean values and characteristic functionals 0.3.4 Covariance characteristics . . 0.3.5 Independent random elements
0.4 Convergence of random elements . 0.4.1 Almost sure convergence .. 0.4.2 Convergence in probability. 0.4.3 Convergence in distribution 0.4.4 T-weak and essentially weak almost sure convergence
0.5 Sums of independent random elements . . . . . . . . . . . . 0.5.1 Inequalities for sums . . . . . . . . . . . . . . . . . . 0.5.2 The weak law of large numbers for sums of independent
random variables . . . ................ .
1 1 1 3 5 7 7 8
11 11 13 16 17 19 19 20 20 21 22 24 24 25 25 27 31 31
35
vi
0.6 Gaussian random elements ..... 0.6.1 Gaussian random variables . 0.6.2 Gaussian random vectors. . 0.6.3 Gaussian random elements.
36 36 39 41
Chapter 1 Series of independebt random elements 47 1.1 The symmetrization principle. Four remarkable theorems on series of
independent random variables . . . . . . . . . . . . . . . . . . . . .. 47 1.2 The Levy theorem in F -spaces . . . . . . . . . . . . . . . . . . . . . . 53 1.3 Equivalence of the strong and weak almost sure convergence of series
of independent symmetric summands . . . . . . . . . . . . . . . . . . 55 1.4 Weak topologies and convergence of series of independent symmetric
summands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.5 Fourier analysis and convergence of series of independent terms in
Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 65 1.6 Series with stable terms in Hilbert spaces . . . . . . . . . . . . . . .. 77 1.7 Integrability of sumS of inde")endent random elements. . . . . . . .. 83 1.8 The Abel transformation and the contraction principle for random
series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 1.9 The majorization principle for random series . . . . . . . . 96 1.10 Sub-Gaussian random variables. Gaussian majorization of
sub-Gaussian series . . . . . . . . . . . . . . . . . . 99 1.11 Random scries in the space of continuous functions . . . . 110
Chapter 2 Linear transformations of independent random elements and series in sequence spaces 123 2.1 Random elements in sequence spaces . . . . . . . . . . . . . . . .. 124 2.2 Linear summability schemes and series in sequence spaces. . . . .. 137 2.3 Stochastic arrays and linear sequences. Oscillation properties of linear
sequences ............................ . 2.4 Oscillation properties of Gaussian sequences . . . . . . . . . . 2.5 Multiplicative transformations of stochastic arrays. Examples 2.6 The contraction principle for stochastic arrays . . . . . . . . . 2.7 Strong laws of large numbers for weighted sums of independent sum-
145 · 161 · 175 · 180
mands . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 2.8 Generalized summability methods .................... 197 2.9 Stability in probability of linear summability schemes ......... 201 2.10 Gaussian majorization for linear transformations of independent sub-
Gaussian random variables and vectors ................. 206
Part II Limit theorems for operator-normed sums of independent random vectors and their applications 215
Chapter 3 Operator-normed sums of independent random vectors 217 3.1 The Prokhorov-Loeve type strong laws of large numbers ....... 218
vii
3.2 Strong laws of large numbers for operator-normed sums of independent random vectors . . . . . . . . . . . . . . . . . . 225
3.3 Strong laws of large numbers for spherically symmetric random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 244
3.4 Almost sure boundedness and the iterated logarithm type laws . . 248 3.5 Almost sure convergence of operator-normed sums of independent
random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 3.6 Operator-normed sums of independent Gaussian and sub-Gaussian
vectors ................................... 261
Chapter 4 Operator-normed sums of independent identically distributed random vectors 4.1 Integral type criteria ...................... . 4.2 Some properties of sums of independent identically distributed
random vectors with finite second moments .......... . 4.3 The equivalence of operator and scalar normalizations for sums of
independent identically distributed random vectors with finite second
269 .270
moments. Integral criteria ........................ 286 4.4 Strong relative stability of linearly transformed sums of independent
identically distributed symmetric randQm vectors ......... . 298
Chapter 5 Asymptotic properties of Gaussian Markov sequences 307 5.1 Gaussian Markov sequences and stochastic recurrence equations . 307 5.2 Enlropy conditions of boundedness and convergence of Gaussian
Markov sequences. . . . . . . . . . . . . . . . . 325 5.3 Onc-dimensional Gaussian Markov sequences. . . . . . . . . . 332
Chapter 6 Continuity of sample paths of Gaussian Markov processes 343 6.1 Oscillations of Gaussian processes . . . . . . . . . . . . . . . . 343 6.2 The equivalence of sample and sequential continuity of Gaussian
processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 9 6.3 A rank criterion of continuity of Gaussian Markov processes . 354 6.4 An entropy criterion of continuity of Markov processes ... . 361
Chapter 7 Asymptotic properties of recurrent random sequences 363 7.1 Convergence to zero of Gaussian Markov sequences in R m • . 364 7.2 A Gaussian majorization principle for solutions of stochastic
recurrence equations with sub-Gaussian perturbations. . . . . 368 7.3 Almost sure convergence to zero of m-th order random recurrent
sequences in R . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 373 7.4 Almost sure boundedness and the iterated logarithm type laws for
normalized m-th order recurrent sequences in R ............ 388 7.5 Asymptotic behaviour of recurrent sequences in Rm .......... 395 7.6 Strong laws of large numbers and the iterated logarithm type laws
for sums of elements of recurrent sequences in R m (m> 1) ...... 397
viii
7.7 Appendix. Inequalities for the norms of the matrices A"H ...... 410
Chapter 8 The interplay between strong and weak limit theorems for sums of independent random variables 417 8.1 A characterization of the law of the iterated logarithm in terms of
asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . 418 8.2 UDPA and UNA: two special classes of sequences of random
variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 8.3 Normalization and strong relative stability of sums of UDPA random
variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 431 8.4 Strong and weak limit theorems for UNA random variables. .. . 432 8.5 Normalization and strong relative stability of weighted sums of
independent identically distributed random variables ... . .. . 436
Comments 443
Bibliography 455
Preface
Limit theorems for random sequences may conventionally be divided into two large parts, one of them dealing with convergence of distributions (weak limit theorems) and the other, with almost sure convergence, that is to say, with asymptotic prop erties of almost all sample paths of the sequences involved (strong limit theorems). Although either of these directions is closely related to another one, each of them has its own range of specific problems, as well as the own methodology for solving the underlying problems. This book is devoted to the second of the above mentioned lines, which means that we study asymptotic behaviour of almost all sample paths of linearly transformed sums of independent random variables, vectors, and elements taking values in topological vector spaces.
In the classical works of P.Levy, A.Ya.Khintchine, A.N.Kolmogorov, P.Hartman, A.Wintner, W.Feller, Yu.V.Prokhorov, and M.Loeve, the theory of almost sure asymptotic behaviour of increasing scalar-normed sums of independent random vari ables was constructed. This theory not only provides conditions of the almost sure convergence of series of independent random variables, but also studies different ver sions of the strong law of large numbers and the law of the iterated logarithm. One should point out that, even in this traditional framework, there are still problems which remain open, while many definitive results have been obtained quite recently. The books by P.Revesz (1967), V.V.Petrov (1975, 1987), and W.F.Stout (1974) give a complete insight into the limit theorems related to the almost sure asymptotic behaviour of increasing scalar-normed sums of independent random variables and provide an extensive bibliography on the subject.
Considerable attention of experts in the probability theory has recently been focused on studying asymptotic properties of weighted sums of independent random variables. The initial efforts were apparently concentrated around the weighted sums which emerge in various generalized summability schemes. However, it was not only due to a natural desire of extending the classical scheme, but also under the important influence of the problems in mathematical statistics, which have to do, for example, with properties of the strong consistency of functional non-parametric statistics, that a more detailed study of the properties of weighted sums was car ried out. An important stimulating effect has also been produced by studying the asymptotic properties of different procedures of stochastic approximation.
Another important problem which also naturally extends the traditional summa biliLy schcme, is a comprehensive investigation of the asymptotic behaviour of opera tor-normed sums of independent random vectors. It has not been until recently when the corresponding limit theorems were introduced into consideration, although it is
ix
x PREFACE
in these theorems that a complete information on the asymptotic behaviour of sums of independent random vectors is contained. This direction is closely related to the asymptotic behaviour of solutions of multi-dimensional stochastic recurrence equations and, at the same time, to the problems of multi-dimensional stochastic approximation and the Kalman filtration theory. Weak limit theorems for sums of operator-normed independent random vectors were considered by N.P.Kandelaki and V.V.Sazonov, M.Hahn, M.Klass, Z.J.Jurek and J.D.Mason, as well as by a number of other authors. Strong limit theorems for these sums are less studied, and the authors hope that this gap is partly filled in by this book. Weighted sums of independent random variables and operator-normed sums of random vectors belong to the so-called non-traditional summability schemes. It is clear that one should involve new approaches for studying these sums. One of these approaches system atically developed in this book is based on representation of the random sequences which result from general linear transformations of independent random variables, by series of independent random elements in the sequence spaces. The theory of series of independent summands in infinite-dimensional topological vector spaces is well-developed, and it proves to be effective as applied to the corresponding repre sentations of random sequences.
One should note that limi t theorems for sums of independent random elements in infinite-dimensional topological vector spaces, in particular in infinite-dimensional Banach spaces, have been and still remain subject of intensive investigations, as well as the distribution theory in these spaces. Systematic investigations in this field, initiated by E.Mourier, R.Fortet, Yu.V.Prokhorov and followed by many mathemati cians, h!l-ve led to a number of remarkabl.e results of importance for the probability theory, functional analysis, theory and statistics of stochastic processes. The vari ety of developments in the probability theory in Banach space is summarized in the fundamental monograph by M.Ledoux and M.Talagrand (1991).
This book consists of nine chapters divided into two parts:
Part I Random series and linear transformations of sequences of independent random elements (Chapters 0-2)
Part II Limit theorems for operator-normed sums of independent random vectors and their applications (Chapters 3-8)
The basic plots of the first part are series of independent terms and infinite summability matrices. Series of independent random elements in the context of separable F -spaces (complete metrizable topological vector spaces which are in du ality with their dual spaces) play an important role in the whole book. We give a scope of the theory of these series in Chapter 1. F -spaces possess rather nice properties in order for the theory of summation in these spaces to be instructive, without having any need of involving especially subtle results from the theory of topological vector spaces. It is in separable F -spaces where the problem of gen eralization of the Ito-Nisio theorem on equivalence of the strong and essentially weak almost sure convergence for series of independent symmetric terms finally fits into its natural margins. However, we gladly use each occasion to switch to Ba nach spaces, and sometimes even to Hilbert spaces, in different applications. The
PREFACE xi
contraction principle for series of independent terms and the theorem on Gaussian majorization of sub-Gaussian series which leans on this principle are also stud ied in Chapter 1. This theorem combined with the theorem on equivalence of strong and essentially weak almost sure convergence constitute the "investigation kit" we employ in what follows. Not all the statements obtained in Chapter 1 find use in the subsequent chapters. However, it is our opinion that the series of in dependent random elements in infinite-dimensional topological spaces are worthy of detailed consideration. The bibliography on different lines of development in the theory of these series and its applications is quite extensive. Let us mention the books by J.-P.Kahane (1968), V.V.Buldygin (1980), M.B.Marcus and G.Pisier (1981), N.N.Vakhania, V.I. Tarieladze , and S.A.Chobanyan (1987), M.Ledoux and M.Talagrand (1991), S.Kwapien and W.A.Woyczynsky (1992), as well as the exten sive paper by J.Hoffmann-Jf2Irgensen (1977), where the series of independent sum mands are studied among other problems. This is why we have only included in Chapter 1 the material poorly covered in literature leaving out the facts which are rather well-known.
In Chapter 2, infinite summability matrices are employed for studying the lin ear transformations of infinite sequences of independent random elements. The sequences obtained by these transformations may naturally be represented as se ries of independent random summands in sequence spaces. This connection enables establishing various contraction principles for summability matrices, a Gaussian ma jorization principle, etc. We study oscillation properties of linear sequences and, in particular, those of Gaussian sequences. As applications, the strong laws of large numbers for weighted sums of random elements and generalized summability meth ods are considered as applied to independent symmetric random elements.
The problems we deal with in Chapters 1 and 2 as well as in the further ex position, requires invoking diverse preliminary notions whose summary is given in Chapter O.
The main topic of the second part of the book is concerned with the strong limit theorems for operator-normed (matrix-normed) sums of independent random vectors in finite-dimensional Euclidean spaces (Chapters 3 and 4) and their applications to Gaussian Markov processes, both one-dimensional and multi-dimensional (Chapters 5 and 6), and to solutions of stochastic recurrence equations (Chapter 7).
In Chapter 3, necessary and sufficient conditions for almost sure convergence, almost sure convergence to zero, almost sure boundedness, and the iterated loga ri thm type laws are established for operator-normed sums of independent symmetric random vectors. We consider scalar normalizations as well. Our main attention is focused on the Prokhorov-Loeve type criteria. In the case of scalar-normed sums of independent symmetric random vectors, the Prokhorov-Loeve type strong law of large numbers reduces the problem of almost sure convergence to zero of these sums to that of the almost sure convergence to zero of a sequence of independent random vectors. The situation becomes more intricate in the case of operator-normed sums: one should check for the almost sure convergence to zero of some set of sequences of independent random vectors. Additional assumptions imposed on the summands, for example, the constraint that the terms are zero-mean Gaussian vectors, enable
xii PREFACE
carrying out a constructive verification of the Prohkorov-Loeve type criteria. More over, given the summands have spherically symmetric distributions (that is, their distributions are invariant with respect to all unitary transformations of the space), operator normalizations become equivalent to those made by means of operator norms.
In Chapter 4, we basically study the same range of problems as in Chapter 3, but under the additional assumption that the summands are identically distributed. This assumption enables establishing some integral-type criteria (that is to say, some criteria expressed in terms of characteristics rather of the individual terms than of their sums) which determine one or another type of asymptotic behaviour of the operator-normed sums. If, moreover, the norms of summands have finite second moments then operator normalizations again become equivalent to those by means of operator norms, which leads to a simple integral-type criteria.
With the results obtained in Chapter 3 we can carry out, in Chapter 5, quite an exhaustive analysis of asymptotic behaviour of almost all sample paths of one dimensional and multi-dimensional zero-mean Gaussian Markov sequences. Con ditions of the almost sure convergence of Gaussian Markov sequences, in turn, combined with the theorem on equivalence of the sample almost sure continuity and sequential almost sure continuity for Gaussian processes lead us, in Chapter 6, to various criteria of sample almost sure continuity of multi-dimensional Gaussian Markov processes.
Chapter 7 is devoted to studying asymptotic behaviour of the random sequences produced by solutions of stochastic recurrence equations. The results of this chapter help us}o judge about the efectiveness <:>f the general methods we have developed in the prevous sections, as applied to concrete problems.
In general, the topic of this book is concentrated around the strong limit theo rems. There are no theorems related to weak convergence of sums of independent random vectors. This gap is somehow filled up in Chapter 8 which deals with the interplay between the central limit theorem and the iterated logarithm type laws for sums of independent random variables.
The first and second part of the book are related to each other. However, the second part may be read independently with appealing to Part I only when necessary. The topics selected for this book naturally reflect the line of investigations carried out by the authors, and this is why many interesting and subtle results related to the sums of independent random elements have been left beyond the book. Some additional information may be found in the bibliography and comments, both at the end of the book. The Comments section refers to the sources in literature either used in writing this book or directly related to its contents.
This book is in general aimed at mathematicians working in probability theory. However, the authors hope that it will be useful to postgraduate and undergraduate students, and some sections in the second part will be of interest to applied scientists.
Acknowledgments
This book has its origin in a small monograph by V.V.Buldygin and S.A.Soln tsev (1989) which appeared in Russian in 1989 giving an account ofthe investigations carried out by the authors in the seventies-eighties. When the translation of this monograph was discussed with the Kluwer publishers, it was a kind offer of the publisher to revise the original book and essentially extend it. The authors are very thankful to the publisher for this possibility, the result of which the reader holds in the hands. The authors express their sincere gratitude to Professor A.V.Skorokhod for attention to their work, for his remarks and advices which have had an essential influence on the contents of many sections and statements. The authors would like to thank their colleagues, especially Professor S.A.Chobanyan, who took part in discussing the results presented in this book, as well as Dr.V.Zaiats for his work on translation and word processing of this book.
The authors would also like to acknowledge a partial support of the work on this book by the International Soros Science Education Program grant number SPU061013.
XI11
Random series and linear transformations of sequences of independent random elements
Chapter 0
Random elements and their convergence (preliminary notions)
This chapter has a double destiny. First, it contains the basic definitions and some preliminary information on ran
dom elements in a sufficiently wide class of topological vector spaces (F-spaces) which we shall need in what follows. We also give a scope of the inequalities for sums of independent random elements and some facts about Gaussian random ele ments.
Second point is that some weakened types of the almost sure convergence (the T-weak and essentially weak convergence) for general sequences of random elements are considered in this chapter. The relation between strong and weak almost sure convergence for series of independent symmetric elements in Banach spaces was dis covered by K. Ito and M. Nisio (1968), and it plays an important role in probability theory in Banach spaces. The structure of Banach space appears to loose its im portance in this case, and general analysis enables to go to the heart of the matter. This is why random elements in F-spaces have been considered. T-weak almost sure convergence turns to good advantage not only in theory of series of independent ran dom elements (Chapter 1): the criterion of weak convergence in probability is an example (Theorem 0.4.4).
The selection of facts and presentation in this section do not pretend to be com plete. We do not prove many general statements and refer the reader to Comments at the end of the book, where we point at the sources in the literature to look for the corresponding proofs.
0.1 F -spaces and separating sets of functionals
0.1.1 Topological vector spaces
Let us denote by (y, r) the topological space constituted by a set Y and some topol ogy r defined on this set (topology means a system of open sets). We shall consider the Hausdorff topologies only. If r is the intrinsic topology, or the context makes
1
V. Buldygin et al., Asymptotic Behaviour of Linearly Transformed Sums of Random Variables © Kluwer Academic Publishers 1997
2 CHAPTER O. RANDOM ELEMENTS AND THEIR CONVERGENCE ...
it clear what topology is considered, then we shall simply refer to the topological space y. If topology l' is generated by some metric d, we shall say that l' and d are consistent and the topological space is metrizable. We denote the corresponding topological space by (y, d).
If 1'1 and 1'2 are two topologies on the same set y, and topology 1'2 majorizes topology 1'1 (that is, 1'2 is not weaker than 1'1) then we write 1'1 ~ 1'2. Convergence in the intrinsic topology of the space (y, 1') will be referred to as the strong convergence, or convergence in the space (y, 1').
A set KeY is compact (1'-compact, compact in topology 1') in the topologi cal space (Y,1') if each covering of this set by open sets from l' contains a finite sub covering.
Now let X be a vector space over the field of real numbers R. Only real vector spaces will be considered in the sequel, which enables simply speaking of the vector space X. If C is a subset of the vector space X then sp C will denote its linear span, that is to say,
sp C = {Z EX: z = t a4ox4o; n ~ 1, a40 E R, X40 E C, k = 1, ... ,n, n ~ I} . 40=1
Suppose that the vector space X is endowed with some topology 1'. If linear operations of the space X are continuous with respect to this topology then we say that (X, 1') is a topological vector space.
For a point x EX, any element of T which contains the point x is called the neigh bourhood of x. The local base of zero is interpreted as a family 0 of neighbourhoods of zero such that any neighbourhood of zero contains some neighbourhood which be longs to O. In a topological vector space X, those and only those sets are open which may be represented as unions of shifts of sets belonging to a local base of zero. The space X is locally convex if there exists a local base of zero constituted by convex sets. Recall that a metric space (y, d) is called separable if it contains a countable everywhere dense subset, and complete if any Cauchy sequence (Yn, n ~ 1) C Y (that is to say, any sequence which satisfies limn,rn_oo d(Yn' Yrn) = 0) has a limit Y = limn_ooYn E Y (which means that limn_ood(Yn,Y) = 0).
A metric d on the vector space X is called invariant if
d(x + Z,Y + z) = d(x,y), (x,y,z EX).
A Hausdorff topological vector space is metrizable if and only if it has a countable local base of zero. In this case, in each metrizable Hausdorff topological vector space (X, T), one can find a function x 1-+ IIxll which maps X into [0,00) such that
(a) if IAI ~ 1 then IAxl ~ Ilxll (x EX);
(b) IIx + YII ~ Ilxll + Ilyll (x,y EX);
(c) IIxll = 0 {::::::} x = 0;
(d) the metric d(x, y) = IIx - yll is invariant and consistent with the topology T.
0.1. F-SPACES AND SEPARATING SETS OF FUNCTIONALS
The function 1\ . 1\ is called quasinorm. The properties (a) and (b) imply that
Ilxll = II-xII (x EX)
and, for all x E X and A :f:. 0,
where Al = (ent (lAI-l) + 1) -1, A2 = ent (IAI) + 1,
and ent (a) denotes the integral part of a E R.
3
Seminorms on a vector space X are defined as nonnegative functions p(x) , x E X such that
(a) p(AX) = IAlp(x) (A E R, x E X)j
(b) p(x + y) ::; p(x) + p(y) (x, y EX).
If moreover
(c) p(x) = 0 <=> x = 0,
then p(.) is called norm, and the space X is said to be normed. It is clear that each norm has all the properties of quasinorms. The sign II . II will denote quasi norms or norms; we shall stipulate what we mean in each concrete case. Once the convergence in metrizable topological vector spaces is considered, we shall assume that the metric driven by the quasi norm of the space is givenj the notation (X, 11·11) will be used in this case.
Let X be a vector space and a, (:J E R. The linear combination aA + (:JB of sets A and B from X is defined to be the set
aA + (:JB = {x EX: x = ay + (:Jz, YEA, z E B} .
If Kl and K2 are two compacts in a topological vector space X then their linear combination aKI + (:JK2 is also a compact set in X.
For a topological vector space (X, T), we denote by (X, T)* the space of all real valued linear functionals defined on the space X and continuous with respect to the topology T. In other words, (X,T)* is the space topologically dual to (X,T).
According to what we have said above, we shall often drop the sign denoting topology and speak of the topological vector space X and its topological dual X*.
The value taken by a functional f (J E X*) on an element x (x E X) is denoted by f(x) or by (x,!).
0.1.2 Separating sets of functionals and weak topologies
Assume that X is a topological vector space. A set T ~ X* is called sepamting (T separates points of the space X) provided that x = 0 if and only if f(x) = 0 for all JET.
If the space X· separates points of X then it is usually said that X and x· are in duality.
Sets of continuous linear functionals enable introducing weak topologies on topo logical vector spaces; in this case, separating sets define the Hausdorff topologies.
Theorem 0.1.1 Let T ~ x· be a sepamting set on a topological vector space (X, T). To each functional f E T and each positive integer n, associate the set
U(J,n) = {x EX: lJ(x) I <~}.
Then the family of all finite intersections of the sets U(J, n) forms a local base of zero in the topology BT , which converts X into a locally convex topological vector space. In this case, the topology OT is majorized by the original topology T, and (X, OT)· = sp T. If, moreover, the set T is countable then the space (X, OT) is metrizable.
The topology OT is called the T-weak topology and is commonly denoted by O'(X, T). Observe that O'(X,T) = O'(X,sp T).
Each element x of a topological vector space X naturally defines a linear func tional on the space X·,
l",(f) = f(x) (J EX·).
In such a way, the space X induces on X· some topology denoted by 0'( X· ,X) and called the *-weak topology. The following two statements going back to S.Banach give a scope of the most important properties of this topology.
Theorem 0.1.2 (Banach-Alaoglu) If U is a neighbourhood of zero in a topolog ical vector space X then its polar
If' = {f E X· : sup If(x)1 ::; I} "'EU
is compact in the *-weak topology. In particular, if X is a normed space then any closed, in the corresponding norm of the dual space, ball in X· is compact in the *-weak topology.
Remark. If the space X is separable then any set in x· which is compact in the *-weak topology, is metrizable in this topology. 6.
Theorem 0.1.3 Suppose that X is a complete sepamble locally convex topological vector space and assume that some linear functional 1 is given on the space X·. If limn_co l(fn) = 0 for each sequence (fn, n ~ 1) C X· which tends to zero in the topology O'(X·, X), then there exists a unique element x E X such that
l(J) = f(x) (f EX·).
0.1.3 F-spaces
A complete metrizable topological vector space X which is in duality with its topo logically dual space will be called the F -space. One can also say that an F -space is a complete quasi-normed topological vector space X, whose dual space X' separates points of the space X.
If an F-space is locally convex then it is called the F'rechet space. Along with the FrecMt spaces, the class of F-spaces includes Banach (complete normed) and Hilbert spaces (those Banach spaces whose normes are driven by scalar products (. ,.), that is to say, 11.11 = (. ,.)1/2).
The pair (.t, II . II) will either stand for an F-space X with quasinorm II . II or for a Banach space X with norm II . II j we shall claim explicitly what we mean in each case.
As an elementary example of some concrete space, consider the space of all real valued sequences RNj this space would become a FrecMt space if we introduce the quasinorm
II xii = f: Tn IXnl (x = (Xn' n ~ 1), Y = (Yn, n ~ 1». n=1 1 + IXnl
The topology induced by this quasinorm is nothing else than the Tychonoff topology of the coordinate-wise convergence. We shall also consider different subspaces in R N , such as eoo • c, and co. Recall that eoo is the space of bounded sequences, c the space of convergent sequences, and Co the space of sequences which converge to zero. Either of these spaces is Banach space with respect to the norm Ilxll = sUPn>1 Ixnl. However, c and Co are separable, while eoo fails to have this property. The- space £2 C RN of square sum mabie sequences is a separable Hilbert space with respect to the scalar product
00
(x, y} = L XnYn (x, Y E £2)' n=1
In this case,
Ilxll = J(X, X} = (~x!) 1/2
To give examples of the F-spaces which are not FrecMt spaces, consider the spaces 11' (0 < p < 1) ofreal-valuedp-th power summable sequences x = (Xn, n ~ 1), endowed with the quasinorm
00
IIxll = L IXkl1' . k=1
One can easily check that the coordinate functionals On, n ~ 1, (here, On(x) = xn) belong to 1;.
Consider the spaces L1'[O, I] (0 < p < 1) whose elements are the Lebesgue measurable [unctions g on [0,1] which satisfy J~ Ig(t)IPdt < 00 (almost everywhere equal functions are interpreted as identic). These spaces are complete metrizable topological vector spaces with respect to the quasinorm
IIgll = 11Ig(t)I1'dt.
However, one can check that L;[O,IJ = {O} which means that Lp[O, IJ (0 < p < 1) are not F -spaces.
The next point we focus on is a property of separating sets in separable F -spaces.
Theorem 0.1.4 Let X be a sepamble F-space and T ~ X" a sepamting set. Then there exists a countable sepamting set Te ~ T.
PROOF. Since X is metrizable then it has a contable base of zero, whose elements will be denoted by Un, n ~ 1.
Let U~ be the polar (see Theorem 0.1.2) of the set Un. By Theorem 0.1.2 and the remark to this theorem, the polar U~ is a compact metrizable space with respect to the topology u(x", X). Consider the set T(n) = T n U~. Since T(n) is a subset of the separable metric space then it is also a separable metric space with respect to the induced metric. Denote by TJn) the separability set in T(n) and put
00
Te = U T~n). n=l
Since each x· belongs to some polar U~ then X· = U~l U~. This is why one can write
00 00
[TeJ. ;2 U [T~n)]. ;2 U (T n U~) = T, n=l n=l
where [ I. denotes closure in the topology u(X", X). This implies immediately that the set Tc separates points of the space X. By the construction, this set is countable .
Suppose that (Xk' II· Ilk) ,k = 1, ... ,n are F-spaces. The space
n
•
of all ordered n-tuples (Xl"'" Xn), where Xk E Xk, k = 1, ... , n, is called the Cartesian product of spaces XII" . ,Xn. If the Xk'S, k = 1, ... , n, are all the same space X then their Cartesian product is called the Cartesian power of the space X and denoted by xn. The Cartesian product Xl x ... X Xn is a vector space with respect to the natural operations of addition and multiplication by scalars:
(XII'" ,xn) + (YlI'" ,Yn) = (Xl + Yt.··· ,xn + Yn),
A (Xli .. ' ,xn ) = (Axl,"" AXn ).
This space is an F-space in the product topology. As a quasinorm, one can take for example
IIxllx1x ... xX" = l2ftn IIXkllk.
The dual space (Xl x ... x Xn)* is identified with the Cartesian product Xi x ... x X~, and the value taken by a functional I = (Ill' .. , In) on an element (XII' .. ,xn)
0.1. F-SPACES AND SEPARATING SETS OF FUNCTIONALS 7
is equal to E~=l f,,(x,,). If T" ~ X: is a separating set for X" then TI x ... X T" is separating in the space X 1 X ••. X Xn. If the X" 's, k = 1, ... , n, are separable F -spaces then XIX ••• X Xn is a separable F -space. If the X" 's, k = 1, ... , n, are FrecMt spaces then X I x ... X Xn is also a FrecMt space. If the X" 's, k = 1, ... , n, are Banach spaces then Xl x ... X Xn is also a Banach space.
The countable Cartesian products of a space X, that is to say, the spaces XN, will be considered in Section 2.1.
0.1.4 Classes of topologies on F -spaces
Assume that (X, r) is an F-space. Let us introduce the classes of topologies on the space X we shall need in what follows. If ~ is a topology on X then we denote by 9Jl{X,~) the class of topologies () on X which satisfy the following assumptions:
(i) () is a Hausdorff topology;
(ii) topology () is consistent with the vector structure of the space Xj
(iii) 0:5 ~j
(iv) (X,O)· sepamtes points of the space X.
If the intrinsic topology r is taken as ~ then the class of topologies 9Jl(X, r) is denoted by 9Jl(X). Along with topology r, the class 9Jl(X) contains a wide range of weak topologies. For example, given any separating set T ~ r, the topology (T(X, T) lies in the class 9Jl(X).
0.1.5 The space Rn
The space Rn (n 2: 1) will have several interpretations in what follows. First of all, the space Rn is interpreted as the space of ordered n-tuples u = (Ul, ... , un) of real numbers. This space is an n-dimensional vector space with respect to the intrinsic coordinate-wise operations of addition and multiplication by scalars. Moreover, this space is Euclidean with respect to the coordinate-wise scalar product
n
with the Euclidean (or h) norm
generated by this scalar product. Since all the finite-dimensional Euclidean spaces are isomorphic to each other,
then we frequently interpret Rn as general n-dimensional Euclidean space endowed with some scalar product (. ,.) and the induced norm II. II = (. , .)1/2.
8 CHAPTER o. RANDOM ELEMENTS AND THEIR CONVERGENCE ...
In some situations related to the use of matrices and matrix products, it may be convenient to interpret the space an as the space of column vectors U = (Ul, ... ,Un) :
where T denotes transposition. In this case, the coordinate-wise scalar product takes the form
(U,v) = uTv.
We shall stipulate which of the above versions we are going to use; otherwise, this will be clear from the context.
0.1.6 Linear operators and matrices
Given two topological vector spaces Xl and X2 , L(Xlo X2 ) is the class of continuous linear operators (maps) from the space Xl into the space X2 . We shall denote by I the identity operator from L(X}, Xl), that is to say, Ix = x, x E Xl. If A: Xl --+ X2 is an invertible linear operator then A-I denotes the inverse. The linear operator A * : X; -> Xt defined by the formula
f(Ax) = (A·f)(x) (x E Xl, f E Xn is commonly called the adjoint operator.
Given two normed vector spaces (Xl, 11·"1) and (X2' 11·112), we define the operator norm of A as
IIAxll2 IIAII = sup -,,-,,-. ",EX! X I ",~o
In this definition, IIAII < 00 if and only if A E L(Xl ,X2 ).
Assume that Ao, Al E L(X}, X2 ) and IIAoll1 < 00. Then, given that IIAII < 1 IllAoll1 ' the operator Ao + A is invertible and II(Ao + A)-III < 00.
Suppose that H is a separable Hilbert space with scalar product (. ,.) and the norm 11·11 = (.,.) 1/2. An operator A E L(H, H) is called the Hilbert-Schmidt operator if, for some orthonormal basis (ek' k ~ 1) in H,
The quantity IIAII2 is invariant with respect to the choice of basis (ek' k ~ 1) and is called the Hilbert-Schmidt norm. In this case,
00
IIAII~ = LILk' k=l
where ILk, k ~ 1, are eigenvalues of the operator A* A. The numbers ViIk, k ~ 1, are often called the singular numbers of the operator A. Observe that
IIAII = VilA· All = m:x.Jiik.
0.1. F-SPACES AND SEPARATING SETS OF FUNCTIONALS 9
A Hilbert-Schmidt operator A is called the nuclear operator (operator of the trace class) if
00
IIAIII = L.Jjii. < 00. le=1
The quantity II A III is called the nuclear norm of the operator A. The trace of a nuclear operator A is defined to be the quantity
00
tr A = L(Aek,ek), k=1
where (en, n ~ 1) is an arbitrary orthonormal basis in H. The trace of operator does not depend on the choice of basis (en, n ~ 1).
An operator A E L(H, H) is called symmetric (self-adjoint) if N = A, that is for all u,v E H
(Ax, y) = (x, Ay).
An operator A is called positive semi-definite if for all u E H
(Au,u) ~ o.
An operator A is called positive definite if for all u E H, u -=f. 0,
(Au,u) > o.
Observe that if a nuclear operator A is symmetric and positive semi-definite then J-tk = A~, k ~ 1, where (Ak, k ~ 1) C [0,00) are eigenvalues of the operator A, and IIAIII = tr A = Lh.1 Ale·
Suppose that A E L(H, H). An operator B is called the square root of the operator A (8 = AI/2 = v'A) if A = BB·. Each nuclear symmetric positive semi definite operator A has the square root v'A which is a Hilbert-Schmidt symmetric positive semi-definite operator.
An operator A is called unitary if it maps the whole of the space H onto the whole of the space H and preserves the scalar product, that is to say, for all u, v E H
{Au, Av} = (u, v).
Observe that this formula already holds given it is true for all u = v E H. If A is unitary then N = A-I and IIAII = 1.
The well-known theorem on polar decomposition states that any Hilbert-Schmidt operator A may be represented as A = UT, where U is a unitary operator and T = (A· A)1/2 a symmetric positive semi-definite Hilbert-Schmidt operator.
Assume that the Hilbert space H is represented as direct sum H = HI EB H2, where HI and H2 are orthogonal subspaces of H, that is to say, each vector h E H is uniquely represented as h = hi + h2, where hi E HI, h2 E H2, and (hi, h2 ) = o. The operator Prul defined by the formula
is called the projection operator (projector) onto the space HI. The projection op erator is symmetric, and moreover Pr~1 = PrHI and IIPrH,1I = 1.
Suppose that Rm (m ~ 1) and Rd (d ~ 1) are finite-dimensional Euclidean spaces with scalar products (. , ')m and (. , ·)d. Since both Rm and Rd are finite dimensional Hilbert spaces, then all the above mentioned statements and facts re main valid for operators from the classes L (Rm, R d ) and L (Rm, Rm) . Moreover,
observe that any linear operator from Rm into Rd is continuous and, since the space L (R m , R d) is a finite-dimensional vector space, all the norms are equivalent in this space.
Let {el, ... , em} and {e~, ... , ed} be some fixed bases in the spaces R m and R d,
correspondingly. Each operator A E L (Rm, R d ) may naturally be represented by a
(d ) t · A- [- lk=l •...• m h - (A ')' 1 d k 1 x m -rna fiX = ajk j=l ....• d ' were ajk = ek, ej ,J = , ... ,' = , ... , m. By associating the vector u E Rm to the column vector
and the vector Au to the column vector
ii' = «(u, e~)d, ... , (u, e~)d) T ,
one has, by virtue of the matrix algebra, that ii' = Aii. This matrix interpretation intrinsically leads to considering the spaces Rm and Rd as spaces of column vectors.
If we define operators via matrices, the adjoint operator corresponds to the trans posed matrix, and the diagonal matrix diag {I, ... ,I} represents the identity op erator I E L (R m , R m) , this is the reason we also denote this matrix by I; a unitary operator U E L (Rm, Rm) corresponds to an orthogonal matrix whose columns con stitute an orthonormal basis in the space of column vectors Rm.
In what follows, Mm denotes the space of square (m x m)-matrices [ajkl~k=\ with real-valued entries. The Euclidean (or Hilbert-Schmidt) norm of matrices [ajkl E Mm may be introduced as follows
This norm is consistent with the Euclidean norm II . II of the space of column vectors Rm, that is to say, ifu = (u\, ... ,Um)T then
As is usually done, we understand the trace of a matrix [ajkJ E Mm as the quantity
m
tr [ajkJ = L akk· k=l
We give a scope of some additional facts from matrix theory in Section 7.7.
0.2. u-ALGEBRAS AND MEASURES. CONVERGENCE OF MEASURES 11
0.2 a-algebras and measures. Convergence of measures
0.2.1 a-algebras
If Y is a topological space then we denote by B(Y) the u-algebra generated by open sets of the space y. This u-algebra is traditionally called the u-algebm of Borel sets or the Borel u-algebm of the space y.
The least u-algebra of sets of a topological vector space X with respect to which all the funcLionals of a family T ~ X· are measurable, will be denoted by C(X, T).
Fix a finite set of functionals (flJ ... , fn) C X·. This set generates au-algebra in the space X whose elements have the form
C/l ..... ,nCD) = {x EX: (f1(X),,,,, fn(x)) ED},
where D E 8(Rn). The union of all these u-algebras, with the tuple (!I, ... , fn) running through all possible finite sets from T, constitute an algebra which is denoted by A (X, T) and called the algebm of T -cylindrical sets of the space X. The u-algebra generated by the algebra A(X, T) is called the u-algebm of T -cylindrical sets; this u-algcbra is nothing else than C(X, T).
By comparing the Borel u-algebra to that of cylindrical sets, one can see that C(X, T) ~ B(X). In general, this inclusion is strict. However, in the case of sepa rable F-spaces, the following statement holds.
Theorem 0.2.1 Assume that X is a sepamble F-space and a set T ~ X" sepamtes points of X. Then C(X, T) = B(X).
Theorem 0.2.1 is based on the following well-known result.
Theorem (Suslin) Let t/J be a one-to-one continuous map of a complete sepamble metric space S into a metric space G. Then, for any set B E 8(S), the image t/J(B) belongs to H(G). In other words, the image of a Borel set under the map t/J is also a Borel set.
PROOF OF THEOREM 0.2.1. If the set T is uncountable then, by Theorem 0.1.4, there exists a countable subset Tc C T which separates points of the space X. If the set T is countable then put T" = T. Denote by X" the topological vector space (X, u(X, Tc)). Since u(X, Tc) $ T, where T is the intrinsic topology of the space X, and since X is separable then the space Xc is also separable. Moreover, since the separating set Tc is countable then the space Xc is a metrizable topological vector space (Theorem 0.1.1).
Let I : X ...... Xc be the intrinsic embedding of the space X into the space Xc, that is, Ix = x. This embedding is continuous since u(X, Tc) $ T. By the Suslin theorem, any Borel set in X is still a Borel set in Xc, that is, B(X) C B(Xc).
Let us show that B(Xc) c C(X, Tc). Since Xc is a metrizable separable topological vector space and the u-algebra C(X, Tc) is closed with respect to set translations,
one should only prove that C(X, Tc) contains all open, in the metric of the space Xc, balls centred at zero. This fact becomes evident once we take as metric consistent with topology a(X, Tc) the metric
d(x, y) = f 2-k I fie (x - y)1 (x, y EX), 1e=1 1 + Ifle(x - y)1
where (fie, k ~ 1) = Tc· The claim of the theorem follows then from the sequence of inclusions
B(X) c B(Xc) C C(X, Tc) c B(X). • Corollary 0.2.1 Assume that X is a separable F-space and IDl(X) the class of topologies on X (see 0.1.4). For any topology () E IDl(X), one has
B(Xo) = B(X),
where Xo = (X, ()).
Let (S, B) be a measurable space (that is to say, S is the basic set and 13 is some a-algebra of subsets of 8) and X be some F -space. A map 9 : 8 -+ X is called strongly measurable if it is (13, B(X))-measurable, that is g-I(B) E 13 for all BE B(X). Further, assume that T C; x*j a map 9 is called T-weakly measurable if, for any functional f E T, the map fog: x -+ f(g(x)) is (13, B(R))-measurable. The following version of the Pettis theorem is immediate from Theorem 0.2.1.
Theorem 0.2.2 If X is a separable F-space and a set T C; X* separates points of the space X then any T -weakly measurable map is strongly measurable and vice versa.
Let (V, B) be a measurable space, with V a vector space. The space (V, B) is called measurable vector space if the operation of addition
(x,y) ~ x + y (x,y E V)
is a (13 x 13, B)-measurable map of V x V into V, and the operation of homothety
(A,x) ~ AX (A E R,x E V)
is a (B(R) x 13, B)-measurable map of R x V into V. If gle, k = 1, ... ,n, are measurable maps of a measurable space (8,:F) into a mea
surable vector space (V, B) and Ale E R, k = 1, ... ,n, then the linear combination E~=1 Alegle is also measurable.
0.2. a-ALGEBRAS AND MEASURES. CONVERGENCE OF MEASURES 13
Let X be a topological vector space (X* f. {O}) and T ~ X*. Then the space (X, C(X, T)) is a measurable vector space. This implies, by Theorem 0.2.1, that if X is a separable F-space then (X, B(X)) is a measurable vector space.
Assume that X 1, ••. ,Xn are F -spaces. In the Cartesian product X I X .•. X
Xn, along with the Borel a-algebra B (XI x ... x Xn) , one can also consider the least a-algebra with respect to which all the maps (XI, ... ,xn ) ~ Xk, k = 1, ... ,n (projections) are measurable. This a-algebra is called the product oj a-algebras and denoted by
n
B(X1) x ... x B(Xn) = IT B(Xk). k=1
Given that Xk = X, k = 1, ... , n, we use the notation Bn(x). It is clear that the product of a-algebras is just the least a-algebra which contains all the sets Al x ... x An, where Ak E B(X), k = 1, ... ,n. Moreover, it is evident that B(XI ) x " . x B(Xn) c B (XI X ... x Xn). If X Il .•. , Xn are separable F-spaces then XI x ... X Xn is also a separable F-space, this is why Theorem 0.2.1 implies that, in this case, one has
0.2.2 Pre-measures, measures and characteristic functionals
Let E be a set where some algebra A ofits subsets is fixed. A pre-measure (or finitely additive measure) on (E, A) is defined to be an arbitrary map J.L : A - [0, +(0) which satisfies the finite additivity condition: J.L(A 1 U A2) = J.L(A 1) + J.L(A 2) for AI, A2 E A such that Al n A2 = 0. A pre-measure is called measure on (E,A) if it possesses the property of a-additivity, that is to say, if
for any countable family (Ak' k ~ 1) of pair-wise disjoint sets from A, whose union also belongs to the algebra A. A measure (pre-measure) J.L on (E, A) is called probability if 11,(E) = 1. We shall mainly focus on the probability measures in what follows.
An important role in measure theory belongs to the Carathcodory extension theorem. Let a(A) be the a-algebra generated by an algebra A. The CaratModory theorem claims that each pre-measure J.L defined on (E, A) may (uniquely) be ex tended to a measure on (E, a(A)) iJ and only if J.L is a measure on the algebra A.
The CaratModory extension theorem and Theorem 0.1.1 imply the next state ment.
Theorem 0.2.3 Assume that X is a separable F-space and a set T ~ X· separates points oj the space X. Each measure defined on the algebra A(X, T) of T -cylindrical sets may be extended in the unique manner to a measure on the a-algebra of Borel sets B (X:).
Measures defined on the u-algebra of Borel sets are traditionally called the Borel measures. An important property of the tightness of Borel measures is stated in the following theorem.
Theorem (Ulam) Each probability Borel measure J.L on a complete sepamble metric space S is tight, that is, for every e > 0, there exists a compact set KE C S such that
J.L(S\ K E ) ~ e.
Let X be a topological vector space which is in duality with its dual space X·. Consider a measure J.L on the u-algebra of B(X·, X)-cylindrical sets. The complex valued functional
c,ol'(u) = Ix exp(iu(x))J.L(dx) (u EX·).
is said to be the chamcteristic functional of the measure J.L. Let us give a scope of some properties of the characteristic functionals:
(a) the normalization condition: if J.L(X) = c then c,ol'(O) = c; in particular, one has c,ol'(O) = 1 in the case of probability measures;
(b) positive semi-definiteness: for any n ~ 1 and given any finite set of linear functionals u), ... , Un E X· and arbitrary complex numbers z), ... ,Zn, one has
n
L ZkZmc,ol'(Uk - Urn) ~ 0; k,m=)
(c) "-weak sequential continuity: if, for all x E X, one has un(x) ---+ u(x) then n_oo
c,ol'(un) ---+ c,ol'(u); n_oo
(d) if measure J.L is the convolution, J.Ll * J.L2, of measures J.Ll and J.L2, that is to say,
then
It is well-known that, in finite-dimensional Euclidian spaces, any Borel measure may be uniquely reconstructed after its characteristic functional. A similar fact also holds for the Borel measures in separable F-spaces. Moreover, a measure is characterized by the values the characteristic functional takes on an arbitrary linear separating set.
Theorem 0.2.4 Assume that J.Ll and J.L2 are Borel measures on a sepamble F -space X and T ~ X· is a sepamting set. If
c,o"'l (u) = c,o"'2(U) (u E sp T)
then J.Ll = J.L2·
The proof of this statement is immediate by virtue of its finite-dimensional ver sion and Theorem 0.2.3.
A Borel measure JI. on an F-space X is called symmetric if for any A E B(X)
JI.{A) = JI.(-A).
An important class of measures in topological vector spaces is constituted by Gaussian measures.
A probability Borel measure JI. on an F-space X is called Gaussian if it has the characteristic functional of the form
<p,..(u) = exp (ia(u) - ~ Q(u)) (u EX"),
where a(u) = Ix u(x)JI.(dx) , Q(u) = Ix (u(x) - a(u))2J1.(dx).
If a(u) = 0 for all u E X" then we say that the measure JI. has mean zero (is centred). Each zero-mean Gaussian measure is symmetric. The class of Gaussian measures is of considerable importance in this book, and the properties of these measures will repeatedly be appealed to in different sections.
Let us point out the simplest form of the Anderson inequality for zero-mean Gaussian measures JI. in a separable Banach space. This inequality asserts that, for any r > 0 and arbitrary u E X, one has
JI.{X: IIx - ull < r} :::; JI.{x: IIxll < r},
that is to say, the measure of a ball centred at zero decreases under translations. The following result is immediate from the Anderson inequality: for any r > 0
JI.{x: Ilxll < r} > O.
Indeed, given that {Xle' k ~ I} is a countable dense set in X, one has for each r > 0
1 = JI.(X) = JI. e.gl {x: IIx - xlell < r}) :::; t. JI.{x: Ilx - xlell < r}.
This is why a number k' should exist such that
JI. {x: IIx - xlc'lI < r} > o.
Apply the Anderson inequality to obtain the result required. Assume that Xl, ... ,Xn are separable F-spaces and Jl.1e are probability Borel
measures on XIc, k = 1, ... , n. The measure JI. on (Xl x ... X Xn, B(XI ) x ... x B(Xn)) which satisfies the equality
n
JI. (AI) X ••• x JI. (An) = n Jl.1e(A Ie ) (Ale E B(XIc), k = 1, ... , n) Ie=l
is called the product-measure and denoted by P.l X ... X p.n' Since (see 0.2.1)
then the product-measure is a probability Borel measure on Xl x ... X Xn. It is clear that for all fk E Xk and k = 1, ... ,n
hlx ... xXn exp (i E fk(Xk») P.(d(Xl"" ,xn» = fl h" exp (ifk(xk)) p.k(dxk).
Moreover, since, by Theorem 0.2.4, characteristic functionals of probability Borel measures on a separable F-space define these measure uniquely, the last formula holds if and only if the Borel measure on Xl x '" X Xn is a product-measure.
0.2.3 Weak convergence of measures
Let 8 be a metric space. A sequence (/-Ln, n;::: 1) of Borel measures on 8 is called weakly convergent if there exists a Borel measure p'oo such that, for any bounded continuous real-valued function (g(x), x E 8), one has
lim ( g(x)/-Ln(dx) = ( g(x)p.oo(dx). n-+ooJs Js
The weak convergence of a sequence of measures (/-Ln, n;::: 1) to a measure p. will be denoted by p'n ~ p., or simply by /-Ln ==} P. if it is clear what space 8 we
n-+oo n-+oo mean.
If 111 is a measurable map of (S, 8(8» into (R, 8(R» and p. is some measure on (8,8(8», then p. 0 111-1 is the distribution (measure) on (R,8(R» induced by the map 111. The weak convergence of /-Ln to P. means that, for each continuous real-valued function 1j; = (1j;(x),x E 8), one has
(0.1)
Given A E 8(8), we denote by 8A the boundary of the set A in what follows.
Lemma 0.2.1 The following assertions are equivalent:
(a) p.n~p.;
(b) for any A E 8(8) such that p.(8A) = 0, one has
/-Ln(A) - p.(A). n-+oo
Let us consider a family of measures {p.a} on (8,8(8», with the index set whose cardinality may be arbitrary. The family {p.a} is called weakly relatively compact if any infinite sequence of measures from {p.a} contains a weakly convergent
subsequence. The family {J.'a} is called tight if, for any e > 0, there exists a compact set KB C S such that
sup J.'a(S \ KB ) ~ e. a
The next theorem reveals the link between the weak conpactness and tightness of a family of measures.
Theorem (Prokhorov) Let {J.'a} be a family of probability Borel measures defined on a metric space S. Then the tightness of the family {J.'a} implies its weak relative compactness. Conversely, if the space S is complete and sepamble then each weakly relatively compact family {J.'a} is tight.
The Prokhorov theorem and Theorem 0.2.4 enable establishing a criterion of the weak convergence in separable F-spaces; we are going to formulate this criterion for probability measures only.
Theorem 0.2.5 Assume that X is a sepamble F-space, T ~ X· a sepamting set, (J.'n, n ;::: 1) a sequence of probability Borel measures, and CPn are chamcteristic functionals of the measures J.'n, n;::: 1. The sequence of measures (J.'n, n;::: 1) converges weakly if an only if:
(a) the sequence of measures (J.'n, n ;::: 1) is tight;
(b) the sequence (CPn(u), n;::: 1) converges for any U E sp T.
0.2.4 T-weak and essentially weak convergence
By reducing the quantity of functionals 'I/J for which formula (0.1) may hold, we can relax the notion of weak convergence of measures. In an F-space X, the functionals 'I/J may run through subsets of the space X·.
Assume that T ~ X*. We shall say that a sequence (J.'n, n ;::: 1) of Borel measures T-weakly converges if there exists a Borel measure J.' such that formula (0.1) holds for each 'I/J E T.
In a similar manner, we shall say that a sequence of measures (J.'n, n ;::: 1) essentially weakly converges if there exists a separating set T ~ X· such that the sequence (J.'n, n ;::: 1) converges (sp T)-weakly.
It is clear that the (sp T)-weak convergence of the sequence (J.'n, n ;::: 1) to a measure J.' is equivalent to that of the characteristic functionals of these measures on the set sp T, that is to say
( exp{if{x»J.'n{dx) -+ ( exp(if(x»J.'(dx) (J E sp T). ix n-ooix In order to proceed to formulating the conditions which lead to essentially weak
convergence of measures, let us introduce the definition of O-tightness of a family of measures.
Assume that 0 is a topology on an F -space X and that this topology is dominated by the intrinsic topology of the space. We shall say that the family {J.'a} of Borel
measures on X is (J-tight if for any e > 0 one can find a (J-compact set K. c X such that
If the topology (J is equivalent to the intrinsic topology of the space X, then the definitions of O-tightness and tightness of a family of measures just coincide.
Theorem 0.2.6 Let X be a sepamble F-space, T ~ r a sepamting set, (/.Ln, n ~ 1) a sequence of Borel measures on Xi CPn, n ~ 1, the chamcteristic functionals of the measures /.Ln, n ~ 1, and rot(X, u(X, T)) the class of topologies on X (see 0.1.4). Let the following assumptions hold:
(i) there exists a topology (J E rot(X, u(X, T)) such that the sequence (ILn, n ~ 1) is (J-ti9hti
(ii) for any U E sp T, the sequence (CPn(u), n ~ 1) is convergent.
Then the sequence of measures (/.Ln, n ~ 1) converges essentially weakly.
PROOF. Consider the topological vector space X, = (X,(J). Since (J ~ u(X,T) then Xo ~ sp T. By definition of the class of topologies rot(X, u(X, T)), the set Xo separates points of the space X and, by Theorem 0.1.4, there exists a countable separating set Te C .to. Theorem 0.1.1 yields that the space
is metrizable. It is clear that B(Xe) ~ B(X) and the measures /Ln, n ~ 1, may be considered as Borel measures on Xc' Since u(X, Te» ~ (J then, by assumption (i) of the theorem, the sequence of measures (/Ln. n ~ 1) is tight in the space Xc' The Prokhorov theorem implies that there exist a Borel measure IL on Xc and a subsequence (/.Lnk' k ~ 1) such that
Hence, for any u E sp Te , one has
tpnk(U) -+ cp(U) , Ic->oo
where cp(u} is characteristic functional of the measure /L. By assumption (ii),
tpn(u} -+ cp(u} (u E sp T). n->oo
One should only observe now that B(Xe} = B(X} by virtue of Theorem 0.2.1. This is why /L is a Borel measure on X. •
Let us make a special emphasis on the case of T = X·.
Corollary 0.2.2 If, under the assumptions of Theorem 0.2.6, one has T = X" then assumption (i) may be replaced by the following one:
(i') there exists a topology (J E rot(X) such that the sequence of measures (/.Ln, n ~ 1) is (J-tight.
0.3. RANDOM ELEMENTS AND THEIR CHARACTERISTICS 19
0.3 Random elements and their characteristics
0.3.1 Random elements
Let (n, F,P) be a probability space, that is to say, n is the basic space (the space of elcmentary events), F be some u-algebra of subsets (events) in OJ P be a probability measure (probability) on the measurable space (0, F). Let us assume, though it may often be redundant, that the u-algebra F is complete with respect to the measure P, which means that F contains all the subsets of the sets of P-measure zero.
Consider an F-space space X. A map X : 0 -+ X is called mndom element in the space X or X-valued mndom element,l if this map is (F,B(X}}-measurable, that is to say, {w : X(w) e B} e F whenever B e B(X). Random elements in a measurable space (8, B) may be defined along similar lines. With rare exception, we usually take separable F-spaces X as space 8 and the Borel u-algebra B(X} as u-algebra B, otherwise we state explicitly what space and what u-algebra should be considered.
If X = R (Rn, n > I) then we shall say, as it usually is, that X is a mndom variable (mndom vector).
The inverse image Fx = X-I B(X} of the u-algebra B(X} under the map X is called the u-algebm genemted (driven) by the mndom element X.
Unless otherwise stated, we assume that all random elements considered in each concrete case are defined on the space (0, F,P). This assumption causes no am biguity when one considers random elements in separable F-spaces, which enables speaking of the basic probability space (n, F,P).
To case notations, we shall sometimes drop the symbol w and denote the event {w : X(w) e B} just by {X e B}. Similar abbreviations will also be used in other formulas. If an event A has probability one then we shall follow the common practice in saying that A occurs almost surely (a.s.).
Random elements X and Y are called equivalent (P-equivalent, stochastically equivalent) if P{X =f Y} = O. Substituting random elements by their equivalents does not alter the properties of random elements we are studying in what follows, so that we shall deal in fact with the classes of equivalent random elements. In this context, the equality X = Y means that X = Y a.s.
When one would like to know whether a map from 0 to X is random element, the following version of Theorem 0.2.2 adapted to the probability context can be a help.
Theorem 0.3.1 Assume that X is a sepamble F-space, T ~ r a sepamting set, and X : n -+ X. The map X is mndom element if and only if, for any f e T, the map f(X) : n -+ R is mndom variable.
Let X h ... ,Xn be random elements in a separable F -space X. The fact that (X, B(X» is a measurable vector space, or if one wishes, Theorem 0.3.1, implies
I We retain the term 'random element' commonly accepted in the Russian mathematical litera ture. The reader should bear in mind that the concept of' X -valued random element' is equivalent to that of 'X-valued random variable.' ('lnmslator's remark.)
that, for any real numbers all ... ,an, the linear combination alXI + ... + a .. X .. is an X-valued random element.
0.3.2 Distributions of random elements
Each X-valued random element X generates the probability measure
Px(A) = P{X E A} (A E B(X)),
which is called the distribution of the random element X. If X is a random element in a separable F -space X then its distribution Px is a
Borel measure on X. Let X and Y be two random elements defined, generally speaking, on two differ
ent probability spaces. We say that X and Y are identically distributed, or similar, or that Y is a copy of X, if Px = Py.
A random element X in a measurable vector space is called symmetric if X and - X are identically distributed. If X is a symmetric random element in an F -space then its distributions is a symmetric measure.
0.3.3 Mean values and characteristic functionals
The symbol EX will denote the mean value (mathematical expectation) of a random variable X. If X = (Xl, X 2 , • •• ,X .. ) is a random vector then EX = (EXI' EX2 , ••• ,
EX .. ). Let X be a separable F-space. Given an X-valued random element X, the mean value of X is defined by means of the Pettis integral.
More precisely, assume that an X-valued random element X satisfies the condi tion
EIJ(X)I < 00 (J E r). (0.2)
If there exists an element m E X such that
J(m) = EJ(X) (J E X*),
then m is called the mean value (mathematical expectation) in the Pettis sense, or the weak mean, and denoted by EX.
If the weak mean exists then it is unique. This is why one should be able to solve the problem of existence of the weak mean and that of its explicit construction. It is sometimes very easy to give an explicit formula for the weak mean. For example, let a random element X be symmetric and assume that condition (0.2) holds. Then it is immediate that EX is just zero element of the space X.
The mean obeys the property of linearity:
(i) if EXI and EX2 exist then for any a, fJ E R
E(aXI + fJX2 ) = aEXI + fJEX2 •
Apart from the property of linearity, mean values of random elements in a separable Banach space (X, II· II) possess the following properties:
(ii) if EX exists and A is a bounded linear operator which maps X into a Banach space Xl then E(AX) also exists and E(AX) = AE(X)j
(iii) if EIIXII < 00 then EX exists and IIEXII ~ EIIXII.
Let (X, 11·11) be a separable Banach space and X be an X-valued random element. If EIIXII < 00 then the Pettis mean coincides with the Bochner strong mean which is often denoted by In X (w)P(dw). The Bochner integral may be constructed following a scheme totally similar to the classic one used in introducing the Lebesgue integral. Observe that, for Bochner means, the Lebesgue theorem on taking the limit under the sign of mean remain valid (see 0.4.1).
For any random element X in a separable F -space X, the chamcteristic functional 'P x is defined as that of the distribution Px, that is
'Px(u) = Eexp(iuX) = Ix exp(iux)Px(dx) (u E X*)
(see 0.2.2).
0.3.4 Covariance characteristics
Consider a random element X of the weak second order. This means that
Elf(XW < 00 (J E X*).
For this random element, the bilinear form Q may be defined,
Q(g, f) = Eg(X)f(X) (g, f E X·)j
this form is called the covariance form of the element X. One can check that this form is:
(i) symmetric: Q(g, f) = Q(J, g) (g, f E x*)j
(ii) positive semi-definite: Q(J, f) ~ 0 (J E X*).
A linear operator C is intrinsically associated with the bilinear form Qj generally speaking, this operator maps the space X* into the space (X·)U of real-valued linear functionals over X·. The operator C is defined by the formula
(Cf)(g) = Q(g,f) (g,/ E X*)
and called the covariance opemtor of the random element X. Covariance operators inherit the properties of the bilinear form Q, that is to say, they are
(i) symmetric: (Cf)(g) = (Cg)(J) (J,g E x*)j
(ii) positive semi-definite: (C f) (f) ~ 0 (f E X*).
Along with covariance forms (operators), correlation forms (operators) may be introduced. A bilinear form Qo defined by the formula
Qo(g, f) = E(g(X) - Eg(x)) (f(X) - Ef(X)) (g, f EX·)
is called the correlation form of the mndom element X. Correlation forms are sym metric and positive semi-definite. A symmetric positive semi-definite linear operator K : X* --+ (X*)U defined by the formula
(Kg)(f) = Qo(g, f)
is called the correlation opemtor of the mndom element X. Much like the covari ance characteristics, correlation forms (operators) are symmetric and positive semi definite. It is clear that, given that a random element X has weak mean EX, the correlation form (operator) of X coincides with the covariance form (operator) of the random element X - EX. If EX = 0 then the covariance and correlation char acteristics are just the same. The covariance operator of a random element X is often denoted by COY X.
In those cases where X· = sp T, which occurs for example in the finite-dimension al spaces Rn, n?: 1, or in the sequence space R N , the covariance form Q may be uniquely reconstructed after the function
Q(f, g) (f, gET).
For example, if X = (XI"'" Xn) is a second order random vector then COY X is conveniently represented in the form of covariance matrix
cov X = [EXkXml~,m=1 = EXT X.
0.3.5 Independent random elements
Let (f2,:F,P) be a probability space and {Ac.} be a nonempty family of events, Aa E :F. We say that the events {Aa} are independent (jointly independent) if for any finite set of indices a!, ... ,an
Now, let {:Fa} be a nonempty family of a-algebras, :Fa ~ :F. We shall say that the a-algebms {:FaJ are independent (jointly independent) if, for any finite set of indices 0'1, ... ,an and any Al E :FOIl' ... , An E :Fan' one has
P COl Ak) = fl P(Ak)'
Let {Xa} be finite or infinite family of random elements in an F-space X. We shall say that random elements {Xa} are independent (jointly independent) if for any finite set of indices a!, ... ,an and any BI, ... ,Bn E B(X)
P (fl{Xal< E Bd) = t1 P{Xal< E Bk }.
In other words, {X",} is a family of independent random "elements if {rx.J is a family of independent O"-algebras, where rXa denotes the O"-algebra generated by the random element X",.
The following lemma will often appear in the proofs of various statements. Recall that, given a sequence of events (Ak, k ~ 1), the event lim Ak = n~1 Uk::n Ak means that infinitely many events occur in the sequence (Ak' k ~ 1).
Lemma (Borel-Cantelli) If
then P (lim Ak) = o.
If (Ak' k ~ 1) is a sequence of independent events and
then
The asymptotic properties of infinite sequences of jointly independent random element are closely related to the Kolmogorov 0-1 law. Let {Xn,n ~ I} be a sequence of independent random elements in an F-space X. Denote by Bm the 0"
algebra generated by the random elements Xn , n ~ m, and consider the so-called tail O"-algebra Boo = n~=IBm. The Kolmogorov 0-1 law asserts that, for any B E Boo, the probability P(B) equals 0 or 1.
Assume that XI, ... , Xn are independent random elements in a separable F -space _~ and gl, ... , gn are measurable maps of (X, B(X)) into (R, B(R». Then
n n
E II 9k(Xk) = II Egk(Xk), k=1 k=1
where we intrinsically assume that all the above means are defined. Since (see 0.2.1) B(xn) = Bn(x) then (XI, ... ,Xn) is random element in the separable F-space xn. Random elements X I, ... , Xn are independent if and only if the distribution of (XI, ... ,Xn) in xn is the product measure PX1 x ... X PXn • This is why random elements X I, ... ,Xn are independent if and only if one has for any II, ... ,fn E XO
Eexp (i E fk(Xk») = fl Eexp(ifk(Xk».
Distribution of the sum Lk=1 Xk of random summands is given by the convolu tion PX1 * ... * px ... Then the characteristic functional has the following form
Eexp (if (E Xk)) = fl Eexp(if(Xk)) (J E XO).
This formula implies immediately that the sum of independent symmetric random clements is a symmetric random element.
0.4 Convergence of random elements
We shall consider the three basic types of convergenre, the almost sure convergence (or, which is the same, convergence with probability one), convergence in probability and in distribution, in studying the convergenre of sequences of random elements.
In a metric space, the almost sure convergenre and that in probability are inter preted as those in the metric of the corresponding spare. If we consider F -spares, the definition and general properties of these types of convergence are similar to those in the real-valued case. Let us recall the basic definitions and notations.
0.4.1 Almost sure convergence
Assume that (X, II· II) is an F-space and (Y", n ~ 1) a sequence of X-valued random elements defined on the basic probability space (0, .r,P). We say that the sequenre (Y", n ~ 1) converges almost surely, or with probability one, to an X-valued mndom element Y if
P {w: lim IIY,,(w) - Y(w)II = o} = 1. "_00 We shall denote the almost sure convergence by one of the folowing notations
Y" ~ Y, Y" -- Y a.s. n ....... oo n ...... oo
We shall say that the sequenre (Y", n ~ 1) converges almost surely, if one can find an X-valued random element Y such that Y" ~ Y. In order to emphasize that "_00 the convergence is interpreted in the intrinsic metric (quasinorm) of the space X, we shall sometimes say that the sequenre (Y", n ~ 1) strongly converges almost surely. Sinre the space X is always complete then almost sure convergenre is equivalent to the almost sure Cauchy property: P(Oc) = 1, where
S1c = {w: lim IIY,,(w) - Ym(w) II = o} n,m ...... oo
is the convergence set of the sequence (Y", n ~ 1). If the sequence (Y", n ~ 1) converges almost surely then the limit lim,,_oo Y,,(w)
fails to be defined for those w's which belong to the set 0 \ Oc of P-measure zero. One can complete the definition of lim.,_oo Y,,(w) for these w's by any fixed element of the space X, say, by zero element, so that we may consider lim,,_oo Y,,(w) to be an X-valued random element.
Suppose that (Y", n ~ 1) is a sequence of independent X-valued random ele ments. The Borel-Cantelli lemma implies that the sequence (Y", n ~ 1) almost surely converges to zero, that is to say, II Y" II ~ 0 if and only if for any e > 0
"_00
00
This criterion will often be used in what follows.
0.4. CONVERGENCE OF RANDOM ELEMENTS 25
Assume that (Yn , n ~ 1) is a sequence of random elements in a separable Banach space (X, II . II), and EllYnll < 00. Then the Bochner means (see 0.3.3) obey the Lebesgue theorem on passage to the limit: given that Yn ~ Y and n_oo E sUPn~ 1 IIYnll < 00, the mean EY exists and limn-oo EYn = EY, that is to say,
lim IIEYn - EYII = O. n--+oo
0.4.2 Convergence in probability
The sequence (Yn, n ~ 1) is said to be convergent in probability to an X-mndom element Y if for any e > 0
lim P {llYn - YII > e} = 0, n--+oo
which is written as Y = P- lim Yn , Yn ~ Y.
n-+oo n-+oo
We shall say that the sequence (Yn , n ~ 1) con verges in probability if there exists an X-valued random element Y such that Y" ~ Y. Convergence in probability is "_00 equivalent to the Cauchy property in probability: for any e > 0
lim P{IIYn-Ymll>e}=O. n,m-+oo
Almost sure convergence implies that in probability. On the other hand, a se quence (Yn , n ~ 1) converges in probability to Y if and only if any subsequence of the sequence (Yn , n ~ 1) contains a subsequence which converges to Y almost surely.
Along with convergence in the original metric of the space X, we shall also con sider convergence in weaker topologies. This is why, in order to emphasize what we mean, we shall speak, for example, of the almost sure convergence, or the conver gence in probability in the norm of a Banach space, etc.
0.4.3 Convergence in distribution
The convergence in distribution of a sequence (Yn , n ~ 1) is interpreted as weak convergence of the sequence (Py", n ~ 1) of the corresponding distributions (see 0.2.3 and 0.3.2).
Given that the sequence of distributions (Py", n ~ 1) weakly converges to dis tribution P of some X-valued random element Y, which need not be defined on (n, ,r,P), we use the notation
V Yn --+ Y. n--+oo If the random elements Yn , n ~ 1, and Y are defined on the general probability
space then the convergence in distribution means, in terms of the mean values, that for any bounded continuous real-valued function g(x), x E X,
Eg(Yn ) --+Eg(Y). n-oo Theorem 0.2.5 implies the following criterion of convergence in distribution.
Theorem 0.4.1 Assume that X is a separable F-space, T ~ X· a separating set, and (Yn , n ~ 1) a sequence of X-valued random elements. In order for the sequence (Yn , n ~ 1) to be convergent in distribution, the following conditions are necessary and sufficient:
(i) the sequence of distributions (Py", n ~ 1) is tight;
(ii) for any f E sp T, the sequence of random variables (f(Yn ), n ~ 1) converges in distribution.
Convergence in probability implies that in distribution. The reverse is false in general. However, there are some situations when the reverse also holds. Let us point out at one of such situations.
Lemma 0.4.1 Let (Yn , n ~ 1) be a sequence of random elements in a separable F-space (X, II . II) and y E X a nonrandom element. If (Yn, n ~ 1) converges in distribution to the element y then (Yn , n ~ 1) converges to y in probability.
PROOF. For any e > 0, set
B,,(y) = {x EX: IIx - yll :::; e},
Consider the point measure fly concentrated on the element y, that is
where 1A denotes indicator function of the set A. By our assumption,
For any e > 0, the boundary of B,,(y) has the form
(}B,,(y) = S,,(y) = {x EX: Ilx - yll = e}.
This is why, for any e > 0,
fly ({}B,,(y)) = 0
and hence, by Lemma 0.2.1, for any e > 0
lim P {llYn - yll :::; e} = lim P y" (B,,(y)) = fly (B,,(y)) = 1, n-+oo n-+oo
that is to say, Yn ~ y. n ..... oo •
0.4.4 T-weak and essentially weak almost sure convergence
Assume that X is a separable F -space and T ~ X·. We shall say that a sequence (Y .. , n ~ 1) T-weakly converges almost surely if there exists an X-valued random element Y such that
I(Yn ) ~ I(Y), (f E T).
In a similar manner, we say that the sequence (Y .. , n ~ 1) essentially weakly con verges almost surely if there exists a separating set T ~ X· such that the sequence (Y .. , n ~ 1) T-weakly converges almost surely.
It is clear that the weak almost sure limit is defined uniquely. Observe also that the T-weak almost sure convergence and (sp T)-weak almost sure convergence are equivalent.
Now we are going to formulate a criterion of T -weak almost sure convergence for countable separating sets T.
Theorem 0.4.2 Let X be a sepamble F-space, TeX· be a countable sepamting set, and (Yn , n ~ 1) be a sequence 01 X-valued mndom elements. The sequence (Yn , n ~ 1) T -weakly converges almost surely il and only il the lollowing conditions hold:
(i) lor any lET, the sequence 01 mndom variables (f(Yn ), n ~ 1) converges almost surely;
(ii) there exists a Borel measure p. on X such that lor any I E sp T
Eexp (i/(Yn )) -t ( exp (i/(x)) p.(dx). n-+OO Jx
PROOF. The necessity of conditions (i) and (ii) is immediate from the definition of T -weak almost sure convergence. The sufficiency may be proved using the method proposed by K. Ito and M. Nisio (1968a).
Consider the space RN of all real-valued sequences and endow this space with the Tychonoff topology. Recall that the a-algebra B(RN) coincides with that of cylindrical sets. Set T = (fk, k ~ 1) and put
ek = lim Ik(Y .. ) a.s., k ~ 1. "-+00
By condition (i), the random variables ek, k ~ 1, are well-defined. Consider the maps
4>: X --+ RN, 4>(x) = (fk(X), k ~ 1);
=:: n --+ RN, 2(w) = (elc(W), k ~ 1).
It is clear that 4> is a continuous linear map and 2 is measurable. Assume that a;, j = 1, ... , m, are arbitrary real numbers. By assumption (ii),
Eexp (ita;e;) = J!..~Eexp ((ta;l;) (Y .. )) = l exp (ta;I;(X)) p.(dx). 3=1 3=1 X 3=1
28 CHAPTER;O. ,RANDOM ELEMENTS AND THEIR CONVERGENCE ...
Both the maps cP and 3 generate the same measure II on (RN, B(RN») ,
Since the set T separates points of the space X then cP is a one-to-one continuous map of X into CP(X). By the Suslin theorem (see 0.2.1), CP(B) E B(RN) whenever BE B(RN). This is why the inverse map cp-l is a measurable map of cp(X) into X. Set
{ cp-l(X), if x E cp(X),
W(x) = 0, if x rt cp(X).
The map W is a measurable map of (RN, B(RN») into (X, B(X», and this is why the superposition Y = W 0 3 is an X-valued random element.
Let us show that the sequence (Yn , n ~ 1) T-weakly converges almost surely to Y. Employ the definition of the map W to observe that, given C = (Clla k ~ 1) E cp(X), one has
Hence for all k ~ 1
Since P{w: 3(w) E cp(X)} = p, (cp-l(cp(X») = p,(X) = 1
then fOl" all fie E T
P { lim fle(Yn) = fle(Y)} = P {(Ie = fle(Y)} = 1. n-+oo
•
Theorem 0.4.3 Assume that X is a sepamble F -space, T ~ x· a sepamting set, and (Yn , n ~ 1) a sequence of X-valued mndom elements. Assume that the following conditions hold:
(i) for any f E T, the sequence of mndom variables (J(Yn) , n ~ 1) converges almost surely;
(ii) there exists a topology (J E !lJt(X,u(X, T» such that the sequence of measures (PYn , n ~ 1) is (J-tight.
Then the sequence (Yn , n ~ 1) essentially weakly converges almost surely.
Corollary 0.4.1 If, given the assumptions of Theorem 0.4.3, one has T = r then condition (ii) of the theorem may be replaced by the following one:
(ii') there exists a topology () E rot{X) such that the sequence of measures (Py",
n ~ 1) is ()-tight.
Apply Theorem 0.1.2 to obtain the following result from Corollary 0.4.1.
Corollary 0.4.2 Let (r, II . II) be a sepamble dual Banach space. Assume that the following conditions hold:
(i) for any f E X, the sequence of mndom variables (f(Y,,) , n ~ 1) converges almost surelYi
(ii) lim supP{IIY,,1I > c} = o. c~oo n
Then the sequence (Y", n ~ 1) essentially weakly converges almost surely.
Theorem 0.4.3 demonstrates the effect over the almost sure convergence of a general sequence of X-valued random elements produced by the concentration of distributions of these elements on the sets which are compact in weak topologies. The statement of Theorem 0.4.3 will be made much more sharp in Chapter 1 for series of independent symmetric random elements. For now, as an application of Theorem 0.4.3, we are going to give a criterion of convergence in probability, similar to that of convergence in distribution.
Theorem 0.4.4 Assume that X is a sepamble F-space, T ~ X· a sepamting set, and (Y", n ~ 1) a sequence of X-valued mndom elements. In order for the sequence (Y", n ~ 1) to be convergent in probability, it is necessary and suffiient that the following condition be satisfied:
(i) the sequence of distributions (Py", n ~ 1) is tighti
(ii) for any f E T, the sequence of mndom variables (f{Y,,) , n ~ 1) converges in probability.
PROOF. The necessity of assumptions (i) and (ii) is immediate. Let us prove that they are also sufficient.
Put J.L" = py". By Theorem 0.1.4, the set T may be assumed to be countable. Apply the diagonal method and take into account assumption (ii) to find a subse quence (Y"Ic' k ~ 1) in the sequence (Y", n ~ 1), such that, for any f E T, the sequence of random variables (f(Y"Ic)' k ~ 1) would converge almost surely. More over, assumption (i) and Theorem 0.2.5 imply that there exists a Borel measure J.L on X such that
This is why for any f E sp T
Eexp (if(Y"Ic)) --+ r exp{if(x))J.L(dx) k-+oo lx
This means that the sequence (Yn ,.., k ~ 1) satisfies all the assumptions of Theo rem 0.4.2. Then this sequence T-weakly converges almost surely to some X-valued random element Y. This fact and assumption (ii) imply that for any f E T
(0.3)
Assume that Zn = Yn - Y, n ~ 1. We are going to show that
(0.4)
Take some c > O. By assumption (i) and the Ulam theorem, one can find compact sets K~ and K: such that
s~pP {Yn ct K:} <~, p {Y ct K:} < ~. Put
Then
supPZn (X \ K,;) = supP {Zn rt. Ke} ~ supP {Yn rt. K;} + P {Y ct K;'} < c. n n n
This is why
(a) the sequence of distributions (PZn , n ~ 1) is tight.
Moreover, the following convergence holds by (0.3)
(b) I{)n(f) -+ 1 = cpa(f) (f E sp T),
n_oo
where I{)n is characteristic functional of the random element Zn, and I{)o is character istic functional of the probability measure concentrated in zero element of the space X. Formula (0.4) follows from assertions (a) and (b), and Theorems 0.2.5 and 0.2.

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