Gaussian Random Functions

346 Pages·1995·9.987 MB·English

by  M. A. Lifshits (auth.)

Gaussian Random Functions
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 322
Gaussian
Random Functions
by
M. A. Lifshits
St. Petersburg University of Finance and Economics,
St. Petersburg, Russia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A CoI.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-90481-4528-7 ISBN 978-94-015-8474-6 (eBook)
DOI 10.1007/978-94-015-8474-6
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1st edition 1995
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.
TABLE OF CONTENTS
Preface ............................................................................................. vii
Section 1. Gaussian distributions and random variables................................. 1
Section 2. Multi-dimensional Gaussian distributions ..................................... 8
Section 3. Covariances ....................................................................... 16
Section 4. Random functions ................................................................ 22
Section 5. Examples of Gaussian random functions..................................... 30
Section 6. Modelling the covariances ...................................................... .41
Section 7. Oscillations ........................................................................ 53
Section 8. Infinite-dimensional Gaussian distributions .................................. 68
Section 9. Linear functionals, admissible shifts, and the kerneL ...................... 84
Section 10. The most important Gaussian distributions .................................. lOl
Section 11. Convexity and the isoperimetric inequality .................................. 108
Section 12. The large deviations principle ................................................. 139
Section 13. Exact asymptotics of large deviations ........................................ 156
Section 14. Metric entropy and the comparison principle ................................ 177
Section 15. Continuity and boundedness .................................................. 211
Section 16. Majorizing measures ........................................................... 230
Section 17. The functional law of the iterated logarithm ................................. 246
Section 18. Small deviations ................................................................ 258
Section 19. Several open problems ......................................................... 276
comments ....................................................................................... 282
References ....................................................................................... 295
Subject Index ................................................................................... 327
List of Basic Notations ..................................................................... 331
v
PREFACE
It is well known that the normal distribution is the most pleasant, one can even say, an
exemplary object in the probability theory. It combines almost all conceivable nice
properties that a distribution may ever have: symmetry, stability, indecomposability, a
regular tail behavior, etc. Gaussian measures (the distributions of Gaussian random
functions), as infinite-dimensional analogues of tht< classical normal distribution, go to
work as such exemplary objects in the theory of Gaussian random functions. When one
switches to the infinite dimension, some "one-dimensional" properties are extended
almost literally, while some others should be profoundly justified, or even must be
reconsidered. What is more, the infinite-dimensional situation reveals important links
and structures, which either have looked trivial or have not played an independent role in
the classical case. The complex of concepts and problems emerging here has become a
subject of the theory of Gaussian random functions and their distributions, one of the
most advanced fields of the probability science. Although the basic elements in this field
were formed in the sixties-seventies, it has been still until recently when a substantial
part of the corresponding material has either existed in the form of odd articles in various
journals, or has served only as a background for considering some special issues in
monographs.
The last decade not only enriched the theory of Gaussian random functions with
several new beautiful and important results, but also land marked a significant shift in the
approach to presenting the material. New, simple and short, proofs of a number of
fundamental statements have appeared, based on a systematic use of the convexity of
measures and the isoperimetric inequalities.t At the moment, a series of the most essen
tial properties of Gaussian random functions and measures enables a coherent, compact,
and mathematically complete presentation, without either demanding cumbersome
calculations or invoking extraneous methods. An attempt of such presentation is made in
this book.
This book is, first of all, intended as a textbook. Therefore, we focus on quite a few
fundamental objects in the theory and try to discover their interrelations. By concentrat
ing on the principal points, we do not aim at covering all the material available by now.
Here are the titles of basic plots presented to the reader in the book:
- the kernel of a Gaussian measure;
- the model of a Gaussian random function;
- oscillations of sample functions;
- the convexity and isoperimetric inequalities;
- studying the regularity of sample functions by means of entropy characteristics and
the majorizing measures;
- functional laws of the iterated logarithm;
Tn.
A similar trend can also be traced in other fields of the probability theory (see [L-
vii
viii Preface
- estimates for the probabilities of large deviations.
The last plot fills up an especially important place. It is given the role of a denoue
ment, a point where other lines of the presentation cross over, a moment when the links
among all basic elements of the theory become revealed.
Let us try to outline briefly a general idea of each of the above mentioned plot lines.
Kernels. The distribution of a random function is naturally interpreted as a measure
in a space of sample functions. In the Gaussian case, one can be surprised by finding a
linear subspace in this basic space (the kernel or, as it is often called, the reproducing
kernel Hilbert space, RKHSt), intrinsically related to the original distribution. More
over, this kernel appears to be equipped with a natural structure of a Hilbert space, in
which all the properties of the distribution are "encoded".
Models. Since a Gaussian random function can be completely described by its co
variance structure, it may be merely treated as a set in the Hilbert space of square
integrable random variables. By the isomorphism of Hilbert spaces, one can translate the
set considered to another Hilbert space, which is more convenient for calculations, or
which is more stimulative for the intuitive perception. As a rule, the result of this trans
lation (it is called the model of a random function) bears no resemblance to a space of
random variables or to other probability objects. Nonetheless, the geometric form of a
model completely characterizes the random function under consideration, and provided
that the method of modelling has been chosen appropriately, it may help in eliciting its
properties.
Oscillations. Let a point t belong to the domain of definition of a function f. Then
the oscillation of f at t is, roughly speaking, the difference between the maximum and
the minimum of f in an infinitesimal neighborhood of the point t (for example, the
oscillation of a continuous function equals zero). When we deal with oscillations of dis
continuous sample paths of a random process, it seems that the oscillation must be a ran
dom variable. For a Gaussian process, however, the oscillation appears to be nonran
dom, and this quantity plays an essential part in studying a number of problems in this
book.
The isoperimetric property of half-spaces and the convexity of measures. A ball is
well known to be a solution to the isoperimetric problem for the Lebesgue measure in a
finite-dimensional Euclidean space, which means that it has the least surface area among
all the sets of equal measure. It turns out that when the Lebesgue measure is replaced by
a Gaussian measure, a half-space, not a ball, becomes a solution to this problem. More
over, provided that the isoperimetric problem is reset appropriately, this half-space will
be an extremal set for the infinite-dimensional Gaussian measure.
Closely related to the solution of the isoperimetric problem is the concentration
principle for a Gaussian measure, asserting that the classical exponential estimates for
the decay of tails of the normal distribution, in a certain sense, remain also valid in the
infinite-dimensional case.
t This tenn coming from a particular situation is unfortunate from the author's point of view.
Preface IX
The isoperimetric problem has much to do with the property of convexity of a
Gaussian measure. This latter means that, given any sets (events) A and B and a
number yE (0,1), the measure (probability) of the set yA + (l-y)B can be bounded
from below in terms of y and the measures of A and B. This statement can be formal
ized in several ways and proves an indispensable tool for estimating the Gaussian proba
bilities.
Regularity of sample functions. This plot includes studying the question of bound
edness and continuity of sample functions; in the continuous case, it is natural to go even
further and search for an optimal modu!us of continuity. The key to these problems is in
endowing the parametric set, where a random function is defined, with an intrinsic metric
and studying the properties of the metric space obtained. The main part in this develop
ment belongs to the tools borrowed from analysis, such as the metric entropy and major
izing measures.
Large deviations. The probabilities of large deviations are the probabilities of
events that sample functions deviate strongly from the corresponding mean. These prob
abilities are of the utmost importance in statistics, where they give rise to various criteria
for matching a probability model chosen to the statistical data available, as well as in
many other applications. In this section, a vital role is played by the principle of large
deviations, which goes far beyond the Gaussian context. This principle asserts that as
soon as a set of sample functions A deviates from the mean function, the probability of
A approximately decreases as
exp {-inf I(X)},
XEA
where I is a certain function (in the Gaussian case, a positive quadratic form); this func
tion is called the action functional, and it characterizes the degree of deviation from the
mean. The above mentioned formula not only provides the clue to calculating the proba
bilities of large deviations, but also shows that only those (usually, a few) elements
x E A for which the value of I (x) is minimal, contribute substantially to these asymp
totic probabilities.
The book is first of all intended for those who have already made their first steps in
studying the theory of stochastic processes and wish to go further, to the modern state of
this theory. To read it, one will need a good command of basic concepts of the measure
theory (a-algebra, measurability, measure, absolute continuity, and the change of vari
ables), probability theory (random variables, distribution functions, the mean and varian
ce of a random variable), the elements of linear algebra (linear spaces, operators, and
scalar products), and the principles of general topology (open, closed, and compact sets;
continuous mappings, metrics). Thus, our presentation is quite accessible for, say, a
university student who has chosen a mathematical specialty and has completed two or
three years of studies.
x Preface
Whichever topic we touch upon in this book, we always use one or another illustra
tive geometric interpretation (although infinite-dimensional) for solving the problems,
which are conceptually probabilistic. It is not surprising that in doing so we enter con
sistently to the area common for the probability theory, functional analysis, and measure
theory. Therefore, the mathematicians who specialize in any of these fields will also find
the issues of interest for them in this book. To make their reading more easy, we give all
necessary information from the theory of random processes (for a more profound
familiarity with it, we recommend the textbook by A.D.Wentzell [Wen2]).
The theory presented in this book is a result of efforts of several generations of
scientists. The first examples of Gaussian random functions go back to N. Wiener and
P. Levy. The concept of a Gaussian measure was introduced by A.N.Kolmogorov.
Substantial contributions to the formation of the theory of Gaussian measures are due to
Yu. K.Belyaev, C. Borell, I. Hajek, I. Feldman, and Yu. A.Rozanov. An important role
of oscillations was clarified by K. Ito and M. Nisio, and B. S.Tsyrelson. Different forms
of the Gaussian isoperimetric inequalities were discovered independently by C. Borell,
H.I. Landau and L.A. Shepp, V. N.Sudakov and B.S. Tsyrelson. Two different types of
the convexity of Gaussian measures are due to C. Borell and A. Ehrhard. Useful and
nontrivial models for the covariances were constructed by N.N. Chentsov and A. Noda.
The merit of discovering the beauty of the functional law of the iterated logarithm
belongs to V. Strassen. The entropy approach to the regularity of sample paths and
estimating the probabilities of large deviations was developed by R. M.Dudley,
X. Fernique, and V.N. Sudakov; important and exact estimates for these probabilities
were also obtained by V.A.Dmitrovskii, M. Talagrand, and B.S. Tsyrelson. At the same
time, X.Fernique also contributed substantially to another approach to the regularity
problem, that based on the concept of a majorizing measure. Later, this approach was
decisively complemented by Talagrand's results, and the Kolmogorov problem, open for
almost thirty years - find necessary and sufficient conditions for both the continuity and
boundedness of sample functions - has finally been solved in general. The above given
list of mathematicians, whose ideas and results were used by the author is, of course, far
from being complete. More references to the sources of particular results are given in
Comments at the end of the book.
The first systematic presentation of the theory of Gaussian random functions
(adequate to the level of development of the theory in the late sixties) is given in the
books by I. Neveu and Yu.A. Rozanov [Nev, Roz2]. Among later publications, let us
point out the books and surveys by A. Badrikian and S. Chevet, X.Fernique, T.Hida and
M.Hitsuda, H.-H.Kuo, I. A.Ibragimov and Yu.A. Rozanov, V.1. Piterbarg, H. Sato,
V.N.Sudakov [B-Ch, Fer4, H-H, Kuo, I-R, Pit2, Pit3, Sat2, Sud3], and especially the
recent paper of R. Adler [Adl], which contains a rich material on the Gaussian large
deviations, regularity of sample functions, and related matters.
The reader who wishes to go beyond the Gaussian paradigm and learn about a more
general viewpoint on the subject should refer to the fundamental work by M. Ledoux and
M. Talagrand [L-T].
Other references concerning various special questions are given in Comments.
Preface xi
Theorems, lemmas, and formulas are numbered separately in each section of the
book. A double number suggests a reference to an object from another section. For
example, formula (1.4) mentioned in Section 2 sends the reader back to formula 4 in
Section 1.
The reader who is familiar with the principles of the theory of random functions, can
merely glance through the introductory Sections 1-5 and then either study the following
material section by section, or proceed directly to a topic of his interest: models of the
covariance functions (Section 6), oscillations (Section 7), infinite-dimensional Gaussian
distributions (Sections 8-9), or the convexity (Section II). The complex of central
topics, large deviations and the regularity of sample functions (Sections 12-16), requires
a preliminary reading of Sections 7-11. Finally, the concluding part of the book
(Sections 17-19) consists rather of illustrations and complements to the main topics of
the book selected according to the author's personal preferences.
It is recommended that one looks through the List of Basic Notations before proceed
ing to a systematic reading of the book.
I am grateful to all of those, whose support, advices, and remarks were of great help
for me when writing this book: to my university professors Yu. A.Davydov, I. A.Ibragi
mov, and V. N.Sudakov, as well as to Ya.I. Belopol'skaya, V.1. Bogachev, A.V. Bulin
skii, V.A. Dmitrovskii, M.1. Gordin, B.Heinkel, A.L. Koldobskii, M. Ledoux, B.A. Lif
shits, W. Linde, V.I.Piterbarg, V.Zaiats, and A.Yu. Zaitsev.