To the memory of my daughter Ann
Series Editor
Nikolaos Limnios
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
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Library of Congress Control Number: 2019946454
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ISBN 978-1-78630-267-0
The study of modern theory of stochastic processes, infinite-dimensional analysis and Malliavin calculus is impossible without a solid knowledge of Gaussian measures on infinite-dimensional spaces. In spite of the importance of this topic and the abundance of literature available for experienced researchers, there is no textbook suitable for students for a first reading.
The present manual is an excellent get-to-know course in Gaussian measures on infinite-dimensional spaces, which has been given by the author for many years at the Faculty of Mechanics & Mathematics of Taras Shevchenko National University of Kyiv, Ukraine. The presentation of the material is well thought out, and the course is self-contained. After reading the book it may seem that the topic is very simple. But that is not true! Apparent simplicity is achieved by careful organization of the book. For experts and PhD students having experience in infinite-dimensional analysis, I prefer to recommend the monograph V. I. Bogachev, Gaussian Measures (1998). But for first acquaintance with the topic, I recommend this new manual.
Prerequisites for the book are only a basic knowledge of probability theory, linear algebra, measure theory and functional analysis. The exposition is supplemented with a bulk of examples and exercises with solutions, which are very useful for unassisted work and control of studied material.
In this book, many delicate and important topics of infinite-dimensional analysis are analyzed in detail, e.g. Borel and cylindrical sigma-algebras in infinite-dimensional spaces, Bochner and Pettis integrals, nuclear operators and the topology of nuclear convergence, etc. We present the contents of the book, emphasizing places where finite-dimensional results need reconsideration (everywhere except Chapters 1).
– Chapter 1. Gaussian distributions on a finite-dimensional space. The chapter is preparatory but necessary. Later on, many analogies with finite-dimensional space will be given, and the places will be visible where a new technique is needed.
– Chapter 2. Space ℝ∞, Kolmogorov theorem about the existence of probability measure, product measures, Gaussian product measures, Gaussian product measures in l2 space. After reading the chapter, the student will start to understand that on infinite-dimensional space there are several ways to define a sigma-algebra (luckily, in our case Borel and cylindrical sigma-algebras coincide). Moreover, it will become clear that infinite-dimensional Lebesgue measure does not exist, hence construction of measure by means of density needs reconsideration.
– Chapter 3. Bochner and Pettis integrals, Hilbert–Schmidt operators and nuclear operators, strong and weak moments. The chapter is a preparation for the definition of the expectation and correlation operator of Gaussian (or even arbitrary) random element. We see that it is not so easy to introduce expectation of a random element distributed in Hilbert or Banach space. As opposed to finite-dimensional space, it is not enough just to integrate over basis vectors and then augment the results in a single vector.
– Chapter 4. Characteristic functionals, Minlos–Sazonov theorem. One of the most important methods to investigate probability measures on finite-dimensional space is the method of characteristic functions. As well-known from the course of probability theory, these will be all continuous positive definite functions equal to one at zero, and only them. On infinite-dimensional space this is not true. For the statement “they and only them”, continuity in the topology of nuclear convergence is required, and this topology is explained in detail.
– Chapter 5. General Gaussian measures. Based on results of previous chapters, we see the necessary and sufficient conditions that have to be satisfied by the characteristic functional of a Gaussian measure in Hilbert space. We realize that we have used all the knowledge from Chapters 2–4 (concerning integration of random elements, about Hilbert–Schmidt and nuclear operators, Minlos–Sazonov theorem, etc.). We notice that for the eigenbasis of the correlation operator, a Gaussian measure is just a product measure which we constructed in Chapter 2. This seems natural; but on our way it was impossible to discard any single step without loss of mathematical rigor. In this chapter, Fernique’s theorem about finiteness of an exponential moment of the norm of a Gaussian random element is proved and the criterion for the weak convergence of Gaussian measures is stated.
– Chapter 6. Equivalence and mutual singularity of measures. Here, Kakutani’s theorem is proven about the equivalence of the infinite product of measures. As we saw in the previous chapter, Gaussian measures on Hilbert spaces are product measures, in a way. Therefore, as a consequence of general theory, we get a criterion for the equivalence of Gaussian measures (Feldman–Hájek theorem). The obtained results are applied to problems of infinite-dimensional statistics. One should be careful here, as due to the absence of the infinite-dimensional Lebesgue measure, the Radon–Nikodym density should be written w.r.t. one of the Gaussian measures.
The author of this book, Professor A.G. Kukush, has been working at the Faculty of Mechanics & Mathematics of Taras Shevchenko National University for 40 years. He is an excellent teacher and a famous expert in statistics and probability theory. In particular, he used to give lectures to students of mathematics and statistics on Measure Theory, Functional Analysis, Statistics and Econometrics. As a student, I was lucky to attend his fascinating course on infinite-dimensional analysis.
Andrey PILIPENKO
Leading Researcher at the Institute of Mathematics of Ukrainian National Academy of Sciences, Professor of Mathematics at the National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute” August 2019
This book is written for graduate students of mathematics and mathematical statistics who know algebra, measure theory and functional analysis (generalized functions are not used here); the knowledge of mathematical statistics is desirable only to understand section 6.4. The topic of this book can be considered as supplementary chapters of measure theory and lies between measure theory and the theory of stochastic processes; possible applications are in functional analysis and statistics of stochastic processes. For 20 years, the author has been giving a special course “Gaussian Measures” at Taras Shevchenko National University of Kyiv, Ukraine, and in 2018–2019, preliminary versions of this book have been used as a textbook for this course.
There are excellent textbooks and monographs on related topics, such as Gaussian Measures in Banach Spaces [KUO 75], Gaussian Measures [BOG 98] and Probability Distributions on Banach Spaces [VAK 87]. Why did I write my own textbook?
In the 1970s, I studied at the Faculty of Mechanics and Mathematics of Taras Shevchenko National University of Kyiv, at that time called Kiev State University. There I attended unforgettable lectures given by Professors Anatoliy Ya. Dorogovtsev (calculus and measure theory), Lev A. Kaluzhnin (algebra), Mykhailo I. Yadrenko (probability theory), Myroslav L. Gorbachuk (functional analysis) and Yuriy M. Berezansky (spectral theory of linear operators). My PhD thesis was supervised by famous statistician A. Ya. Dorogovtsev and dealt with the weak convergence of measures on infinite-dimensional spaces. For long time, I was a member of the research seminar “Stochastic processes and distributions in functional spaces” headed by classics of probability theory Anatoliy V. Skorokhod and Yuriy L. Daletskii. My second doctoral thesis was about asymptotic properties of estimators for parameters of stochastic processes. Thus, I am somewhat tied up with measures on infinite-dimensional spaces.
In 1979, Kuo’s fascinating textbook was translated into Russian. Inspired by this book, I started to give my lectures on Gaussian measures for graduate students. The subject seemed highly technical and extremely difficult. I decided to create something like a comic book on this topic, in particular to divide lengthy proofs into small understandable steps and explain the ideas behind computations.
It is impossible to study mathematical courses without solving problems. Each section ends with several problems, some of which are original and some are taken from different sources. A separate chapter contains detailed solutions to all the problems.
I would like to thank my colleagues at Taras Shevchenko National University of Kyiv who supported my project, especially Yuliya Mishura, Oleksiy Nesterenko and Ivan Feshchenko. Also I wish to thank my students of different generations who followed up on the ideas of the material and helped me to improve the presentation. I am grateful to Fedor Nazarov (Kent State University, USA) who communicated the proof of theorem 3.9. In particular, I am grateful to Oksana Chernova and Andrey Frolkin for preparing the manuscript for publication. I thank Sergiy Shklyar for his valuable comments.
My wife Mariya deserves the most thanks for her encouragement and patience.
Alexander KUKUSH
Kyiv, Ukraine
September 2019
The theory of Gaussian measures lies on the junction of theory of stochastic processes, functional analysis and mathematical physics. Possible applications are in quantum mechanics, statistical physics, financial mathematics and other branches of science. In this field, the ideas and methods of probability theory, nonlinear analysis, geometry and theory of linear operators interact in an elegant and intriguing way.
The aim of this book is to explain the construction of Gaussian measure in Hilbert space, present its main properties and also outline possible applications in statistics.
Chapter 1 deals with Euclidean space, where the invariance of Lebesgue measure is explained and Gaussian vectors and Gaussian measures are introduced. Their properties are stated in such a form that (later on) they can be extended to the infinite-dimensional case. Furthermore, it is shown that on an infinite-dimensional Hilbert space there is no non-trivial measure, which is invariant under all translations (the same concerning invariance under all unitary operators); hence on such a space there is no measure analogous to the Lebesgue one.
In Chapter 2, a product measure is constructed on the sequence space ℝ∞ based on Kolmogorov extension theorem. For standard Gaussian measure μ on ℝ∞, Kolmogorov–Khinchin criterion is established. In particular, it is shown that μ is concentrated on certain weighted sequence spaces l2,a, and based on isometry between l2,a, and 12, a Gaussian product measure is constructed on the latter sequence space.
Chapter 3 introduces important classes of operators in a separable infinite-dimensional Hilbert space H, in particular S-operators, i.e. self-adjoint, positive and nuclear ones. Theorem 3.9 shows that the convergence of S-operators is equivalent to certain convergence of corresponding quadratic forms. Also the weak (Pettis) and strong (Bochner) integrals are defined for a function valued in a Banach space.
Borel probability measures on H and a normed space X are studied with examples. The boundedness of moment forms of such measures is shown, with simple proof based on the classical Banach–Steinhaus theorem. Corollary 3.3 and remark 3.8 give mild conditions for the existence of mean value of a probability measure μ as Pettis integral, and if the underlying space is a separable Banach space B and μ has a strong first moment, then its mean value exists as Bochner integral.
In Chapter 4, properties of characteristic functionals of Borel probability measures on H are studied. A special linear topology, S-topology, is introduced in H with a neighborhood system consisting of ellipsoids. Classical Minlos–Sazonov theorem is proven and properly extends Bochner’s theorem from ℝn to H. According to Minlos–Sazonov theorem, the characteristic functional of a Borel probability measures on H should be continuous in S-topology. A part of proof of this theorem (see lemma 4.9) suggests the way to construct a probability measure by its characteristic functional.
In Chapter 5, theorem 5.1 uses the Minlos–Sazonov theorem to describe a Gaussian measure on H of general form. It turns out that the correlation operator of such a measure is always an S-operator. It is shown that each Gaussian measure on H is just a product of one-dimensional Gaussian measures w.r.t. the eigenbasis of the correlation operator. Thus, every Gaussian measure on H can be constructed along the way, as demonstrated in Chapter 2.
The support of Gaussian measure is studied. It is shown that a centered Gaussian measure is invariant under quite a rich group of linear transforms (see theorem 5.5). Hence, a Gaussian measure in Hilbert space can be considered as a natural infinite-dimensional analogue of (invariant) Lebesgue measure.
A criterion for the weak convergence of Gaussian measures is stated, where (due to theorem 3.9) we recognize the convergence of correlation operators in nuclear norm.
In section 5.5, we study Gaussian measures on a separable normed space X. Important example 5.3 shows that a Gaussian stochastic process generates a measure on the path space Lp [0,T], hence in case p = 2, we obtain a Gaussian measure on Hilbert space. Lemma 5.9 presents a characterization of Gaussian random element in X.
The famous theorem of Fernique is proven, which states that certain exponential moments of a Gaussian measure on X are finite. In particular, every Gaussian measure on a separable Banach space B has mean value as Bochner integral and its correlation operator is well-defined. Theorem 5.10 derives the convergence of moments of weakly convergent Gaussian measures.
In Chapter 6, Kakutani’s remarkable dichotomy for product measures on ℝ∞ is proven. In particular, two such product measures with absolutely continuous components are either absolutely continuous or mutually singular. This implies the dichotomy for Gaussian measures on ℝ∞: two such measures are either equivalent or mutually singular. Section 6.3 proves the famous Feldman–Hájek dichotomy for Gaussian measures on H, and in case of equivalent measures, expressions for Radon–Nikodym derivatives are provided.
In section 6.4, the results of Chapter 6 are applied in statistics. Based on a single observation of Gaussian random element in H, we construct unbiased estimators for its mean and for parameters of its correlation operator; also we check a hypothesis about the mean and the correlation operator (the latter hypothesis is in the case where the Gaussian element is centered). In view of example 5.3 with p = 2, these statistical procedures can be used for a single observation of a Gaussian process on finite time interval.
The book is aimed for advanced undergraduate students and graduate students in mathematics and statistics, and also for theoretically interested students from other disciplines, say physics.
Prerequisites for the book are calculus, algebra, measure theory, basic probability theory and functional analysis (we do not use generalized functions). In section 6.4, the knowledge of basic mathematical statistics is required.
Some words about the structure of the book: we present the results in lemmas, theorems, corollaries and remarks. All statements are proven. Important and illustrative examples are given. Furthermore, each section ends with a list of problems. Detailed solutions to the problems are provided in Chapter 7.
The abbreviations and notation used in the book are defined in the corresponding chapters; an overview of them is given in the following list.
= A
, the latter is Lebesgue integral
with norm ‖x‖p = 
if 1 ≤ p < ∞, and ‖x‖∞ = supn≥1 |xn| if p = ∞. For p = 2, l2 is Hilbert space with inner product 
.
