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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
At the nexus of probability theory, geometry and statistics, a Gaussian measure is constructed on a Hilbert space in two ways: as a product measure and via a characteristic functional based on Minlos-Sazonov theorem. As such, it can be utilized for obtaining results for topological vector spaces. Gaussian Measures contains the proof for Fernique s theorem and its relation to exponential moments in Banach space. Furthermore, the fundamental Feldman-Hajek dichotomy for Gaussian measures in Hilbert space is investigated. Applications in statistics are also outlined. In addition to chapters devoted to measure theory, this book highlights problems related to Gaussian measures in Hilbert and Banach spaces. Borel probability measures are also addressed, with properties of characteristic functionals examined and a proof given based on the classical Banach Steinhaus theorem. Gaussian Measures is suitable for graduate students, plus advanced undergraduate students in mathematics and statistics. It is also of interest to students in related fields from other disciplines. Results are presented as lemmas, theorems and corollaries, while all statements are proven. Each subsection ends with teaching problems, and a separate chapter contains detailed solutions to all the problems. With its student-tested approach, this book is a superb introduction to the theory of Gaussian measures on infinite-dimensional spaces.
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Seitenzahl: 353
Veröffentlichungsjahr: 2019
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Table of Contents
Cover
Foreword
Preface
Introduction
Abbreviations and Notation
1 Gaussian Measures in Euclidean Space
1.1. The change of variables formula
1.2. Invariance of Lebesgue measure
1.3. Absence of invariant measure in infinite-dimensional Hilbert space
1.4. Random vectors and their distributions
1.5. Gaussian vectors and Gaussian measures
2 Gaussian Measure in l2 as a Product Measure
2.1. Space ℝ∞
2.2. Product measure in ℝ∞
2.3. Standard Gaussian measure in ℝ∞
2.4. Construction of Gaussian measure in l2
3 Borel Measures in Hilbert Space
3.1. Classes of operators in H
3.2. Pettis and Bochner integrals
3.3. Borel measures in Hilbert space
4 Construction of Measure by its Characteristic Functional
4.1. Cylindrical sigma-algebra in normed space
4.2. Convolution of measures
4.3. Properties of characteristic functionals in H
4.4. S-topology in H
4.5. Minlos–Sazonov theorem
5 Gaussian Measure of General Form
5.1. Characteristic functional of Gaussian measure
5.2. Decomposition of Gaussian measure and Gaussian random element
5.3. Support of Gaussian measure and its invariance
5.4. Weak convergence of Gaussian measures
5.5. Exponential moments of Gaussian measure in normed space
6 Equivalence and Singularity of Gaussian Measures
6.1. Uniformly integrable sequences
6.2. Kakutani’s dichotomy for product measures on ℝ∞
6.3. Feldman–Hájek dichotomy for Gaussian measures on H
6.4. Applications in statistics
7 Solutions
7.1. Solutions for Chapter 1
7.2. Solutions for Chapter 2
7.3. Solutions for Chapter 3
7.4. Solutions for Chapter 4
7.5. Solutions for Chapter 5
7.6. Solutions for Chapter 6
Summarizing Remarks
References
Index
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
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To the memory of my daughter Ann
Series Editor
Nikolaos Limnios
Gaussian Measures in Hilbert Space
Construction and Properties
Alexander Kukush
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:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2019
The rights of Alexander Kukush to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2019946454
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-267-0
Foreword
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 l2space. 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
Preface
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.
Acknowledgments
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
