Cover image for Functional analysis for probability and stochastic processes : an introduction
Title:
Functional analysis for probability and stochastic processes : an introduction
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Publication Information:
Cambridge, UK : Cambridge University Press, 2005
ISBN:
9780521831666

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30000004599498 QA320 B62 2005 Open Access Book Book
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Summary

Summary

This text is designed both for students of probability and stochastic processes, and for students of functional analysis. For the reader not familiar with functional analysis a detailed introduction to necessary notions and facts is provided. However, this is not a straight textbook in functional analysis; rather, it presents some chosen parts of functional analysis that can help understand ideas from probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook or for self-study.


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Choice Review

Bobrowski (Lublin Univ. of Technology) lucidly presents this material for those with good backgrounds in functional analysis but little knowledge of probability, and for statisticians with no knowledge of functional analysis. Readers need a good knowledge of measure theory, some exposure to solving ordinary differential equations, and some knowledge of abstract algebra and topology--all briefly sketched in chapter 1. Other chapters discuss linear spaces, Banach spaces, and the space of bounded linear operators; conditional expectations and their properties; Brownian motion, Wiener's proof of the existence of Brownian motion by using Hilbert Space theory, and the Ito integral; various modes of convergence of probability measures, the notion of a Banach limit, dual spaces, and compact sets; Gelfand transforms; and Markov processes: Levy processes (a particular type of Markov process) and the Hille-Yosida theorem, according to which there is a 1-1 correspondence between Markov processes and a class of linear operators--the class of generators of corresponding semi groups. Three appendixes contain bibliographical notes, solutions and hints for exercises, and a list of some commonly used notations. An attractive feature is the numerous solved examples. A good addition to the statistical literature. Useful index. ^BSumming Up: Recommended. Graduate students; faculty. D. V. Chopra Wichita State University


Table of Contents

Prefacep. xi
1 Preliminaries, notations and conventionsp. 1
1.1 Elements of topologyp. 1
1.2 Measure theoryp. 3
1.3 Functions of bounded variation. Riemann-Stieltjes integralp. 17
1.4 Sequences of independent random variablesp. 23
1.5 Convex functions. Holder and Minkowski inequalitiesp. 29
1.6 The Cauchy equationp. 33
2 Basic notions in functional analysisp. 37
2.1 Linear spacesp. 37
2.2 Banach spacesp. 44
2.3 The space of bounded linear operatorsp. 63
3 Conditional expectationp. 80
3.1 Projections in Hilbert spacesp. 80
3.2 Definition and existence of conditional expectationp. 87
3.3 Properties and examplesp. 91
3.4 The Radon-Nikodym Theoremp. 101
3.5 Examples of discrete martingalesp. 103
3.6 Convergence of self-adjoint operatorsp. 106
3.7 ... and of martingalesp. 112
4 Brownian motion and Hilbert spacesp. 121
4.1 Gaussian families & the definition of Brownian motionp. 123
4.2 Complete orthonormal sequences in a Hilbert spacep. 127
4.3 Construction and basic properties of Brownian motionp. 133
4.4 Stochastic integralsp. 139
5 Dual spaces and convergence of probability measuresp. 147
5.1 The Hahn-Banach Theoremp. 148
5.2 Form of linear functionals in specific Banach spacesp. 154
5.3 The dual of an operatorp. 162
5.4 Weak and weak* topologiesp. 166
5.5 The Central Limit Theoremp. 175
5.6 Weak convergence in metric spacesp. 178
5.7 Compactness everywherep. 184
5.8 Notes on other modes of convergencep. 198
6 The Gelfand transform and its applicationsp. 201
6.1 Banach algebrasp. 201
6.2 The Gelfand transformp. 206
6.3 Examples of Gelfand transformp. 208
6.4 Examples of explicit calculations of Gelfand transformp. 217
6.5 Dense subalgebras of C(S)p. 222
6.6 Inverting the abstract Fourier transformp. 224
6.7 The Factorization Theoremp. 231
7 Semigroups of operators and Levy processesp. 234
7.1 The Banach-Steinhaus Theoremp. 234
7.2 Calculus of Banach space valued functionsp. 238
7.3 Closed operatorsp. 240
7.4 Semigroups of operatorsp. 246
7.5 Brownian motion and Poisson process semigroupsp. 265
7.6 More convolution semigroupsp. 270
7.7 The telegraph process semigroupp. 280
7.8 Convolution semigroups of measures on semigroupsp. 286
8 Markov processes and semigroups of operatorsp. 294
8.1 Semigroups of operators related to Markov processesp. 294
8.2 The Hille-Yosida Theoremp. 309
8.3 Generators of stochastic processesp. 327
8.4 Approximation theoremsp. 340
9 Appendixp. 363
9.1 Bibliographical notesp. 363
9.2 Solutions and hints to exercisesp. 366
9.3 Some commonly used notationsp. 383
Referencesp. 385