Functional Analysis for Probability and Stochastic Processes

An Introduction

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Hardcover | 6 October 2005

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408 Pages

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16.0 x 23.7 x 3.1

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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.

Industry Reviews

"My impression is that this text might well succeed as an attractive introduction to, or even as propaganda for the subject of probability and stochastic processes for a well-educated analyst without a probabilistic background." N.H. Bingham, Journal of the American Statistical Association

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Preface	p. xi
Preliminaries, notations and conventions	p. 1
Elements of topology	p. 1
Measure theory	p. 3
Functions of bounded variation. Riemann-Stieltjes integral	p. 17
Sequences of independent random variables	p. 23
Convex functions. Holder and Minkowski inequalities	p. 29
The Cauchy equation	p. 33
Basic notions in functional analysis	p. 37
Linear spaces	p. 37
Banach spaces	p. 44
The space of bounded linear operators	p. 63
Conditional expectation	p. 80
Projections in Hilbert spaces	p. 80
Definition and existence of conditional expectation	p. 87
Properties and examples	p. 91
The Radon-Nikodym Theorem	p. 101
Examples of discrete martingales	p. 103
Convergence of self-adjoint operators	p. 106
... and of martingales	p. 112
Brownian motion and Hilbert spaces	p. 121
Gaussian families & the definition of Brownian motion	p. 123
Complete orthonormal sequences in a Hilbert space	p. 127
Construction and basic properties of Brownian motion	p. 133
Stochastic integrals	p. 139
Dual spaces and convergence of probability measures	p. 147
The Hahn-Banach Theorem	p. 148
Form of linear functionals in specific Banach spaces	p. 154
The dual of an operator	p. 162
Weak and weak* topologies	p. 166
The Central Limit Theorem	p. 175
Weak convergence in metric spaces	p. 178
Compactness everywhere	p. 184
Notes on other modes of convergence	p. 198
The Gelfand transform and its applications	p. 201
Banach algebras	p. 201
The Gelfand transform	p. 206
Examples of Gelfand transform	p. 208
Examples of explicit calculations of Gelfand transform	p. 217
Dense subalgebras of C(S)	p. 222
Inverting the abstract Fourier transform	p. 224
The Factorization Theorem	p. 231
Semigroups of operators and Levy processes	p. 234
The Banach-Steinhaus Theorem	p. 234
Calculus of Banach space valued functions	p. 238
Closed operators	p. 240
Semigroups of operators	p. 246
Brownian motion and Poisson process semigroups	p. 265
More convolution semigroups	p. 270
The telegraph process semigroup	p. 280
Convolution semigroups of measures on semigroups	p. 286
Markov processes and semigroups of operators	p. 294
Semigroups of operators related to Markov processes	p. 294
The Hille-Yosida Theorem	p. 309
Generators of stochastic processes	p. 327
Approximation theorems	p. 340
Appendix	p. 363
Bibliographical notes	p. 363
Solutions and hints to exercises	p. 366
Some commonly used notations	p. 383
References	p. 385
Table of Contents provided by Ingram. All Rights Reserved.

Functional Analysis for Probability and Stochastic Processes

An Introduction

At a Glance

Hardcover

Industry Reviews

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Defective items

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