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Searching... | 30000004599498 | QA320 B62 2005 | Open Access Book | Book | Searching... |
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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.
Reviews 1
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
Preface | p. xi |
1 Preliminaries, notations and conventions | p. 1 |
1.1 Elements of topology | p. 1 |
1.2 Measure theory | p. 3 |
1.3 Functions of bounded variation. Riemann-Stieltjes integral | p. 17 |
1.4 Sequences of independent random variables | p. 23 |
1.5 Convex functions. Holder and Minkowski inequalities | p. 29 |
1.6 The Cauchy equation | p. 33 |
2 Basic notions in functional analysis | p. 37 |
2.1 Linear spaces | p. 37 |
2.2 Banach spaces | p. 44 |
2.3 The space of bounded linear operators | p. 63 |
3 Conditional expectation | p. 80 |
3.1 Projections in Hilbert spaces | p. 80 |
3.2 Definition and existence of conditional expectation | p. 87 |
3.3 Properties and examples | p. 91 |
3.4 The Radon-Nikodym Theorem | p. 101 |
3.5 Examples of discrete martingales | p. 103 |
3.6 Convergence of self-adjoint operators | p. 106 |
3.7 ... and of martingales | p. 112 |
4 Brownian motion and Hilbert spaces | p. 121 |
4.1 Gaussian families & the definition of Brownian motion | p. 123 |
4.2 Complete orthonormal sequences in a Hilbert space | p. 127 |
4.3 Construction and basic properties of Brownian motion | p. 133 |
4.4 Stochastic integrals | p. 139 |
5 Dual spaces and convergence of probability measures | p. 147 |
5.1 The Hahn-Banach Theorem | p. 148 |
5.2 Form of linear functionals in specific Banach spaces | p. 154 |
5.3 The dual of an operator | p. 162 |
5.4 Weak and weak* topologies | p. 166 |
5.5 The Central Limit Theorem | p. 175 |
5.6 Weak convergence in metric spaces | p. 178 |
5.7 Compactness everywhere | p. 184 |
5.8 Notes on other modes of convergence | p. 198 |
6 The Gelfand transform and its applications | p. 201 |
6.1 Banach algebras | p. 201 |
6.2 The Gelfand transform | p. 206 |
6.3 Examples of Gelfand transform | p. 208 |
6.4 Examples of explicit calculations of Gelfand transform | p. 217 |
6.5 Dense subalgebras of C(S) | p. 222 |
6.6 Inverting the abstract Fourier transform | p. 224 |
6.7 The Factorization Theorem | p. 231 |
7 Semigroups of operators and Levy processes | p. 234 |
7.1 The Banach-Steinhaus Theorem | p. 234 |
7.2 Calculus of Banach space valued functions | p. 238 |
7.3 Closed operators | p. 240 |
7.4 Semigroups of operators | p. 246 |
7.5 Brownian motion and Poisson process semigroups | p. 265 |
7.6 More convolution semigroups | p. 270 |
7.7 The telegraph process semigroup | p. 280 |
7.8 Convolution semigroups of measures on semigroups | p. 286 |
8 Markov processes and semigroups of operators | p. 294 |
8.1 Semigroups of operators related to Markov processes | p. 294 |
8.2 The Hille-Yosida Theorem | p. 309 |
8.3 Generators of stochastic processes | p. 327 |
8.4 Approximation theorems | p. 340 |
9 Appendix | p. 363 |
9.1 Bibliographical notes | p. 363 |
9.2 Solutions and hints to exercises | p. 366 |
9.3 Some commonly used notations | p. 383 |
References | p. 385 |