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Probability, Random Processes, and Ergodic Properties (Gray)

Probability, Random Processes, and Ergodic Properties...

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Probability, Random Processes, and Ergodic Properties January 2, 2010
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Probability, Random Processes, and Ergodic Properties Robert M. Gray Information Systems Laboratory Electrical Engineering Department Stanford University Springer-Verlag New York
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iv c 1987 by Springer Verlag. Revised 2001, 2006, 2007, 2008 by Robert M. Gray.
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v This book is affectionately dedicated to the memory of Elizabeth Dubois Jordan Gray 1906–1998 R. Adm. Augustine Heard Gray, U.S.N. 1888–1981 Sara Jean Dubois 1811–? and William “Old Billy” Gray 1750–1825
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vi
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Contents Contents vii Preface ix 1 Probability and Random Processes 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Probability Spaces and Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Random Processes and Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Standard alphabets 21 2.1 Extension of Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Standard Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Some properties of standard spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Simple standard spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Extension in Standard Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7 The Kolmogorov Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8 Extension Without a Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Borel Spaces and Polish alphabets 45 3.1 Borel Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Polish Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Polish Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Averages 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Discrete Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5 Time Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Convergence of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.7 Stationary Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 vii
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viii CONTENTS 5 Conditional Probability and Expectation 91 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Measurements and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Restrictions of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Elementary Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6 The Radon-Nikodym Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.7 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.8 Regular Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.9 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.10 Independence and Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6 Ergodic Properties 119 6.1 Ergodic Properties of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 119 6.2 Some Implications of Ergodic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3 Asymptotically Mean Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . 127 6.4 Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.5 Asymptotic Mean Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.6 Limiting Sample Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.7 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7 Ergodic Theorems 149 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2 The Pointwise Ergodic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.3 Block AMS Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4 The Ergodic Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.5 The Subadditive Ergodic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8 Process Metrics and the Ergodic Decomposition 169 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.2 A Metric Space of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3 The Rho-Bar Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.4 Measures on Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.5 The Ergodic Decomposition Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.6 The Ergodic Decomposition of Markov Processes . . . . . . . . . . . . . . . . . . . . 186 8.7 Barycenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.8 Affine Functions of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 8.9 The Ergodic Decomposition of Affine Functionals . . . . . . . . . . . . . . . . . . . . 194 Bibliography 197 Index 201
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Preface History and Goals This book has been written for several reasons, not all of which are academic. This material was for many years the first half of a book in progress on information and ergodic theory. The intent was and is to provide a reasonably self-contained advanced treatment of measure theory, probability theory, and the theory of discrete time random processes with an emphasis on general alphabets and on ergodic and stationary properties of random processes that might be neither ergodic nor stationary. The intended audience was mathematically inclined engineering graduate students and visiting scholars who had not had formal courses in measure theoretic probability.
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