This book is a self-contained treatment of the theory of probability, random processes. It is intended to lay solid theoretical foundations for advanced probability, that is, for measure and integration theory, and to develop in depth the long term time average behavior of measurements made on random processes with general output alphabets. Unlike virtually all texts on the topic, considerable space is devoted to processes that violate the usual assumptions of stationarity and ergodicity, yet which still possess the fundamental properties of convergence of long term averages to appropriate expectations. The theory of asymtotically mean stationary processes and the ergodic decomposition are both treated in depth for both one-sided and two-sided random processes. In addition, the book treats many of the fundamental results such as the Kolmogorov extension theorem and the ergodic decomposition theorem. Much of the material has not previously appeared in book form, and the treatment takes advantage of many recent generalizations and simplifications.
Author(s): Robert M. Gray
Edition: 1
Year: 1987
Language: English
Pages: 295
Tags: Математика;Теория вероятностей и математическая статистика;Теория случайных процессов;
Contents......Page 7
Preface......Page 9
Probability Spaces and Random Variables......Page 13
Random Processes and Dynamical Systems......Page 18
Distributions......Page 20
Extension......Page 25
Isomorphism......Page 31
Extension of Probability Measures......Page 33
Standard Spaces......Page 34
Some properties of standard spaces......Page 38
Simple standard spaces......Page 41
Metric Spaces......Page 43
Extension in Standard Spaces......Page 48
The Kolmogorov Extension Theorem......Page 49
Extension Without a Basis......Page 50
Borel Spaces......Page 57
Polish Spaces......Page 60
Polish Schemes......Page 66
Discrete Measurements......Page 73
Quantization......Page 76
Expectation......Page 79
Time Averages......Page 89
Convergence of Random Variables......Page 92
Stationary Averages......Page 99
Measurements and Events......Page 103
Elementary Conditional Probability......Page 107
Projections......Page 110
The Radon-Nikodym Theorem......Page 113
Conditional Probability......Page 116
Regular Conditional Probability......Page 118
Conditional Expectation......Page 121
Independence and Markov Chains......Page 127
Ergodic Properties of Dynamical Systems......Page 131
Some Implications of Ergodic Properties......Page 134
Asymptotically Mean Stationary Processes......Page 139
Recurrence......Page 146
Asymptotic Mean Expectations......Page 150
Limiting Sample Averages......Page 152
Ergodicity......Page 154
The Pointwise Ergodic Theorem......Page 161
Block AMS Processes......Page 166
The Ergodic Decomposition......Page 168
The Subadditive Ergodic Theorem......Page 172
Introduction......Page 181
A Metric Space of Measures......Page 182
The Rho-Bar Distance......Page 188
Measures on Measures......Page 194
The Ergodic Decomposition Revisited......Page 195
The Ergodic Decomposition of Markov Processes......Page 198
Barycenters......Page 200
Affine Functions of Measures......Page 203
The Ergodic Decomposition of Affine Functionals......Page 206
Bibliography......Page 207
Index......Page 209
