Probability, Random Processes, and Ergodic Properties (Gray)

Probability, Random Processes, and Ergodic Properties...

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Unformatted text preview: Probability, Random Processes, and Ergodic Properties January 2, 2010 ii Probability, Random Processes, and Ergodic Properties Robert M. Gray Information Systems Laboratory Electrical Engineering Department Stanford University Springer-Verlag New York iv c 1987 by Springer Verlag. Revised 2001, 2006, 2007, 2008 by Robert M. Gray. v This book is aectionately dedicated to the memory of Elizabeth Dubois Jordan Gray 19061998 R. Adm. Augustine Heard Gray, U.S.N. 18881981 Sara Jean Dubois 1811? and William Old Billy Gray 17501825 vi 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 viii...
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This note was uploaded on 10/02/2011 for the course 334323 232 at Stanford.

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Probability, Random Processes, and Ergodic Properties...

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