Markov Chains and Invariant Probabilities

This book concerns discrete-time homogeneous Markov chains that admit an invariant probability measure. The main objective is to give a systematic, self-contained presentation on some key issues about the ergodic behavior of that class of Markov chains. These issues include, in particular, the various types of convergence of expected and pathwise occupation measures, and ergodic decompositions of the state space.

Acknowledgements

ix

Preface

xi

Abbreviations and List of Symbols

xv

Preliminaries

1

(18)

Introduction

1

(1)

Measures and Functions

1

(3)

Weak Topologies

4

(2)

Convergence of Measures

6

(7)

Complements

13

(4)

Notes

17

(2)

I Markov Chains and Ergodicity

19

(74)

Markov Chains and Ergodic Theorems

21

(20)

Introduction

21

(1)

Basic Notation and Definitions

22

(6)

Ergodic Theorems

28

(5)

The Ergodicity Property

33

(3)

Pathwise Results

36

(3)

Notes

39

(2)

Countable Markov Chains

41

(6)

Introduction

41

(1)

Classification of States and Class Properties

41

(3)

Limit Theorems

44

(2)

Notes

46

(1)

Harris Markov Chains

47

(16)

Introduction

47

(1)

Basic Definitions and Properties

47

(4)

Characterization of Harris recurrence

51

(5)

Sufficient Conditions for P.H.R.

56

(4)

Harris and Doeblin Decompositions

60

(1)

Notes

61

(2)

Markov Chains in Metric Spaces

63

(20)

Introduction

63

(1)

The Limit in Ergodic Theorems

64

(4)

Yosida's Ergodic Decomposition

68

(5)

Pathwise Results

73

(1)

Proofs

74

(8)

Notes

82

(1)

Classification of Markov Chains via Occupation Measures

83

(10)

Introduction

83

(1)

A Classification

84

(4)

On the Birkhoff Individual Ergodic Theorem

88

(4)

Notes

92

(1)

II Further Ergodicity Properties

93

(40)

Feller Markov Chains

95

(8)

Introduction

95

(1)

Weak- and Strong-Feller Markov Chains

96

(3)

Quasi Feller Chains

99

(3)

Notes

102

(1)

The Poisson Equation

103

(18)

Introduction

103

(1)

The Poisson Equation

103

(2)

Canonical Pairs

105

(5)

The Cesaro-Averages Approach

110

(4)

The Abelian Approach

114

(5)

Notes

119

(2)

Strong and Uniform Ergodicity

121

(12)

Introduction

121

(1)

Strong and Uniform Ergodicity

122

(5)

Weak and Weak Uniform Ergodicity

127

(4)

Notes

131

(2)

III Existence and Approximation of Invariant Probability Measures

133

(60)

Existence of Invariant Probability Measures

135

(22)

Introduction and Statement of the Problems

135

(1)

Notation and Definitions

136

(2)

Existence Results

138

(5)

Markov Chains in Locally Compact Separable Metric Spaces

143

(2)

Other Existence Results in Locally Compact Separable Metric Spaces

145

(2)

Technical Preliminaries

147

(2)

Proofs

149

(6)

Notes

155

(2)

Existence and Uniqueness of Fixed Points for Markov Operators

157

(18)

Introduction and Statement of the Problems

157

(1)

Notation and Definitions

158

(2)

Existence Results

160

(7)

Proofs

167

(7)

Notes

174

(1)

Approximation Procedures for Invariant Probability Measures

175

(18)

Introduction

175

(1)

Statement of the Problem and Preliminaries

176

(2)

An Approximation Scheme

178

(5)

A Moment Approach for a Special Class of Markov Chains

183

(7)

Notes

190

(3)

Bibliography

193

(10)

Index

203

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Markov Chains and Invariant Probabilities

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Markov Chains and Invariant Probabilities

Summary

Table of Contents

Supplemental Materials

Digital License