Lectures on the Coupling Method
by Lindvall, TorgnyFormat: Paperback
Pub. Date: 2002-08-30
Publisher(s): Dover Publications
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Summary
This text represents a detailed, comprehensive examination of the coupling method and its broad variety of applications. Readers progress from simple to advanced topics, with end-of-discussion notes that reinforce the preceding material. Topics include renewal theory, Markov chains, Poisson approximation, ergodicity, and Strassen's theorem. 1992 edition.
Table of Contents
| Introduction | p. 1 |
| Three Examples | p. 1 |
| An Outline | p. 6 |
| Notes | p. 7 |
| Preliminaries | p. 9 |
| What Is a Coupling? | p. 9 |
| The Coupling Inequality | p. 10 |
| Rates of Convergence | p. 14 |
| Weak Coupling | p. 17 |
| The [gamma] Coupling | p. 18 |
| The Polish Assumption | p. 20 |
| Notes | p. 20 |
| Discrete Theory | p. 21 |
| Renewal Theory | p. 21 |
| Basics | p. 21 |
| Stationarity. The Coupling | p. 22 |
| The Discrete Renewal Theorem | p. 24 |
| Finite Moments of T | p. 25 |
| Renewal Sequences | p. 31 |
| Notes | p. 34 |
| Markov Chains | p. 34 |
| Notation | p. 34 |
| Positive Recurrent Chains | p. 34 |
| Null-Recurrent Chains | p. 39 |
| An Observation | p. 40 |
| Notes | p. 41 |
| Random Walk | p. 41 |
| The Ornstein Coupling | p. 41 |
| Null-Recurrent Markov Chains | p. 43 |
| The Mineka Coupling | p. 44 |
| Blocks | p. 47 |
| The Harris Random Walk | p. 48 |
| A Multidimensional Random Walk | p. 51 |
| Notes | p. 51 |
| Card Shuffling | p. 52 |
| Basics | p. 52 |
| "Top to Random" Shuffling | p. 54 |
| Notes | p. 57 |
| Poisson Approximation | p. 57 |
| Basics | p. 57 |
| Another Simple Coupling | p. 57 |
| The Stein--Chen Method | p. 60 |
| An Example | p. 64 |
| Notes | p. 66 |
| Continuous Theory | p. 67 |
| Renewal Theory | p. 67 |
| Basics | p. 67 |
| Stationarity | p. 69 |
| Blackwell's Renewal Theorem | p. 73 |
| Bounds for U | p. 77 |
| An Exact Coupling | p. 79 |
| Finite Moments of T. Rate Results | p. 83 |
| Notes | p. 88 |
| Harris Chains | p. 88 |
| Basics | p. 88 |
| Harris Chains | p. 91 |
| Regeneration and Stationarity | p. 92 |
| Ergodicity | p. 95 |
| Random Walk | p. 97 |
| Notes | p. 98 |
| Maximal Coupling | p. 99 |
| The Coupling. Goldstein's Theorem | p. 99 |
| From Weak to Strong Coupling | p. 105 |
| Notes | p. 107 |
| Regenerative Processes | p. 107 |
| Basics. Stationarity | p. 107 |
| Coupling of Regenerative Processes | p. 110 |
| Notes | p. 112 |
| On Markov Processes | p. 113 |
| Some Remarks | p. 113 |
| Ergodicity | p. 118 |
| Notes | p. 125 |
| Diffusions | p. 207 |
| One-Dimensional Processes | p. 207 |
| Basics | p. 207 |
| Ergodicity. I Closed | p. 209 |
| Ergodicity. I Not Closed | p. 211 |
| The Strong Feller Property | p. 214 |
| Domination | p. 216 |
| Notes | p. 218 |
| Multidimensional Processes | p. 219 |
| Basics | p. 219 |
| Brownian Motion | p. 219 |
| Radial Drift | p. 223 |
| Another Reflection Coupling | p. 226 |
| Notes | p. 227 |
| Appendix | p. 229 |
| Polish Spaces | p. 229 |
| A Quick Survey | p. 229 |
| The Banach Space bM[subscript s] | p. 232 |
| Notes | p. 233 |
| Epilogue | p. 233 |
| Some History | p. 233 |
| Frequently Used Notation | p. 237 |
| References | p. 241 |
| Index | p. 255 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
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