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## Overview

An important tool in probability theory and its applications, the coupling method is primarily used in estimates of total variation distances. The method also works well in establishing inequalities, and it has proven highly successful in the study of Markov and renewal process asymptotics. This text represents a detailed, comprehensive examination of the 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. A practical and easy-to-use reference, this volume will accommodate the diverse needs of professionals in the fields of statistics, mathematics, and operational research, as well as those of teachers and students.

## Product Details

ISBN-13: | 9780486153247 |
---|---|

Publisher: | Dover Publications |

Publication date: | 07/18/2012 |

Series: | Dover Books on Mathematics |

Sold by: | Barnes & Noble |

Format: | NOOK Book |

Pages: | 272 |

File size: | 15 MB |

Note: | This product may take a few minutes to download. |

## Table of Contents

Introduction1. Three Examples

2. An Outline

3. Notes

Chapter I. Preliminaries

Appendix II

Bibliography

Index

1. What Is a Coupling?

2. The Coupling Inequality

3. Rates of Convergence

4. Weak Coupling

5. The gamma Coupling

6. The Polish Assumption

7. Notes

Chapter II. Discrete Theory

1. Renewal Theory

1. Basics

2. Stationarity. The Coupling

3. The Discrete Renewal Theorem

4. Finite Moments of T

5. Renewal Sequences

6. Notes

2. Markov Chains

7. Notation

8. Positive Recurrent Chains

9. Null-Recurrent Chains

10. An Observation

11. Notes

3. Random Walk

12. The Ornstein Coupling

13. Null-Recurrent Markov Chains

14. The Mineka Coupling

15. Blocks

16. The Harris Random Walk

17. A Multidimensional Random Walk

18. Notes

4. Card Shuffling

19. Basics

20. "Top to Random" Shuffling

21. Notes

5. Poisson Approximation

22. Basics

23. Another Simple Coupling

24. The Stein-Chen Method

25. An Example

26. Notes

Chapter III. Continuous Theory

1. Renewal Theory

2. Basics

3. Stationarity

4. Blackwell's Renewal Theorem

4. Bounds for U

5. An Exact Coupling

6. Finite Moments of T. Rate Results

7. Notes

2. Harris Chains

8. Basics

9. Harris Chains

10. Regeneration and Stationarity

11. Ergodicity

12. Random Walk

13. Notes

3. Maximal Coupling

14. The Coupling. Goldstein's Theorem

15. From Weak to Strong Coupling

16. Notes

4. Regenerative Processes

17. Basics. Stationarity

18. Coupling of Regenerative Processes

19. Notes

5. On Markov Processes

20. Some Remarks

21. Ergodicity

22. Notes

Chapter IV. Inequalities

1. Strassen's Theorem

1. Basics

2. The Theorem

3. Alternative Formulations

4. Notes

2. Domination

5. The General Result

6. Monotonicity and Convergence

7. Notes

3. Domination and Monotonicity of Markov Processes

8. Basics

9. A Monotonicity Result

4. Examples of Domination

10. Direct Constructions

11. Percolation

12. Bernstein Polynomials

13. Increasing Power Functions

14. Cox Processes

15. Notes

Chapter V. Intensity-Governed Processes

1. Birth and Death Processes

1. Basics

2. Ergodicity

3. Rates

4. Domination and Monotonicity

5. Notes

2. General Birth and Death Processes

6. Basics

7. Ergodicity

8. Networks

9. Propagations

10. Notes

3. Interacting Particle Systems

11. A Signpost. Basics and Examples

12. The Vasershtein Coupling

13. Attractiveness and Monotonicity

14. On the Example Processes

15. Notes

4. Embedding in Poisson Processes

16. A Multivariate Exponential Distribution

17. Embedding in a Bivariate Poisson Process

18. Urns and Boxes

19. On Free Parking Spaces

20. Notes

5. More Renewal Theory

21. Basics

22. The DFR Case

23. The IFR Case

24. Notes

6. On a Class of Point Processes

25. Basics

26. On the FDR Concept

27. The (A, m) Processes

28. Notes

Chapter VI. Diffusions

1. One-Dimensional Processes

1. Basics

2. Ergodicity. I Closed

3. Ergodicity. I Not Closed

4. The Strong Feller Property

5. Domination

6. Notes

2. Multidimensional Processes

7. Basics

8. Brownian Motion

9. Radial Drift

10. Another Reflection Coupling

11. Notes

Appendix 1. Polish Spaces

Appendix 1. A Quick survey

Appendix 2. The Banach space bM subscript s

Appendix 3. Notes

Appendix 4. Epilogue

Appendix 5. Some History

Frequently Used Notation; References; Index

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