Lectures on the Coupling Method

by Torgny Lindvall

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:	eBook
Pages:	272
File size:	15 MB
Note:	This product may take a few minutes to download.

Table of Contents

Introduction
1. 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