Lectures on the Coupling Method: (Dover Books on Mathema 1.4tics)

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.

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