This is a book on coupling, including self-contained treatments of station arity and regeneration. Coupling is the central topic in the first half of the book, and then enters as a tool in the latter half. The ten chapters are grouped into four parts as follows: Chapters 1-2 form an introductory part presenting basic elemen tary couplings (Chapter 1 on random variables) and the classical tri umphs of the coupling method (Chapter 2 on Markov chains, random walks, and renewal theory). Chapters 3-7 present a general coupling theory highlighting max imal couplings and convergence characterizations for random ele ments, stochastic processes, random fields, and random elements un der the action of a transformation semigroup. Chapters 8-9 present Palm theory of stationary stochastic processes associated with a simple point process. Chapter 8 treats the one dimensional case and Chapter 9 the higher-dimensional case. Chapter 10 deals with regeneration, both classical regenerative pro cesses and three generalizations: wide-sense regeneration (as in Harris chains); time-inhomogeneous regeneration (as in time-inhomogeneous recurrent Markov chains); and taboo regeneration (as in transient Markov chains). It ends with a section on perfect simulation ( cou pling from-the-past). This enormous chapter is thrice the size of a normal chapter, and is really a book within the book.

"What the book does offer is a areful, stimulating, and original discussion of major themes in coupling. As such, it will be invaluable to probabilists and also to the increasing number of statisticians working on Markov Chain Monte Carlo and especially perfect simulation."W.S. Kendall in "Short Book Reviews", Vol. 21/1, April 2001

1 Random Variables.- 1 Introduction.- 2 The i.i.d. Coupling — Positive Correlation.- 3 Quantile Coupling — Stochastic Domination.- 4 Coupling Event — Maximal Coupling.- 5 Poisson Approximation — Total Variation.- 6 Convergence of Discrete Random Variables.- 7 Continuous Variables — Hitting the Limit.- 8 Convergence in Distribution and Pointwise.- 9 Quantile Coupling — Dominated Convergence.- 10 Impossible Coupling — Quantum Physics.- 2 Markov Chains and Random Walks.- 1 Introduction.- 2 Classical Coupling — Birth and Death Processes.- 3 Classical Coupling — Recurrent Markov Chains.- 4 Classical Coupling — Rates and Uniformity.- 5 Ornstein Coupling — Random Walk on the Integers.- 6 Ornstein Coupling — Recurrent Markov Chains.- 7 Epsilon-Coupling —Nonlattice Random Walk.- 8 Epsilon-Coupling —Blackwell’s Renewal Theorem.- 9 Renewal Processes — Stationarity.- 10 Renewal Processes — Asymptotic Stationarity.- 3 Random Elements.- 1 Introduction.- 2 Back to Basics — Definition of Coupling.- 3 Extension Techniques.- 4 Conditioning — Transfer.- 5 Splitting.- 6 Random Walk with Spread-Out Step-Lengths.- 7 Coupling Event — Maximal Coupling.- 8 Maximal Coupling Two Elements — Total Variation.- 9 Hitting the Limit.- 10 Convergence in Distribution and Pointwise.- 4 Stochastic Processes.- 1 Introduction.- 2 Preliminaries — What Is a Stochastic Process?.- 3 Exact Coupling — Distributional Exact Coupling.- 4 Distributional Coupling.- 5 Exact Coupling — Inequality and Asymptotics.- 6 Exact Coupling — Maximality.- 7 Coupling with Respect to a Sub-a-Algebra.- 8 Exact Coupling — Another Proof of Theorem 6.1.- 9 Exact Coupling — Tail a-Algebra — Equivalences.- 5 Shift-Coupling.- 1 Introduction.- 2 Shift-Coupling — Distributional Shift-Coupling.- 3 Shift-Coupling — Inequality and Asymptotics.- 4 Shift-Coupling — Maximality.- 5 Shift-Coupling — Invariant a-Algebra — Equivalences.- 6 E-Coupling — Distributional E-Coupling.- 7 e-Coupling — Inequality and Asymptotics.- 8 E-Coupling — Maximality.- 9 e-Coupling — Smooth Tail a-algebra — Equivalences.- 6 Markov Processes.- 1 Introduction.- 2 Mixing and Triviality of a Stochastic Process.- 3 Markov Processes — Preliminaries.- 4 Exact Coupling.- 5 Shift-Coupling.- 6 Epsilon-Coupling.- 7 Stationary Measure.- 7 Transformation Coupling.- 1 Introduction.- 2 Shift-Coupling Random Fields.- 3 Transformation Coupling.- 4 Inequality and Asymptotics.- 5 Maximality.- 6 Invariant a-Algebra and Equivalences.- 7 Topological Transformation Groups.- 8 Self-Similarity — Exchangeability — Rotation.- 9 Exact Transformation Coupling.- 8 Stationarity, The Palm Dualities.- 1 Introduction.- 2 Preliminaries — Measure-Free Part of the Dualities.- 3 Key Stationarity Theorem.- 4 The Point-at-Zero Duality.- 5 Interpretation — Point-Conditioning.- 6 Application — Perfect Simulation.- 7 The Invariant a-Algebras I and J.- 8 The Randomized-Origin Duality.- 9 Interpretation — Cesaro Limits and Shift-Coupling.- 10 Comments on the Two Palm Dualities.- 9 The Palm Dualities in Higher Dimensions.- 1 Introduction.- 2 The Point-Stationarity Problem.- 3 Definition of Point-Stationarity.- 4 Palm Characterization of Point-Stationarity.- 5 Point-Stationarity Characterized by Randomization.- 6 Point-Stationarity and the Invariant a-Algebras.- 7 The Point-at-Zero Duality.- 8 The Randomized-Origin Duality.- 9 Comments.- 10 Regeneration.- 1 Introduction.- 2 Preliminaries — Stationarity.- 3 Classical Regeneration.- 4 Wide-Sense Regeneration — Harris Chains — GI/GI/k.- 5 Time-Inhomogeneous Regeneration.- 6 Classical Coupling.- 7 The Coupling Time — Rates and Uniformity.- 8 Asymptotics From-the-Past.- 9 Taboo Regeneration.- 10 Taboo Stationarity.- 11 Perfect Simulation — Coupling From-the-Past.- Notes.- References.- Notation.