Coupling, stationarity, and regeneration
著者
書誌事項
Coupling, stationarity, and regeneration
（Probability and its applications）
Springer-Verlag, c2000
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
Coupling is a general method of establishing properties of random variables and processes through a joint construction on a common probability space. This method has relevance to all areas of probabilistic inquiry including quantum physics, self-similarity, relativity, and queueing theory. In addition to providing new developments in coupling, this book also includes self-contained treatments of Markov chains, stationarity, regeneration, perfect simulation, and quasi-stationarity.
目次
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.
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