Thorisson - Coupling, Stationarity, and Regeneration

Coupling, Stationarity, and Regeneration

Medium: Buch

ISBN: 978-0-387-98779-8

Verlag: Springer Netherlands

Erscheinungstermin: 21.01.2000

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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.