Coupling, Stationarity, and Regeneration
(Sprache: Englisch)
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,...
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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.
Klappentext zu „Coupling, Stationarity, and Regeneration “
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.
This is a book on coupling, the method of establishing properties of random variables and processes (or any random things) through a joint construction on a common probability space. This is a general method relevant in all fields of probabilistic inquiry but the thrust is towards characterizations, approximations and asymptotics. The book also includes self-contained treatments of stationarity (Palm theory) and regeneration (classical, wide-sense, time-inhomogeneous, and taboo regeneration). Other topics discussed are perfect simulation (MCMC) and quasi-stationarity. Links are made to several fields such as quantum physics and non-locality, self-similarity, exchangeability, relativity, and queueing theory. The book is organized in chapters as follows:
1. Random Variables
2. Markov Chains and Random Walks
3. Random Elements
4. Stochastic Processes
5. Shift-Coupling
6. Markov Processes
7. Transformation Coupling
8. Stationarity, the Palm Dualities
9. The Palm Dualities in Higher Dimensions
10. Regeneration
The book should be of interest to students and researchers in probability, stochastic modeling, and mathematical statistics. It is written with a Ph.D. student in mind but the first two chapters can be read at master level and even at an advanced undergraduate level. The book is self-contained, only relying on the measure theoretic basics and on elementary Markov chain theory. Hermann Thorisson received his Ph.D. from the Department of
1. Random Variables
2. Markov Chains and Random Walks
3. Random Elements
4. Stochastic Processes
5. Shift-Coupling
6. Markov Processes
7. Transformation Coupling
8. Stationarity, the Palm Dualities
9. The Palm Dualities in Higher Dimensions
10. Regeneration
The book should be of interest to students and researchers in probability, stochastic modeling, and mathematical statistics. It is written with a Ph.D. student in mind but the first two chapters can be read at master level and even at an advanced undergraduate level. The book is self-contained, only relying on the measure theoretic basics and on elementary Markov chain theory. Hermann Thorisson received his Ph.D. from the Department of
Inhaltsverzeichnis zu „Coupling, Stationarity, and Regeneration “
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 -
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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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Bibliographische Angaben
- Autor: Hermann Thorisson
- 2000, 2000. Aufl., XIV, 517 Seiten, Maße: 16,2 x 24,2 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 0387987797
- ISBN-13: 9780387987798
Sprache:
Englisch
Rezension zu „Coupling, Stationarity, and Regeneration “
"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
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