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Markov Chains and Stochastic Stability

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ISBN-10: 0521731828

ISBN-13: 9780521731829

Edition: 2nd 2009

Authors: Sean Meyn, Richard L. Tweedie, Peter W. Glynn

List price: $120.00
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Book details

List price: $120.00
Edition: 2nd
Copyright year: 2009
Publisher: Cambridge University Press
Publication date: 4/2/2009
Binding: Paperback
Pages: 624
Size: 6.75" wide x 9.50" long x 1.25" tall
Weight: 2.486
Language: English

Richard L. Tweedie was Professor and Head of the Division of Biostatistics at the University of Minnesota before his death in 2001.

List of figures
Prologue to the Second edition, Peter W. Glynn
Preface to the second edition, Sean Meyn
Preface to the first edition
Communication and Regeneration
A range of Markovian environments
Basic models in practice
Stochastic stability for Markov models
Markov models
Markov models in time series
Nonlinear state space models
Models in control and systems theory
Markov models with regeneration times
Transition probabilities
Defining a Markovian Process
Foundations on a countable space
Specific transition matrices
Foundations for general state space chains
Building transition kernels for specific models
Communication and irreducibility: Countable spaces
�-Irreducibility for random walk models
�-Irreducible linear models
Splitting �-irreducible chains
Small sets
Small sets for specific models
Cyclic behavior
Petite sets and sampled chains
Topology and continuity
Feller properties and forms of stability
Continuous components for specific models
The nonlinear state space model
Forward accessibility and continuous components
Minimal sets and irreducibility
Periodicity for nonlinear state space models
Forward accessible examples
Equicontinuity and the nonlinear state space model
Stability Structures
Transience and recurrence
Classifying chains on countable spaces
Classifying �-irreducible chains
Recurrence and transience relationships
Classification using drift criteria
Classifying random walk on R+
Harris and topological recurrence
Harris recurrence
Non-evanescent and recurrent chains
Topologically recurrent and transient states
Criteria for stability on a topological space
Stochastic comparison and increment analysis
The existence of �
Stationarity and invariance
The existence of �: chains with atoms
Invariant measures for countable space models
The existence of �: �-irreducible chains
Invariant measures for general models
Drift and regularity
Regular chains
Drift, hitting times and deterministic models
Drift, criteria for regularity
Using the regularity criteria
Evaluating non-positivity
Invariance and tightness
Chains bounded in probability
Generalized sampling and invariant measures
The existence of a �-finite invariant measure
Invariant measures for e-chains
Establishing boundedness in probability
Ergodic chains on countable spaces
Renewal and regeneration
Ergodicity of positive Harris chains
Sums of transition probabilities
f-Ergodicity and f-regularity
f-Properties: chains with atoms
f-Regularity and drift
f-Ergodicity for general chains
f-Ergodicity of specific models
A key renewal theorem
Geometric ergodicity
Geometric properties: chains with atoms
Kendall sets and drift criteria
f-Geometric regularity of � and its skeleton
f-Geometric ergodicity for general chains
Simple random walk and linear models
V-Uniform ergodicity
Operator norm convergence
Uniform ergodicity
Geometric ergodicity and increment analysis
Models from queueing theory
Autoregressive and state space models
Sample paths and limit theorems
Invariant �-fields and the LLN
Ergodic theorems for chains possessing an atom
General Harris chains
The functional CLT
Criteria for the CLT and the LIL
Null recurrent chains
Characterizing positivity using Pn
Positivity and T-chains
Positivity and e-chains
The LLN for e-chains
Generalized classification criteria
State-dependent drifts
History-dependent drift criteria
Mixed drift conditions
Epilogue to the second edition
Geometric ergodicity and spectral theory
Simulation and MCMC
Continuous time models
Mud maps
Recurrence versus transience
Positivity versus nullity
Convergence properties
Testing for stability
Glossary of drift conditions
The Scalar SETAR model: a complete classification
Glossary of models assumptions
Regenerative models
State space models
Some mathematical background
Some measure theory
Some probability theory
Some topology
Some real analysis
Convergence concepts for measures
Some martingale theory
Some results on sequences and numbers
General index