作者简介 Sean Meyn,Richard L.Tweedie是概率统计领域的著名学者,他们二人合著的马尔科夫链和随机稳定性》,曾获得1994年度ORSA/TIMS“应用概率最优秀出版物奖”。
目录 List of figures Prologue to the second edition, Peter W.Glynn Preface to the second edition, Sean Meyn. Preface to the first edition Ⅰ COMMUNICATION and REGENERATION 1 Heuristics 1.1 A range of Markovian environments 1.2 Basic models in practice 1.3 Stochastic stability for Markov models 1.4 Commentary 2 Markov models 2.1 Markov models in time series 2.2 Nonlinear state space models 2.3 Models in control and systems theory 2.4 Markov models with regeneration times 2.5 Commentary 3 Transition probabilities 3.1 Defining a Markovian process 3.2 Foundations on a countable space 3.3 Specific transition matrices 3.4 Foundations for general state space chains 3.5 Building transition kernels for specific models 3.6 Commentary 4 Irreducibility 4.1 Communication and irreducibility: Countable spaces 4.2 ψ—Irreducibility 4.3 ψ—Irreducibility for random walk models 4.4 ψ—Irreducible linear models 4.5 Commentary 5 Pseudo—atoms 5.1 Splitting ψ—irreducible chains 5.2 Small sets 5.3 Small sets for specific models 5.4 Cyclic behavior 5.5 Petite sets and sampled chains 5.6 Commentary 6 Topology and continuity 6.1 Feller properties and forms of stability 6.2 T—chains 6.3 Continuous components for specific models 6.4 e—Chains 6.5 Commentary 7 The nonlinear state space model 7.1 Forward accessibility and continuous components 7.2 Minimal sets and irreducibility 7.3 Periodicity for nonlinear state space models 7.4 Forward accessible examples 7.5 Equicontinuity and the nonlinear state space model 7.6 Commentary Ⅱ STABILITY STRUCTURES 8 Transience and recurrence 8.1 Classifying chains on countable spaces 8.2 Classifying ψ—irreducible chains 8.3 Recurrence and transience relationships 8.4 Classification using drift criteria 8.5 Classifying random walk on R+ 8.6 Commentary 9 Harris and topological recurrence 9.1 Harris recurrence 9.2 Non—evanescent and recurrent chains 9.3 Topologically recurrent and transient states 9.4 Criteria for stability on a topological space 9.5 Stochastic comparison and increment analysis 9.6 Commentary 10 The existence of π 10.1 Stationarity and invariance 10.2 The existence of π: chains with atoms 10.3 Invariant measures for countable space models 10.4 The existence of π:ψ—irreducible chains 10.5 Invariant measures for general models 10.6 Commentary 11 Drift and regularity 11.1 Regular chains 11.2 Drift, hitting times and deterministic models 11.3 Drift criteria for regularity 11.4 Using the regularity criteria 11.5 Evaluating non—positivity 11.6 Commentary 12 Invariance and tightness 12.1 Chains bounded in probability 12.2 Generalized sampling and invariant measures 12.3 The existence of a σ—finite invariant measure 12.4 Invariant measures for e—chains 12.5 Establishing boundedness in probability 12.6 Commentary Ⅲ CONVERGENCE 13 Ergodicity 13.1 Ergodic chains on countable spaces 13.2 Renewal and regeneration 13.3 Ergodicity of positive Harris chains 13.4 Sums of transition probabilities 13.5 Commentary 14 f—Ergodicity and f—regularity 14.1 f—Properties: chains with atoms 14.2 f—Regularity and drift 14.3 f—Ergodicity for general chains 14.4 f—Ergodicity of specific models 14.5 A key renewal theorem 14.6 Commentary 15 Geometric ergodicity 15.1 Geometric properties: chains with atoms 15.2 Kendall sets and drift criteria 15.3 f—Geometric regularity of Φ and its skeleton 15.4 f—Geometric ergodicity for general chains 15.5 Simple random walk and linear models 15.6 Commentary 16 V—Uniform ergodicity 16.1 Operator norm convergence 16.2 Uniform ergodicity 16.3 Geometric ergodicity and increment analysis 16.4 Models from queueing theory 16.5 Autoregressive and state space models 16.6 Commentary 17 Sample paths and limit theorems 17.1 Invariant σ—fields and the LLN 17.2 Ergodic theorems for chains possessing an atom 17.3 General Harris chains 17.4 The functional CLT 17.5 Criteria for the CLT and the LIL 17.6 Applications 17.7 Commentary 18 Positivity 18.1 Null recurrent chains 18.2 Characterizing positivity using pn, 18.3 Positivity and T—chains 18.4 Positivity and e—chains 18.5 The LLN for e—chains 18.6 Commentary 19 Generalized classification criteria 19.1 State—dependent drifts 19.2 History—dependent drift criteria 19.3 Mixed drift conditions 19.4 Commentary 20 Epilogue to the second edition 20.1 Geometric ergodicity and spectral theory 20.2 Simulation and MCMC 20.3 Continuous time models Ⅳ APPENDICES A Mud maps A:1 Recurrence versus transience A.2 Positivity versus nullity A.3 Convergence properties B Testing for stability B.1 Glossary of drift conditions B.2 The scalar SETAR model: a complete classification C Glossary of model assumptions C.l Regenerative models C.2 State space models D Some mathematical background D.1 Some measure theory D.2 Some probability theory D.3 Some topology D.4 Some real analysis D.5 Convergence concepts for measures D.6 Some martingale theory D.7 Some results on sequences and numbers Bibliography Indexes General index Symbols
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