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[资源] Markov Chains and Stochastic Stability

Contents
Asterisks (*) mark sections from the first edition that have been revised or augmented
in the second edition.
List of figures xi
Prologue to the second edition, Peter W. Glynn xiii
Preface to the second edition, Sean Meyn xvii
Preface to the first edition xxi
I COMMUNICATION and REGENERATION 1
1 Heuristics 3
1.1 A range of Markovian environments 3
1.2 Basic models in practice 6
1.3 Stochastic stability for Markov models 13
1.4 Commentary 19
2 Markov models 21
2.1 Markov models in time series 22
2.2 Nonlinear state space models* 26
2.3 Models in control and systems theory 33
2.4 Markov models with regeneration times 38
2.5 Commentary* 46
3 Transition probabilities 48
3.1 Defining a Markovian process 49
3.2 Foundations on a countable space 51
3.3 Specific transition matrices 54
3.4 Foundations for general state space chains 59
3.5 Building transition kernels for specific models 67
3.6 Commentary 72
v
vi Contents
4 Irreducibility 75
4.1 Communication and irreducibility: Countable spaces 76
4.2 ψ-Irreducibility 81
4.3 ψ-Irreducibility for random walk models 87
4.4 ψ-Irreducible linear models 89
4.5 Commentary 93
5 Pseudo-atoms 96
5.1 Splitting ϕ-irreducible chains 97
5.2 Small sets 102
5.3 Small sets for specific models 106
5.4 Cyclic behavior 110
5.5 Petite sets and sampled chains 115
5.6 Commentary 121
6 Topology and continuity 123
6.1 Feller properties and forms of stability 125
6.2 T-chains 130
6.3 Continuous components for specific models 134
6.4 e-Chains 139
6.5 Commentary 144
7 The nonlinear state space model 146
7.1 Forward accessibility and continuous components 147
7.2 Minimal sets and irreducibility 154
7.3 Periodicity for nonlinear state space models 157
7.4 Forward accessible examples 161
7.5 Equicontinuity and the nonlinear state space model 163
7.6 Commentary* 165
II STABILITY STRUCTURES 169
8 Transience and recurrence 171
8.1 Classifying chains on countable spaces 173
8.2 Classifying ψ-irreducible chains 177
8.3 Recurrence and transience relationships 182
8.4 Classification using drift criteria 187
8.5 Classifying random walk on R + 193
8.6 Commentary* 197
9 Harris and topological recurrence 199
9.1 Harris recurrence 201
9.2 Non-evanescent and recurrent chains 206
9.3 Topologically recurrent and transient states 208
9.4 Criteria for stability on a topological space 213
9.5 Stochastic comparison and increment analysis 218
9.6 Commentary 228
Contents vii
10 The existence of π 229
10.1 Stationarity and invariance 230
10.2 The existence of π: chains with atoms 234
10.3 Invariant measures for countable space models* 236
10.4 The existence of π: ψ-irreducible chains 241
10.5 Invariant measures for general models 247
10.6 Commentary 253
11 Drift and regularity 256
11.1 Regular chains 258
11.2 Drift, hitting times and deterministic models 261
11.3 Drift criteria for regularity 263
11.4 Using the regularity criteria 272
11.5 Evaluating non-positivity 278
11.6 Commentary 285
12 Invariance and tightness 288
12.1 Chains bounded in probability 289
12.2 Generalized sampling and invariant measures 292
12.3 The existence of a σ-finite invariant measure 298
12.4 Invariant measures for e-chains 300
12.5 Establishing boundedness in probability 305
12.6 Commentary 308
III CONVERGENCE 311
13 Ergodicity 313
13.1 Ergodic chains on countable spaces 316
13.2 Renewal and regeneration 320
13.3 Ergodicity of positive Harris chains 326
13.4 Sums of transition probabilities 329
13.5 Commentary* 334
14 f-Ergodicity and f-regularity 336
14.1 f-Properties: chains with atoms 338
14.2 f-Regularity and drift 342
14.3 f-Ergodicity for general chains 349
14.4 f-Ergodicity of specific models 352
14.5 A key renewal theorem 354
14.6 Commentary* 359
15 Geometric ergodicity 362
15.1 Geometric properties: chains with atoms 364
15.2 Kendall sets and drift criteria 372
15.3 f-Geometric regularity of Φ and its skeleton 380
15.4 f-Geometric ergodicity for general chains 384
15.5 Simple random walk and linear models 388
viii Contents
15.6 Commentary* 390
16 V -Uniform ergodicity 392
16.1 Operator norm convergence 395
16.2 Uniform ergodicity 400
16.3 Geometric ergodicity and increment analysis 407
16.4 Models from queueing theory 411
16.5 Autoregressive and state space models 414
16.6 Commentary* 418
17 Sample paths and limit theorems 421
17.1 Invariant σ-fields and the LLN 423
17.2 Ergodic theorems for chains possessing an atom 428
17.3 General Harris chains 433
17.4 The functional CLT 443
17.5 Criteria for the CLT and the LIL 450
17.6 Applications 454
17.7 Commentary* 456
18 Positivity 462
18.1 Null recurrent chains 464
18.2 Characterizing positivity using P n 469
18.3 Positivity and T-chains 471
18.4 Positivity and e-chains 473
18.5 The LLN for e-chains 477
18.6 Commentary 480
19 Generalized classification criteria 482
19.1 State-dependent drifts 483
19.2 History-dependent drift criteria 491
19.3 Mixed drift conditions 498
19.4 Commentary* 508
20 Epilogue to the second edition 510
20.1 Geometric ergodicity and spectral theory 510
20.2 Simulation and MCMC 521
20.3 Continuous time models 523
IV APPENDICES 529
A Mud maps 532
A.1 Recurrence versus transience 532
A.2 Positivity versus nullity 534
A.3 Convergence properties 536
Contents ix
B Testing for stability 538
B.1 Glossary of drift conditions 538
B.2 The scalar SETAR model: a complete classification 540
C Glossary of model assumptions 543
C.1 Regenerative models 543
C.2 State space models 546
D Some mathematical background 552
D.1 Some measure theory 552
D.2 Some probability theory 555
D.3 Some topology 556
D.4 Some real analysis 557
D.5 Convergence concepts for measures 558
D.6 Some martingale theory 561
D.7 Some results on sequences and numbers 563
Bibliography 567
Indexes 587
General index 587
Symbols 593

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附件 1 : Markov_Chains_and_Stochastic_Stability_-_Sean_Meyn,_Richard_L._Tweedie_-_(_Cambridge_University_Press_-_2nd_Ed.2009_(1Ã‚Â°Ed.1993)_-_pp.624).pdf

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