Markov chains and stochastic stability:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London [u.a.]
Springer
1996
|
| Edition: | 3. print. |
| Series: | Communications and control engineering series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [528] - 541 |
| Physical Description: | XVI, 550 S. |
| ISBN: | 3540198326 0387198326 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Table of Contents
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 15
1.4 Commentary 21
2 Markov Models 23
2.1 Markov Models In Time Series . . . . , 24
2.2 Nonlinear State Space Models 28
2.3 Models In Control And Systems Theory 36
2.4 Markov Models With Regeneration Times 43
2.5 Commentary 52
3 Transition Probabilities 54
3.1 Defining a Markovian Process 55
3.2 Foundations on a Countable Space 57
3.3 Specific Transition Matrices 60
3.4 Foundations for General State Space Chains 65
3.5 Building Transition Kernels For Specific Models 73
3.6 Commentary 78
4 Irreducibility 80
4.1 Communication and Irreducibility: Countable Spaces 81
4.2 ^ Irreducibility 86
4.3 ^ Irreducibility For Random Walk Models 92
4.4 y Irreducible Linear Models 94
4.5 Commentary 98
5 Pseudo atoms 100
5.1 Splitting ^ Irreducible Chains 101
5.2 Small Sets 106
5.3 Small Sets for Specific Models 110
5.4 Cyclic Behavior 114
5.5 Petite Sets and Sampled Chains 119
xiv Table of Contents
5.6 Commentary 125
6 Topology and Continuity 126
6.1 Feller Properties and Forms of Stability 128
6.2 T chains 133
6.3 Continuous Components For Specific Models 137
6.4 e Chains 143
6.5 Commentary 147
7 The Nonlinear State Space Model 149
7.1 Forward Accessibility and Continuous Components 150
7.2 Minimal Sets and Irreducibility 157
7.3 Periodicity for nonlinear state space models 160
7.4 Forward Accessible Examples 164
7.5 Equicontinuity and the nonlinear state space model 166
7.6 Commentary 168
II STABILITY STRUCTURES 171
8 Transience and Recurrence 173
8.1 Classifying chains on countable spaces 175
8.2 Classifying ^ irreducible chains 179
8.3 Recurrence and transience relationships 184
8.4 Classification using drift criteria 189
8.5 Classifying random walk on 1R+ 194
8.6 Commentary 199
9 Harris and Topological Recurrence 200
9.1 Harris recurrence 201
9.2 Non evanescent and recurrent chains 206
9.3 Topologically recurrent and transient states 209
9.4 Criteria for stability on a topological space 214
9.5 Stochastic comparison and increment analysis 219
9.6 Commentary 228
10 The Existence of tt 229
10.1 Stationarity and Invariance 230
10.2 The existence of ir: chains with atoms 234
10.3 Invariant measures: countable space models 236
10.4 The existence of ir: ^ irreducible chains 240
10.5 Invariant Measures: General Models 246
10.6 Commentary 252
11 Drift and Regularity 255
11.1 Regular chains 257
11.2 Drift, hitting times and deterministic models 259
11.3 Drift criteria for regularity 262
11.4 Using the regularity criteria 270
11.5 Evaluating non positivity 276
Table of Contents xv
11.6 Commentary 282
12 Invarian.ee and Tightness 285
12.1 Chains bounded in probability 286
12.2 Generalized sampling and invariant measures 289
12.3 The existence of a cr finite invariant measure 294
12.4 Invariant Measures for e Chains 296
12.5 Establishing boundedness in probability 301
12.6 Commentary 305
III CONVERGENCE 307
13 Ergodicity 309
13.1 Ergodic chains on countable spaces 312
13.2 Renewal and regeneration 316
13.3 Ergodicity of positive Harris chains 322
13.4 Sums of transition probabilities 325
13.5 Commentary 328
14 / Ergodicity and / Regularity 330
14.1 / Properties: chains with atoms 332
14.2 / Regularity and drift 336
14.3 / Ergodicity for general chains 342
14.4 / Ergodicity of specific models 346
14.5 A Key Renewal Theorem 347
14.6 Commentary 352
15 Geometric Ergodicity 354
15.1 Geometric properties: chains with atoms 356
15.2 Kendall sets and drift criteria 363
15.3 / Geometric regularity of # and #n 371
15.4 / Geometric ergodicity for general chains 374
15.5 Simple random walk and linear models 378
15.6 Commentary 381
16 F Uniform Ergodicity 382
16.1 Operator norm convergence 385
16.2 Uniform ergodicity 390
16.3 Geometric ergodicity and increment analysis 396
16.4 Models from queueing theory 400
16.5 Autoregressive and state space models 403
16.6 Commentary 407
17 Sample Paths and Limit Theorems 410
17.1 Invariant tr Fields and the LLN 412
17.2 Ergodic Theorems for Chains Possessing an Atom 417
17.3 General Harris Chains 421
17.4 The Functional CLT 431
17.5 Criteria for the CLT and the LIL 43S
xvi Table of Contents
17.6 Applications 441
17.7 Commentary 444
18 Positivity 446
18.1 Null recurrent chains 448
18.2 Characterizing positivity using Pn 452
18.3 Positivity and T chains 455
18.4 Positivity and e Chains 457
18.5 The LLN for e Chains 461
18.6 Commentary 463
19 Generalized Classification Criteria 465
19.1 State dependent drifts 466
19.2 History dependent drift criteria 474
19.3 Mixed drift conditions 481
19.4 Commentary 490
IV APPENDICES 493
A Mud Maps 496
A.I Recurrence versus transience 496
A.2 Positivity versus nullity 498
A.3 Convergence Properties 500
B Testing for Stability 501
B.I A Glossary of Drift Conditions 501
B.2 The scalar SETAR Model: a complete classification 503
C A Glossary of Model Assumptions 506
C.I Regenerative Models 506
C.2 State Space Models 509
D Some Mathematical Background 515
D.I Some Measure Theory 515
D.2 Some Probability Theory 518
D.3 Some Topology 519
D.4 Some Real Analysis 520
D.5 Some Convergence Concepts for Measures 521
D.6 Some Martingale Theory 524
D.7 Some Results on Sequences and Numbers 526
References 528
Index 543
Symbols Index 549
|
| adam_txt |
Table of Contents
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 15
1.4 Commentary 21
2 Markov Models 23
2.1 Markov Models In Time Series . . . . , 24
2.2 Nonlinear State Space Models 28
2.3 Models In Control And Systems Theory 36
2.4 Markov Models With Regeneration Times 43
2.5 Commentary 52
3 Transition Probabilities 54
3.1 Defining a Markovian Process 55
3.2 Foundations on a Countable Space 57
3.3 Specific Transition Matrices 60
3.4 Foundations for General State Space Chains 65
3.5 Building Transition Kernels For Specific Models 73
3.6 Commentary 78
4 Irreducibility 80
4.1 Communication and Irreducibility: Countable Spaces 81
4.2 ^ Irreducibility 86
4.3 ^ Irreducibility For Random Walk Models 92
4.4 y Irreducible Linear Models 94
4.5 Commentary 98
5 Pseudo atoms 100
5.1 Splitting ^ Irreducible Chains 101
5.2 Small Sets 106
5.3 Small Sets for Specific Models 110
5.4 Cyclic Behavior 114
5.5 Petite Sets and Sampled Chains 119
xiv Table of Contents
5.6 Commentary 125
6 Topology and Continuity 126
6.1 Feller Properties and Forms of Stability 128
6.2 T chains 133
6.3 Continuous Components For Specific Models 137
6.4 e Chains 143
6.5 Commentary 147
7 The Nonlinear State Space Model 149
7.1 Forward Accessibility and Continuous Components 150
7.2 Minimal Sets and Irreducibility 157
7.3 Periodicity for nonlinear state space models 160
7.4 Forward Accessible Examples 164
7.5 Equicontinuity and the nonlinear state space model 166
7.6 Commentary 168
II STABILITY STRUCTURES 171
8 Transience and Recurrence 173
8.1 Classifying chains on countable spaces 175
8.2 Classifying ^ irreducible chains 179
8.3 Recurrence and transience relationships 184
8.4 Classification using drift criteria 189
8.5 Classifying random walk on 1R+ 194
8.6 Commentary 199
9 Harris and Topological Recurrence 200
9.1 Harris recurrence 201
9.2 Non evanescent and recurrent chains 206
9.3 Topologically recurrent and transient states 209
9.4 Criteria for stability on a topological space 214
9.5 Stochastic comparison and increment analysis 219
9.6 Commentary 228
10 The Existence of tt 229
10.1 Stationarity and Invariance 230
10.2 The existence of ir: chains with atoms 234
10.3 Invariant measures: countable space models 236
10.4 The existence of ir: ^' irreducible chains 240
10.5 Invariant Measures: General Models 246
10.6 Commentary 252
11 Drift and Regularity 255
11.1 Regular chains 257
11.2 Drift, hitting times and deterministic models 259
11.3 Drift criteria for regularity 262
11.4 Using the regularity criteria 270
11.5 Evaluating non positivity 276
Table of Contents xv
11.6 Commentary 282
12 Invarian.ee and Tightness 285
12.1 Chains bounded in probability 286
12.2 Generalized sampling and invariant measures 289
12.3 The existence of a cr finite invariant measure 294
12.4 Invariant Measures for e Chains 296
12.5 Establishing boundedness in probability 301
12.6 Commentary 305
III CONVERGENCE 307
13 Ergodicity 309
13.1 Ergodic chains on countable spaces 312
13.2 Renewal and regeneration 316
13.3 Ergodicity of positive Harris chains 322
13.4 Sums of transition probabilities 325
13.5 Commentary 328
14 / Ergodicity and / Regularity 330
14.1 / Properties: chains with atoms 332
14.2 / Regularity and drift 336
14.3 / Ergodicity for general chains 342
14.4 / Ergodicity of specific models 346
14.5 A Key Renewal Theorem 347
14.6 Commentary 352
15 Geometric Ergodicity 354
15.1 Geometric properties: chains with atoms 356
15.2 Kendall sets and drift criteria 363
15.3 / Geometric regularity of # and #n 371
15.4 / Geometric ergodicity for general chains 374
15.5 Simple random walk and linear models 378
15.6 Commentary 381
16 F Uniform Ergodicity 382
16.1 Operator norm convergence 385
16.2 Uniform ergodicity 390
16.3 Geometric ergodicity and increment analysis 396
16.4 Models from queueing theory 400
16.5 Autoregressive and state space models 403
16.6 Commentary 407
17 Sample Paths and Limit Theorems 410
17.1 Invariant tr Fields and the LLN 412
17.2 Ergodic Theorems for Chains Possessing an Atom 417
17.3 General Harris Chains 421
17.4 The Functional CLT 431
17.5 Criteria for the CLT and the LIL 43S
xvi Table of Contents
17.6 Applications 441
17.7 Commentary 444
18 Positivity 446
18.1 Null recurrent chains 448
18.2 Characterizing positivity using Pn 452
18.3 Positivity and T chains 455
18.4 Positivity and e Chains 457
18.5 The LLN for e Chains 461
18.6 Commentary 463
19 Generalized Classification Criteria 465
19.1 State dependent drifts 466
19.2 History dependent drift criteria 474
19.3 Mixed drift conditions 481
19.4 Commentary 490
IV APPENDICES 493
A Mud Maps 496
A.I Recurrence versus transience 496
A.2 Positivity versus nullity 498
A.3 Convergence Properties 500
B Testing for Stability 501
B.I A Glossary of Drift Conditions 501
B.2 The scalar SETAR Model: a complete classification 503
C A Glossary of Model Assumptions 506
C.I Regenerative Models 506
C.2 State Space Models 509
D Some Mathematical Background 515
D.I Some Measure Theory 515
D.2 Some Probability Theory 518
D.3 Some Topology 519
D.4 Some Real Analysis 520
D.5 Some Convergence Concepts for Measures 521
D.6 Some Martingale Theory 524
D.7 Some Results on Sequences and Numbers 526
References 528
Index 543
Symbols Index 549 |
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| id | DE-604.BV021937163 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:06:41Z |
| indexdate | 2024-07-09T20:47:47Z |
| institution | BVB |
| isbn | 3540198326 0387198326 |
| language | English |
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| physical | XVI, 550 S. |
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| publishDateSearch | 1996 |
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| publisher | Springer |
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| series2 | Communications and control engineering series |
| spelling | Meyn, S. P. Verfasser aut Markov chains and stochastic stability S. P. Meyn and R. L. Tweedie 3. print. London [u.a.] Springer 1996 XVI, 550 S. txt rdacontent n rdamedia nc rdacarrier Communications and control engineering series Literaturverz. S. [528] - 541 Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Zustandsraum (DE-588)4132647-7 gnd rswk-swf Markov-Kette (DE-588)4037612-6 gnd rswk-swf Zeitreihenanalyse (DE-588)4067486-1 gnd rswk-swf Markov-Kette (DE-588)4037612-6 s DE-604 Zeitreihenanalyse (DE-588)4067486-1 s Zustandsraum (DE-588)4132647-7 s Markov-Prozess (DE-588)4134948-9 s Tweedie, R. L. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015152316&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Meyn, S. P. Tweedie, R. L. Markov chains and stochastic stability Markov-Prozess (DE-588)4134948-9 gnd Zustandsraum (DE-588)4132647-7 gnd Markov-Kette (DE-588)4037612-6 gnd Zeitreihenanalyse (DE-588)4067486-1 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4132647-7 (DE-588)4037612-6 (DE-588)4067486-1 |
| title | Markov chains and stochastic stability |
| title_auth | Markov chains and stochastic stability |
| title_exact_search | Markov chains and stochastic stability |
| title_exact_search_txtP | Markov chains and stochastic stability |
| title_full | Markov chains and stochastic stability S. P. Meyn and R. L. Tweedie |
| title_fullStr | Markov chains and stochastic stability S. P. Meyn and R. L. Tweedie |
| title_full_unstemmed | Markov chains and stochastic stability S. P. Meyn and R. L. Tweedie |
| title_short | Markov chains and stochastic stability |
| title_sort | markov chains and stochastic stability |
| topic | Markov-Prozess (DE-588)4134948-9 gnd Zustandsraum (DE-588)4132647-7 gnd Markov-Kette (DE-588)4037612-6 gnd Zeitreihenanalyse (DE-588)4067486-1 gnd |
| topic_facet | Markov-Prozess Zustandsraum Markov-Kette Zeitreihenanalyse |
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