Markov Chains: Gibbs Fields, Monte Carlo simulation, and queues
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer New York
2010
|
| Edition: | 1. ed. 1999. Softcover reprint of the hardcover 1st edition 1999 |
| Series: | Texts in Applied Mathematics
31 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Erscheint: 2. November 2010 |
| Physical Description: | XVIII, 444 Seiten 64 schw.-w. Ill. 235 mm x 178 mm |
| ISBN: | 9781441931313 1441931317 |
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MARC
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| 100 | 1 | |a Brémaud, Pierre |e Verfasser |0 (DE-588)1060028131 |4 aut | |
| 245 | 1 | 0 | |a Markov Chains |b Gibbs Fields, Monte Carlo simulation, and queues |c Pierre Bremaud |
| 250 | |a 1. ed. 1999. Softcover reprint of the hardcover 1st edition 1999 | ||
| 264 | 1 | |a New York, NY |b Springer New York |c 2010 | |
| 300 | |a XVIII, 444 Seiten |b 64 schw.-w. Ill. |c 235 mm x 178 mm | ||
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Contents
Series Preface
vii
Preface ix
1 Probability Review 1
1 Basic Concepts . . . . . . . . . . . . . 1
1.1 Events . . . . . 1
1.2 Random Variables . . . . 3
1.3 Probability. . . . . 4
2 Independence and Conditional Probability . . . . . . . . . . . . . . . 7
2.1 Independence of Events and of Random Variables . . . . 7
2.2 Bayes’sRules. . . . . . . . . . 9
2.3 Markov Property . . 11
3 Expectation . . . . 14
3.1 Cumulative Distribution Function. 14
3.2 Expectation, Mean, and Variance . . . . . . . . . . . 15
3.3 Famous Random Variables . 20
4 Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Absolutely Continuous Random Vectors . . 25
4.2 Discrete Random Vectors . . . . . . . . . 28
4.3 Product Formula for Expectation . 29
5 Transforms of Probability Distributions . 29
5.1 Generating Functions . 29
5.2 Characteristic Functions. . 34
6 Transformations of Random Vectors. . . 36
6.1 Smooth Change of Variables . . . . . . 36
6.2 Order Statistics . . . . . . . . . . . . . . . 38
7 Conditional Expectation of Discrete Variables . 39
7.1 Definition and Basic Properties . 39
7.2 Successive Conditioning . . . . . . . 41
8 The Strong Law of Large Numbers . . . . . . . . . . 42
8.1 Borel-Cantelli Lemma . 42
8.2 Almost-Sure Convergence. 43
xiii
xiv Contents
8.3 Markov’s Inequality . 45
8.4 Proof of Kolmogorov’s SLLN . 47
2 Discrete-Time Markov Models 53
1 The Transition Matrix . 53
1.1 Markov Property . 53
1.2 Distribution of an HMC . . . . . . . . . 56
2 Markov Recurrences . 58
2.1 A Canonical Representation. 58
2.2 A Few Famous Examples. 59
3 First-Step Analysis. 65
3.1 Absorption Probability . . 65
3.2 Mean Time to Absorption . . . 68
4 Topology of the Transition Matrix . 71
4.1 Communication . 71
4.2 Period . 72
5 SteadyState . . . . . . . . . 75
5.1 Stationarity. 75
5.2 Examples 76
6 Time Reversal . . 80
6.1 Reversed Chain . . 80
6.2 Time Reversibility . . 81
7 Regeneration . 83
7.1 Strong Markov Property . 83
7.2 Regenerative Cycles . 86
3 Recurrence and Ergodicity 95
1 Potential Matrix Criterion . 95
1.1 Recurrent and Transient States . . . 95
1.2 Potential Matrix . . . . . . . . . . 97
1.3 Structure of the Transition Matrix . . . . 100
2 Recurrence and Invariant Measures . 100
3 Positive Recurrence .104
3.1 Stationary Distribution Criterion . ’. 104
3.2 Examples . 105
4 Empirical Averages . 110
4.1 Ergodic Theorem . . 110
4.2 Examples . 113
4.3 Renewal Reward Theorem .117
4 Long Run Behavior 125
1 Coupling . . . . . . . . . . 125
1.1 Convergence in Variation . . . . . . . . . . . 125
1.2 The Coupling Method . . . . . . . 128
Contents
xv
2 Convergence to Steady State . . . . . . . . . . . 130
2.1 Positive Recurrent Case . . . . . . . . . . . . . . . 130
2.2 Null Recurrent Case . . . . . . . . . . . . . . .131
2.3 Thermodynamic Irreversibility . . . . . . . . . . 133
2.4 Convergence Rates via Coupling . . . . . 136
3 Discrete-Time Renewal Theory . . . . 137
3.1 Renewal Equation . . . . . . . . . 137
3.2 Renewal Theorem . . . . . . . . . . . 140
3.3 Defective Renewal Sequences . . . . . . . . . . . . . . . 142
4 Regenerative Processes . . . . . . . . . . . .-. . . . . . . . . 145
4.1 Renewal Equation of a Regenerative Process . . . . . . 145
4.2 Regenerative Theorem . . . . . . . . 146
5 Life Before Absorption . . . . . . . . . . . . . . . 149
5.1 Infinite Sojourns . . . . . . . . . . . . . . . . . 149
5.2 Time to Absorption . . . . . . . 153
6 Absorption . . . . . . . . 154
6.1 Fundamental Matrix . . . . . . . . . . . . . . . . . 154
6.2 Absorption Matrix . . . . . 156
5 Lyapunov Functions and Martingales 167
1 Lyapunov Functions . . . . . . . . . . . . 167
1.1 Foster’s Theorem . . . . . . . . . 167
1.2 Queuing Applications . . . . . . . . . . . . . . . . 173
2 Martingales and Potentials . . . . . . . . . . . .178
2.1 Harmonic Functions and Martingales . . . . . . , . 178
2.2 The Maximum Principle . . . . . . . 180
3 Applications of Martingales to HMCs . . . . . . . . . . . . . . . . . 185
3.1 The Two Pillars of Martingale Theory . . . . . . . . 185
3.2 Transience and Recurrence via Martingales . 186
3.3 Absorption via Martingales . . . . . . . . . . . 189
6 Eigenvalues and Nonhomogeneous Markov Chains 195
1 Finite Transition Matrices . . . . . . . . . . . . . . . . . . 195
1.1 Perron-Frobenius Theorem . . . . . . . . . . . . . 195
1.2 Quasi-stationary Distributions . . . . . . . . . . 199
2 Reversible Transition Matrices . . . . . . . . . . . . . . . . . 201
2.1 Eigenstructure and Diagonalization . . . . . . . . . 201
2.2 Spectral Theorem . . . . . . . . . . . . . . . . . 204
3 Convergence Bounds Without Eigenvectors . . . . . . . . . . . 207
3.1 Basic Botmds, Reversible Case . . . . . . . . . . . . . 207
3.2 Nonreversible Case . . . . . . . . . . . . 211
4 Geometric Bounds . . . . . . . . . . . . . . . . 212
4.1 Weighted Paths . . . . . . . . 212
4.2 Conductance . . . . . . . . . . . 215
XVI
Contents
5 Probabilistic Bounds . • • • • • • • • • • • • • 219
5.1 Separation and Strong Stationary Times . . .219
5.2 Convergence Rates via Strong Stationary Times . . 223
6 Fundamental Matrix of Recurrent Chains . . . 226
6.1 Definition of the Fundamental Matrix . . . . . . . . 226
6.2 Mutual Time-Distance Matrix . . 230
6.3 Variance of Ergodic Estimates . . . . 232
7 The Ergodic Coefficient . . . . . . . . 235
7.1 Dobrushin’s Inequality . 235
7.2 Interaction Coefficients and Coincidence . . . .238
8 Nonhomogeneous Markov Chains . . . . . . . . . . . 239
8.1 Ergodicity of Nonhomogeneous Markov Chains . . 239
8.2 Block Criterion of Weak Ergodicity . . . . . . 241
8.3 Sufficient Condition of Strong Ergodicity 242
7 Gibbs Fields and Monte Carlo Simulation 253
1 Markov Random Fields . . . 253
1.1 Neighborhoods and Local Specification.253
1.2 Cliques, Potential, and Gibbs Distributions. 256
2 Gibbs-Markov Equivalence . 260
2.1 From the Potential to the Local Specification . 260
2.2 From the Local Specification to the Potential. 261
3 Image Models . . . . . . . . . . 268
3.1 Textures . 268
3.2 Lines and Points . . . . . . . . . . . . 270
4 Bayesian Restoration of Images . 275
4.1 MAP Likelihood Estimation . 275
4.2 Penalty Methods. 279
5 Phase Transitions . 280
5.1 Spontaneous Magnetization . . 280
5.2 Peierls’s Argument. .281
6 Gibbs Sampler . . . . . .285
6.1 Simulation of Random Fields . 285
6.2 Convergence Rate of the Gibbs Sampler . . . . 288
7 Monte Carlo Markov Chain Simulation .290
7.1 General Principle . . . . . . . . . 290
7.2 Convergence Rates in MCMC . 295
7.3 Variance of Monte Carlo Estimators .299
8 Simulated Annealing . . 305
8.1 Stochastic Descent and Cooling .305
8.2 Convergence of Simulated Annealing .311
Contents xvii
8 Continuous-Time Markov Models 323
1 Poisson Processes . . . . . . . . . 323
1.1 Point Processes . . . . . . . . . . . . . . . . 323
1.2 Counting Process of an HPP . . . . . . .'. . . . . . 324
1.3 Competing Poisson Processes . . 327
2 Distribution of a Continuous-Time HMC . 329
2.1 Transition Semigroup . . . . . . . . . . . 329
2.2 Infinitesimal Generator . . . . . . 333
3 Kolmogorov’s Differential Systems . . 338
3.1 Finite State Space . . . . 338
3.2 General Case. . . .-. . . . . . . . . . . . . . . 340
3.3 Regular Jumps . 344
4 The Regenerative Structure . . . 345
4.1 Strong Markov Property . 345
4.2 Embedded Chain and Transition Times . . 348
4.3 Explosions. . 350
5 Recurrence. . . . . . 357
5.1 Stationary Distribution Criterion of Ergodicity. . . . . . 357
5.2 Time Reversal . . . . . . . . , —. . 361
6 Long-Run Behavior. . . . . . . 363
6.1 Ergodic Chains . . . . 363
6.2 Absorbing Chains. . . . . .364
9 Poisson Calculus and Queues 369
1 Continuous-Time Markov Chains as Poisson Systems . 369
1.1 Strong Markov Property of HPPs . . . . . . . . . . 369
1.2 From Generator to Markov Chain . 372
2 Stochastic Calculus of Poisson Processes. . 375
2.1 Counting Integrals and the Smoothing Formula . . . . . 375
2.2 Kolmogorov’s Forward System via Poisson Calculus . . . . . 378
2.3 Watanabe’s Characterization of Poisson Processes . . 380
3 Poisson Systems . . . 383
3.1 The Purely Poissonian Description. . . . . . . . . . . . . . . 383
3.2 The GSMP construction . 385
3.3 Markovian Queues as Poisson Systems . . . . . . . . . . . 388
4 Markovian Queuing Theory . 394
4.1 Isolated Markovian Queues. 394
4.2 The M/GI/1/oo/FIFO Queue . . . . . . . . . 398
4.3 The GI/M/1/oo/FIFO Queue . . . . . . . 402
4.4 Markovian Queuing Networks . . . . . . . . . . 407
xviii_
Appendix 417
1 Number Theory and Calculus . . . , . . . . . . . . . . . . . . 417
1.1 Greatest Common Divisor . . . . . . . . . . . . . . 417
1.2 Abel’s Theorem . . . . . . . . 418
1.3 Lebesgue’s Theorems for Series . . . . . .420
1.4 Infinite Products . . . . . . . . . . 422
1.5 Tychonov’s Theorem. . . . . . . . . . . . . . . . . . 423
1.6 Subadditive Functions . 423
2 Linear Algebra . . . . . . . . . . . .424
2.1 Eigenvalues and Eigenvectors . . . . . . . 424
2.2 Exponential of a Matrix . . . . . . . . . . 426
2.3 Gershgorin’s Bound . . . . . . . . . . . . . . . . . 427
3 Probability . . . .428
3.1 Expectation Revisited . . . . .428
3.2 Lebesgue’s Theorems for Expectation .430
Bibliography 433
Author Index 439
Subject Index 441 |
| any_adam_object | 1 |
| author | Brémaud, Pierre |
| author_GND | (DE-588)1060028131 |
| author_facet | Brémaud, Pierre |
| author_role | aut |
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| building | Verbundindex |
| bvnumber | BV044870865 |
| ctrlnum | (OCoLC)1031047186 (DE-599)DNB1008416398 |
| discipline | Mathematik |
| edition | 1. ed. 1999. Softcover reprint of the hardcover 1st edition 1999 |
| format | Book |
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| series | Texts in Applied Mathematics |
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| spelling | Brémaud, Pierre Verfasser (DE-588)1060028131 aut Markov Chains Gibbs Fields, Monte Carlo simulation, and queues Pierre Bremaud 1. ed. 1999. Softcover reprint of the hardcover 1st edition 1999 New York, NY Springer New York 2010 XVIII, 444 Seiten 64 schw.-w. Ill. 235 mm x 178 mm txt rdacontent n rdamedia nc rdacarrier Texts in Applied Mathematics 31 Erscheint: 2. November 2010 Markov-Kette (DE-588)4037612-6 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Markov-Kette (DE-588)4037612-6 s Stochastischer Prozess (DE-588)4057630-9 s DE-604 Texts in Applied Mathematics 31 (DE-604)BV002476038 31 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3598139&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030265310&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brémaud, Pierre Markov Chains Gibbs Fields, Monte Carlo simulation, and queues Texts in Applied Mathematics Markov-Kette (DE-588)4037612-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4037612-6 (DE-588)4057630-9 (DE-588)4123623-3 |
| title | Markov Chains Gibbs Fields, Monte Carlo simulation, and queues |
| title_auth | Markov Chains Gibbs Fields, Monte Carlo simulation, and queues |
| title_exact_search | Markov Chains Gibbs Fields, Monte Carlo simulation, and queues |
| title_full | Markov Chains Gibbs Fields, Monte Carlo simulation, and queues Pierre Bremaud |
| title_fullStr | Markov Chains Gibbs Fields, Monte Carlo simulation, and queues Pierre Bremaud |
| title_full_unstemmed | Markov Chains Gibbs Fields, Monte Carlo simulation, and queues Pierre Bremaud |
| title_short | Markov Chains |
| title_sort | markov chains gibbs fields monte carlo simulation and queues |
| title_sub | Gibbs Fields, Monte Carlo simulation, and queues |
| topic | Markov-Kette (DE-588)4037612-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Markov-Kette Stochastischer Prozess Lehrbuch |
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