Markov chains: Gibbs fields, Monte Carlo simulation, and queues
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; Berlin ; Heidelberg
Springer
[1999]
|
| Schriftenreihe: | Texts in applied mathematics
31 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xviii, 444 Seiten Diagramme |
| ISBN: | 0387985093 |
Internformat
MARC
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| 264 | 4 | |c © 1999 | |
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| 490 | 1 | |a Texts in applied mathematics |v 31 | |
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| 650 | 7 | |a Monte Carlo-methode |2 gtt | |
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Datensatz im Suchindex
| _version_ | 1804127243285299200 |
|---|---|
| adam_text | 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 s Rules
........................... 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
Borei-Cantelli
Lemma
....................... 42
8.2
Almost-Sure Convergence
..................... 43
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
Steady State
................................ 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
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 Bounds, Reversible Case
.................. 207
3.2 Nonreversible
Case
........................ 211
4
Geometric Bounds
............................. 212
4.1
WeightedPaths
.......................... 212
4.2
Conductance
............................ 215
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
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
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
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/l/oo/FIFO Queue
.................... 398
4.3
The GI/M/l/oo/FIFO Queue
.................... 402
4.4
Markovian Queuing Networks
................... 407
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 |
| author_sort | Brémaud, Pierre |
| author_variant | p b pb |
| building | Verbundindex |
| bvnumber | BV012593714 |
| callnumber-first | Q - Science |
| callnumber-label | QA274 |
| callnumber-raw | QA274.7 |
| callnumber-search | QA274.7 |
| callnumber-sort | QA 3274.7 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 |
| classification_tum | MAT 629 MAT 607 |
| ctrlnum | (OCoLC)38853973 (DE-599)BVBBV012593714 |
| dewey-full | 519.2/33 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/33 |
| dewey-search | 519.2/33 |
| dewey-sort | 3519.2 233 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV012593714 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:30:15Z |
| institution | BVB |
| isbn | 0387985093 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008552064 |
| oclc_num | 38853973 |
| open_access_boolean | |
| owner | DE-824 DE-739 DE-384 DE-703 DE-92 DE-29T DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-473 DE-BY-UBG DE-706 DE-634 DE-11 DE-188 DE-578 DE-20 DE-83 |
| owner_facet | DE-824 DE-739 DE-384 DE-703 DE-92 DE-29T DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-473 DE-BY-UBG DE-706 DE-634 DE-11 DE-188 DE-578 DE-20 DE-83 |
| physical | xviii, 444 Seiten Diagramme |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Brémaud, Pierre (DE-588)1060028131 aut Markov chains Gibbs fields, Monte Carlo simulation, and queues Pierre Brémaud New York ; Berlin ; Heidelberg Springer [1999] © 1999 xviii, 444 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 31 Markov, Processus de Markov-processen gtt Monte Carlo-methode gtt Processos markovianos larpcal Stochastische processen gtt Wachttijdproblemen gtt Markov processes Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Markov-Kette (DE-588)4037612-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Markov-Kette (DE-588)4037612-6 s Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 DE-604 Texts in applied mathematics 31 (DE-604)BV002476038 31 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008552064&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Brémaud, Pierre Markov chains Gibbs fields, Monte Carlo simulation, and queues Texts in applied mathematics Markov, Processus de Markov-processen gtt Monte Carlo-methode gtt Processos markovianos larpcal Stochastische processen gtt Wachttijdproblemen gtt Markov processes Stochastischer Prozess (DE-588)4057630-9 gnd Markov-Kette (DE-588)4037612-6 gnd |
| subject_GND | (DE-588)4057630-9 (DE-588)4037612-6 (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 Brémaud |
| title_fullStr | Markov chains Gibbs fields, Monte Carlo simulation, and queues Pierre Brémaud |
| title_full_unstemmed | Markov chains Gibbs fields, Monte Carlo simulation, and queues Pierre Brémaud |
| 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, Processus de Markov-processen gtt Monte Carlo-methode gtt Processos markovianos larpcal Stochastische processen gtt Wachttijdproblemen gtt Markov processes Stochastischer Prozess (DE-588)4057630-9 gnd Markov-Kette (DE-588)4037612-6 gnd |
| topic_facet | Markov, Processus de Markov-processen Monte Carlo-methode Processos markovianos Stochastische processen Wachttijdproblemen Markov processes Stochastischer Prozess Markov-Kette Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008552064&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT bremaudpierre markovchainsgibbsfieldsmontecarlosimulationandqueues |