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BK

New York : Springer, c1999

xviii,444 s.

ISBN 0-387-98509-3 (váz.)

Texts in applied mathematics ; 31

Obsahuje předmluvy, autorský a věcný rejstřík

Bibliografie: s. 433-437

Markovovy procesy - studie

Metoda Monte Carlo - studie

000059991

I Probability Review // 1 Basic Concepts // 1.1 Events // 1.2 Random Variables // 1.3 Probability’ // 2 Independence and Conditional Probability // 2.1 Independence of Events and of Random Variables // 2.2 Bayes’s Rules // 2.3 Markov Property // 3 Expectation // 3.1 Cumulative Distribution Function // 3.2 Expectation, Mean, and Variance 15 // 3.3 Famous Random Variables // 4 Random Vectors // 4.1 Absolutely Continuous Random Vectors // 4.2 Discrete Random Vectors // 4.3 Product Formula for Expectation // 5 Transforms of Probability Distributions // 5.1 Generating Functions // 5.2 Characteristic Functions // 6 Transformations of Random Vectors // 6.1 Smooth Change of Variables // 6.2 Order Statistics // 7 Conditional Expectation of Discrete Variables // 7.1 Definition and Basic Properties 39 // 7.2 Successive Conditioning 41 // 8 The Strong Law of Large Numbers 42 // 8.1 Borcl-Cantclli 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 Weighted Paths 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 // 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 // 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 // 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