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目录
Chapter 10 ALGEBRAIC METHODS IN MARKOV CHAINS
1. Prelimmanes 1
2. Relations of Eigenvalues and Recurrence Classes 3
3. Periodic Classes 6
4. Special Computational Methods in Markov Chains 10
5. Examples 14
6. Applications to Coin Tossing 18
Elementary Problems 23
Problems 25
Notes 30
References 30
Chapter 11 RATIO THEOREMS OF TRANSITION PROBABILITIES AND APPLICATIONS
1. Taboo Probabdiues 31
2. Ratio Theorems 33
3. Existence of Generalized Stationary Distributions 37
4. Interpretation of Generalized Stationary Distributions 42
5. Regular, Superregular, and Subregular Sequences for Markov Chains 44
6. Stopping Rule Problems 50
Elementary Problems 65
Problems 65
Notes 70
References 71
Chapter 12 SUMS OF INDEPENDENT RANDOM VARIABLES AS A MARKOV CHAIN
1. Recurrence Properties of Sums of Independent Random Variables 72
2. Local Limit Theorems 76
3. Right Regular Sequences for the Markov Chain {Sn} 83
4. The Discrete Renewal Theorem 93
Elementary Problems 95 Problems 96
Notes 99
References 99
Chapter 13 ORDER STATISTICS, POISSON PROCESSES, AND APPLICATIONS
1. Order Statistics and Their Relation to Poisson Processes 100
2. The Ballot Problem 107
3. Empirical Distribution Functions and Order Statistics 113
4. Some Limit Distributions for Empirical Distribution Functions 119
Elementary Problems 124
Problems 125
Notes 137
References 137
Chapter 14 CONTINUOUS TIME MARKOV CHAINS
1. Differentiability Properties of Transition Probabilities 138
2. Conservative Processes and the Forward and Backward Differential Equations 143
3. Construction of a Continuous Time Markov Chain from Its Infinitesimal Parameters 145
4. Strong Markov Property 149
Problems 152
Notes 156
References 156
Chapter 15 DIFFUSION PROCESSES
1. General Description 157
2. Examples of Diffusion 169
3. Differential Equations Associated with Certain Functionals 191
4. Some Concrete Cases of the Functional Calculations 205
5. The Nature of Backward and Forward Equations and Calculation of Stationary Measures 213
6. Boundary Classification for Regular Diffusion Processes 226
7. Some Further Characterization of Boundary Behavior 242
8. Some Constructions of Boundary Behavior of Diffusion Processes 251
9. Conditioned Diffusion Processes 261
10. Some Natural Diffusion Models with Killing 272
11. Semigroup Formulation of Continuous Time Markov Processes 285
12. Further Topics in the Semigroup Theory of Markov Processes and Applications to Diffusions 305
13. The Spectral Representation of the Transition Density for a Diffusion 330
14. The Concept of Stochastic Differential Equations 340
15. Some Stochastic Differential Equation Models 358
16. A Preview of Stochastic Differential Equations and Stochastic Integrals 368
Elementary Problems 377
Problems 382
Notes 395
References 395
Chapter 16 COMPOUNDING STOCHASTIC PROCESSES
1. Multidimensional Homogeneous Poisson Processes 398
2. An Application of Multidimensional Poisson Processes to Astronomy 404
3. Immigration and Population Growth 405
4. Stochastic Models of Mutation and Growth 408
5. One-Dimensional Geometric Population Growth 413
6. Stochastic Population Growth Model in Space and Time 416
7. Deterministic Population Growth with Age Distribution 419
8. A Discrete Aging Model 425
9. Compound Poisson Processes 426
Elementary Problems 441
Problems 41
Notes 450
References 450
Chapter 17 FLUCTUATION THEORY OF PARTIAL SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES
1. The Stochastic Process of Partial Sums 451
2. An Equivalence Principle 453
3. Some Fundamental Identities of Fluctuation Theory and Direct Applications 459
4. The Important Concept of Ladder Random Variables 464
5. Proof of the Main Fluctuation Theory Identities 468
6. More Applications of Fluctuation Theory 473
Problems 484
Notes 488
References 488
Chapter 18 QUEUEING PROCESSES
1. General Description 489
2. The Simplest Queueing Processes(M/M/l) 490
3. Some General One-Server Queueing Models 492
4. Embedded Markov Chain Method Applied to the Queueing Model(M/GI/l) 497
5. Exponential Service Times(G/M/1) 504
6. Gamma Amval Dtstnbutlon and Generalizations(Ek/M/1) 506
7. Exponential Service with s Servers(GI/M/s) 511
8. The Virtual Waiting Time and the Busy Period 513
Problems 519
Notes 23
References 525
MISCELLANEOUS PROBLEMS 527
Index 539
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