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Karlin & Taylor 两本关于stochastic processes的经典著作：
A First Course in Stochastic Processes
A Second Course in Stochastic Processes
book1名称：A First Course in Stochastic Processes
大小：556页
格式：djvu
目录：
Chapter 1
ELEMENTS OF STOCHASTIC PROCESSES
1. Review of Basic Terminology and Properties of Random Variables
and Distribution Functions \ . . . . . . 1
2. Two Simple Examples of Stochastic Processes . . . . .. 20
3. Classification of General Stochastic Processes . . . . . . 26
4. Denning a Stochastic Process . . . . . . . . . . 32
Elementary Problems .. . . . . .. .. 33
Problems . . . . . . . . , . . . 36
Notes . . . . .. .. . . . . .. 44
References . . . . .. . . .. .. 44
Chapter 2
MARKOV CHAINS
1. Definitions . . . . .. •. . . .. 45
2. Examples of Markov Chains . . . . . . . . . . 47
3. Transition Probability Matrices of a Markov Chain . . . . . . 58
4. Classification of States of a Markov Chain . . . . . . . . 59
5. Recurrence .. .. .. .. .. .. 62
6. Examples of Recurrent Markov Chains . . .. .. .. 67
7. More on Recurrence .. .. .. . . .. .. 72
Elementary Problems . . . . . . . . . . 73
Problems . . .. .. . . .. .. 77
Chapter 3
THE BASIC LIMIT THEOREM OF MARKOV CHAINS AND
APPLICATIONS
1. Discrete Renewal Equation . . .. . . . . . . 81
2. Proof of Theorem 1.1 .. .. .. .. .. 87
3. Absorption Probabilities . . . . . . . . . . 89
4. Criteria for Recurrence . . . . . . . . . . 94
5. A Queueing Example . . .. .. .. . . 96
6. Another Queueing Model .. .. . . . . .. 102
7. Random Walk . . .. .. . . . . .. 106
Elementary Problems . . . . . . . . . . 108
Problems . . . . . . . . . . . . 112
Notes . . .. . . .. . . . . .. 116
Reference . . .. .. . . . . .. 116
Chapter 4
CLASSICAL EXAMPLES OF CONTINUOUS TIME MARKOV
CHAINS
1. General Pure Birth Processes and Poisson Processes .. . . .. 117
2. More about Poisson Processes . . . . . . . . . . 123
3. A Counter Model .. . . .. .. .. . . 128
4. Birth and Death Processes .. .. .. .. .. 131
5. Differential Equations of Birth and Death Processes .. . . .. 135
6. Examples of Birth and Death Processes .. .. . . . . 137
7. Birth and Death Processes with Absorbing States .. .. . . 145
8. Finite State Continuous Time Markov Chains . . . . . . 150
Elementary Problems .. .. .. .. .. 152
Problems .. . . .. .. . . . . 158
Notes .. .. .. .. .. .. .. 165
References .. .. .. .. .. .. 166
Chapter 5
RENEWAL PROCESSES
1. Definition of a Renewal Process and Related Concepts . . . . 167
2. Some Examples of Renewal Processes .. .. .. .. 170
3. More on Some Special Renewal Processes . . .. . . . . 173
4. Renewal Equations and the Elementary Renewal Theorem . . . . 181
5. The Renewal Theorem .. .. .. . . . . 189
6. Applications of the Renewal Theorem . . . . . . . . 192
7. Generalizations and Variations on Renewal Processes . . . . . . 197
8. More Elaborate Applications of Renewal Theory . . . . . . 212
9. Superposition of Renewal Processes . . . . . . . . 221
Elementary Problems .. .. .. .. .. 228
Problems . . .. . . . . . . . . 230
Reference . . .. .. .. . . .. 237
Chapter 6
MARTINGALES
1. Preliminary Definitions and Examples . . . . . . . . 238
2. Supermartingales and Submartingales . . . . .. . . 248
3. The Optional Sampling Theorem . . . . . . . . 253
4. Some Applications of the Optional Sampling Theorem .. . . . . 263
5. Martingale Convergence Theorems . . . . . . . . 278
6. Applications and Extensions of the Martingale Convergence Theorems . . 287
7. Martingales with Respect to cr-Fields .. .. . . . . 297
8. Other Martingales . . . . . . . . . . . . 313
Elementary Problems . . . . . . . . . . 325
Problems . . . . . . . . . . . . 330
References .. . . . . .. .. . . 339
Chapter 7
BROWNIAN MOTION
1. Background Material . . . . . . . . . . 340
2. Joint Probabilities for Brownian Motion . . . . . . . . 343
3 Continuity of Paths and the Maximum Variables . . . . . . 345
4. Variations and Extensions .. . . .. . . .. 351
5. Computing Some Functionals of Brownian Motion by Martingale Methods . . 357
0. Multidimensional Brownian Motion . . . . . . . . 365
7. Brownian Paths .. .. .. .. .. .. 371
Elementary Problems .. .. . . .. .. 383
Problems .. .. .. .. .. .. 386
References .. .. .. . . .. .. 391
Chapter 8
BRANCHING PROCESSES
1. Discrete Time Branching Processes 392
2. Generating Function Relations for Branching Processes .. .. 394
3. Extinction Probabilities 396
4. Examples .. . . .. .. . . .. 400
5. Two-Type Branching Processes .. .. . . .. .. 404
6. Multi-Type Branching Processes .. . . . . .. 411
7. Continuous Time Branching Processes . . . . . . . . 412
8. Extinction Probabilities for Continuous Time Branching Processes . . . . 416
9. Limit Theorems for Continuous Time Branching Processes . . .. 419
10. Two-Type Continuous Time Branching Process . . .. .. 424
11. Branching Processes with General Variable Lifetime . . . . . . 431
Elementary Problems .. .. .. .. .. 436
Problems . . . . . . .. .. .. 438
Notes . . . . .. .. .. . . .. 442
Reference .. .. .. . . . . .. 442
Chapter 9
STATIONARY PROCESSES
1. Definitions and Examples . . . . .. . . .. 443
2. Mean Square Distance . . . . . . . . .. 451
3. Mean Square Error Prediction . . . . . . . . . . 461
4. Prediction of Covariance Stationary Processes . . . . .. 470
5. Ergodic Theory and Stationary Processes . . .. . . . . 474
6. Applications of Ergodic Theory . . . . . . . . 489
7. Spectral Analysis of Covariance Stationary Processes . . . . . . 502
8. Gaussian Systems .. .. .. . . . . . . 510
9. Stationary Point Processes . . . . . . .. . . 516
10. The Level-Crossing Problem .. .. . . .. .. 519
Elementary Problems . . . . . . . . . . 524
Problems . . .. .. .. . . .. 527
Notes .. .. .. .. .. .. .. 534
References . . .. .. . . . . .. 535
Appendix
REVIEW OF MATRIX ANALYSIS
1. The Spectral Theorem .. . . . . .. . . 536
2. The Frobenius Theory of Positive Matrices . . . . . . . . 542
book2名称：A Second Course in Stochastic Processes
大小：542页
格式：djvu
目录：
Chapter 10
ALGEBRAIC METHODS IN MARKOV CHAINS
1. Preliminaries 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. Tuboo Probabilities .. .. .. .. .. 31
2. Ratio Theorems .. .. .. .. „ 33
3. Existence of Gcnerulized 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 .. u 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 Continuoui Time Markov Chiiin from Its
Inrmitesimal Parameters .. ., .. .. I4S
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 Functional 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
Chapter 17
FLUCTUATION THEORY OF PARTIAL SUMS OF INDEPENDENT
IDENTICALLY DISTRIBUTED RANDOM VARIABLES
Chapter 18
QUEUEING PROCESSES
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