A First Course in Stochastic Processes

✍ Scribed by Samuel Karlin, Howard M. Taylor

• Publisher: Academic Press
• Year: 1975
• Tongue: English
• Leaves: 574
• Edition: 2°
• Category: Library

✦ Synopsis

The purpose, level, and style of this new edition conform to the tenets set forth in the original preface. The authors continue with their tack of developing simultaneously theory and applications, intertwined so that they refurbish and elucidate each other.The authors have made three main kinds of changes. First, they have enlarged on the topics treated in the first edition. Second, they have added many exercises and problems at the end of each chapter. Third, and most important, they have supplied, in new chapters, broad introductory discussions of several classes of stochastic processes not dealt with in the first edition, notably martingales, renewal and fluctuation phenomena associated with random sums, stationary stochastic processes, and diffusion theory.

✦ Table of Contents

Cover......Page 1
Title......Page 2
Copyright Page......Page 3
Contents......Page 4
Preface......Page 10
Preface to First Edition......Page 14
1. Review of Basic Terminology and Properties of Random Variables and Distribution Functions......Page 18
2. Two Simple Examples of Stochastic Processes......Page 37
3. Classification of General Stochastic Processes......Page 43
4. Defining a Stochastic Process......Page 49
Elementary Problems......Page 50
Problems......Page 53
References......Page 61
1. Definitions......Page 62
2. Examples of Markov Chains......Page 64
3. Transition Probability Matrices of a Markov Chain......Page 75
4. Classification of States of a Markov Chain......Page 76
5. Recurrence......Page 79
6. Examples of Recurrent Markov Chains......Page 84
7. More on Recurrence......Page 89
Elementary Problems......Page 90
Problems......Page 94
Notes......Page 96
References......Page 97
1. Discrete Renewal Equation......Page 98
2. Proof of Theorem 1.1......Page 104
3. Absorption Probabilities......Page 106
4. Criteria for Recurrence......Page 111
5. A Queueing Example......Page 113
6. Another Queueing Model......Page 119
7. Random Walk......Page 123
Elementary Problems......Page 125
Problems......Page 129
Reference......Page 133
1. General Pure Birth Processes and Poisson Processes......Page 134
2. More about Poisson Processes......Page 140
3. A Counter Model......Page 145
4. Birth and Death Processes......Page 148
5. Differential Equations of Birth and Death Processes......Page 152
6. Examples of Birth and Death Processes......Page 154
7. Birth and Death Processes with Absorbing States......Page 162
8. Finite State Continuous Time Markov Chains......Page 167
Elementary Problems......Page 169
Problems......Page 175
Notes......Page 182
References......Page 183
1. Definition of a Renewal Process and Related Concepts......Page 184
2. Some Examples of Renewal Processes......Page 187
3. More on Some Special Renewal Processes......Page 190
4. Renewal Equations and the Elementary Renewal Theorem......Page 198
5. The Renewal Theorem......Page 206
6. Applications of the Renewal Theorem......Page 209
7. Generalizations and Variations on Renewal Processes......Page 214
8. More Elaborate Applications of Renewal Theory......Page 229
9. Superposition of Renewal Processes......Page 238
Elementary Problems......Page 245
Problems......Page 247
Reference......Page 254
1. Preliminary Definitions and Examples......Page 255
2. Supermartingales and Submartingales......Page 265
3. The Optional Sampling Theorem......Page 270
4. Some Applications of the Optional Sampling Theorem......Page 280
5. Martingale Convergence Theorems......Page 295
6. Applications and Extensions of the Martingale Convergence Theorems......Page 304
7. Martingales with Respect to igma-Fields......Page 314
8. Other Martingales......Page 330
Elementary Problems......Page 342
Problems......Page 347
References......Page 356
1. Background Material......Page 357
2. Joint Probabilities for Brownian Motion......Page 360
3. Continuity of Paths and the Maximum Variables......Page 362
4. Variations and Extensions......Page 368
5. Conlputing Some Functionals of Brownian Motion by Martingale Methods......Page 374
6. Multidirncnsional Brownian Motion......Page 382
7. Brownian Paths......Page 388
Elementary Problems......Page 400
Problems......Page 403
References......Page 408
1. Discrete Time Branching Processes......Page 409
2. Generating Function Relations for Branching Processes......Page 411
3. Extinction Probabilities......Page 413
4. Examples......Page 417
5. Two-Type Branching Processes......Page 421
6. Multi-Type Branching Processes......Page 428
7. Continuous Time Branching Processes......Page 429
8. Extinction Probabilities for Continuous Time Branching Processes......Page 433
9. Limit Theorems for Continuous Time Branching Processes......Page 436
10. Two-Type Continuous Time Branching Process......Page 441
11. Branching Processes with General Variable Lifetime......Page 448
Elementary Problems......Page 453
Problems......Page 455
Reference......Page 459
1. Definitions and Examples......Page 460
2. Mean Square Distance......Page 468
3. Mean Square Error Prediction......Page 478
4. Prediction of Covariance Stationary Processes......Page 487
5. Ergodic Theory and Stationary Processes......Page 491
6. Applications of Ergodic Theory......Page 506
7. Spectral Analysis of Covariance Stationary Processes......Page 519
8. Gaussian Systems......Page 527
9. Stationary Point Processes......Page 533
10. The Level-Crossing Problem......Page 536
Elementary Problems......Page 541
Problems......Page 544
Notes......Page 551
References......Page 552
1. The Spectral Theorem......Page 553
2. The Frobenius Theory of Positive Matrices......Page 559
Index......Page 570

✦ Subjects

Математика;Теория вероятностей и математическая статистика;Теория случайных процессов;