This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of specialization worthwhile, because of the theory of such random walks is far more complete than that of any larger class of Markov chains. The book will present no technical difficulties to the readers with some solid experience in analysis in two or three of the following areas: probability theory, real variables and measure, analytic functions, Fourier analysis, differential and integral operators. There are almost 100 pages of examples and problems.
Author(s): F. Spitzer
Series: Graduate Texts in Mathematics
Edition: 2nd
Publisher: Springer
Year: 2001
Language: English
Pages: 427
Tags: Математика;Теория вероятностей и математическая статистика;Теория случайных процессов;
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
PREFACE TO THE SECOND EDITION......Page 6
PREFACE TO THE FIRST EDITION......Page 8
TABLE OF CONTENTS......Page 12
INTERDEPENDENCE GUIDE......Page 14
1. Introduction ......Page 16
2. Periodicity and recurrence behavior ......Page 29
3. Some measure theory ......Page 39
4. The range of a random walk ......Page 50
5. The strong ratio theorem ......Page 55
Problems ......Page 66
6. Characteristic functions and moments ......Page 69
7. Periodicity ......Page 79
8. Recurrence criteria and examples ......Page 97
9. The renewal theorem ......Page 110
Problems ......Page 116
CHAPTER III. Two-DIMENSIONAL RECURRENT RANDOM WALK ......Page 120
10. Generalities 105......Page 0
11. The hitting probabilities of a finite set ......Page 128
12. The potential kernel I(x,y) ......Page 136
13. Some potential theory ......Page 143
14. The Green function of a finite set ......Page 155
15. Simple random walk in the plane ......Page 163
16. The time dependent behavior ......Page 172
Problems ......Page 186
17. The hitting probability of the right half-line ......Page 189
18. Random walk with finite mean ......Page 205
19. The Green function and the gambler's ruin problem ......Page 220
20. Fluctuations and the arc-sine law ......Page 233
Problems ......Page 246
21. Simple random walk ......Page 252
22. The absorption problem with mean zero, finite variance ......Page 259
23. The Green function for the absorption problem ......Page 273
Problems ......Page 285
24. The Green function G(x,y) ......Page 289
25. Hitting probabilities ......Page 305
26. Random walk in three-space with mean zero and finite second moments ......Page 322
27. Applications to analysis ......Page 337
Problems ......Page 354
28. The existence of the one-dimensional potential kernel ......Page 358
29. The asymptotic behavior of the potential kernel ......Page 367
30. Hitting probabilities and the Green function ......Page 374
31. The uniqueness of the recurrent potential kernel ......Page 383
32. The hitting time of a single point ......Page 392
Problems ......Page 407
BIBLIOGRAPHY ......Page 410
SUPPLEMENTARY BIBLIOGRAPHY ......Page 416
INDEX ......Page 418
Back Cover......Page 427
