Principles of Random Walk
Published on 1. March 2001
Book
Paperback/Softback
XIV, 408 pages
978-0-387-95154-6 (ISBN)
Description
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. I considered this high degree of specialization worth while, because the theory of such random walks is far more complete than that of any larger class of Markov chains. Random walk occupies such a privileged position primarily because of a delicate interplay between methods from harmonic analysis on one hand, and from potential theory on the other. The relevance of harmonic analysis to random walk of course stems from the invariance of the transition probabilities under translation in the additive group which forms the state space. It is precisely for this reason that, until recently, the subject was dominated by the analysis of characteristic functions (Fourier transforms of the transition probabilities). But if harmonic analysis were the central theme of this book, then the restriction to random walk on the integers (rather than on the reals, or on o'ther Abelian groups) would be quite unforgivable. Indeed it was the need for a self contained elementary exposition of the connection of harmonic analysis with the much more recent developments in potential theory that dictated the simplest possible setting.
More details
Series
Edition
Softcover reprint of the original 1st ed. 1964
Language
English
Place of publication
NY
United States
Target group
College/higher education
Professional and scholarly
Research
Edition type
Revised edition
Product notice
Paperback (trade)
Illustrations
1, black & white illustrations
Dimensions
Height: 22.9 cm
Width: 15.2 cm
Thickness: 22 mm
Weight
1320 gr
ISBN-13
978-0-387-95154-6 (9780387951546)
DOI
10.1007/978-1-4684-6257-9
Schweitzer Classification
Content
I. The Classification of Random Walk.- 1. Introduction.- 2. Periodicity and recurrence behavior.- 3. Some measure theory.- 4. The range of a random walk.- 5. The strong ratio theorem.- Problems.- II. Harmonic Analysis.- 6. Characteristic functions and moments.- 7. Periodicity.- 8. Recurrence criteria and examples.- 9. The renewal theorem.- Problems.- III. Two-Dimensional Recurrent Random Walk.- 10. Generalities.- 11. The hitting probabilities of a finite set.- 12. The potential kernel A(x,y).- 13. Some potential theory.- 14. The Green function of a finite set.- 15. Simple random walk in the plane.- 16. The time dependent behavior.- Problems.- IV. Random Walk on a Half-Line.- 17. The hitting probability of the right half-line.- 18. Random walk with finite mean.- 19. The Green function and the gambler's ruin problem.- 20. Fluctuations and the arc-sine law.- Problems.- V. Random Walk on a Interval.- 21. Simple random walk.- 22. The absorption problem with mean zero, finite variance.- 23. The Green function for the absorption problem.- Problems.- VI. Transient Random Walk.- 24. The Green function G(x,y).- 25. Hitting probabilities.- 26. Random walk in three-space with mean zero and finite second moments.- 27. Applications to analysis.- Problems.- VII. Recurrent Random Walk.- 28. The existence of the one-dimensional potential kernel.- 29. The asymptotic behavior of the potential kernel.- 30. Hitting probabilities and the Green function.- 31. The uniqueness of the recurrent potential kernel.- 32. The hitting time of a single point.- Problems.