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First Steps in Random WalksFrom Tools to Applications$
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J. Klafter and I. M. Sokolov

Print publication date: 2011

Print ISBN-13: 9780199234868

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199234868.001.0001

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Generating functions and applications

Generating functions and applications

Chapter:
(p.17) 2 Generating functions and applications
Source:
First Steps in Random Walks
Author(s):

J. Klafter

I.M. Sokolov

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199234868.003.0002

This chapter concentrates on the properties of the walks which do not immediately follow from the distribution of the particle's position after n steps, like the first passage and return probabilities. These are most elegantly obtained via generating functions of the corresponding distributions. Again, the necessary mathematical toolbox is first introduced and discussed, including definition and main properties of generating functions and Tauberian theorems giving a nice method of approximate restoration of probability distributions from their generating functions. Then it concentrates on explicit calculations of first passage and return probabilities of the lattice walks, as well on the mean number of distinct sites visited by a walk of n steps. Some properties of off‐lattice walks are discussed as well.

Keywords:   generating functions, Tauberian theorems, first passage and return probabilities, distinct visited sites, Sparre Andersen theorem

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