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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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Characteristic functions

Characteristic functions

(p.1) 1 Characteristic functions
First Steps in Random Walks

J. Klafter

I.M. Sokolov

Oxford University Press

Since simple random walk is a process with independent increments, its properties are represented in the most simple way by using the techniques based on characteristic functions. This chapter introduces the necessary mathematical instruments, and then use them to discuss general expressions for the distribution of the walker's displacement after a given number of steps in one dimension and in higher dimensions. It moreover discusses moments of displacement, provided these moments exist. The chapter then considers a simple approach to the central limit theorem, and discusses situations, when this breaks down (corresponding to the cases when the second moment of step lengths diverges).

Keywords:   lattice random walks, characteristic functions, off‐lattice walks, moments, independent increments, central limit theorem, long jumps

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