Potential Functions of Random Walks in ℤ with Infinite Variance: Estimates and Applications

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Kôhei Uchiyama

Potential Functions of Random Walks in ℤ with Infinite Variance

Estimates and Applications

Kôhei Uchiyama

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English

Springer International Publishing AG
29 September 2023

Probability & statistics; Stochastics

Series: Lecture Notes in Mathematics

This book studies the potential functions of one-dimensional recurrent random walks on the lattice of integers with step distribution of infinite variance. The central focus is on obtaining reasonably nice estimates of the potential function. These estimates are then applied to various situations, yielding precise asymptotic results on, among other things, hitting probabilities of finite sets, overshoot distributions, Green functions on long finite intervals and the half-line, and absorption probabilities of two-sided exit problems.

The potential function of a random walk is a central object in fluctuation theory. If the variance of the step distribution is finite, the potential function has a simple asymptotic form, which enables the theory of recurrent random walks to be described in a unified way with rather explicit formulae. On the other hand, if the variance is infinite, the potential function behaves in a wide range of ways depending on the step distribution, which the asymptotic behaviour of many functionals of the random walk closely reflects.

In the case when the step distribution is attracted to a strictly stable law, aspects of the random walk have been intensively studied and remarkable results have been established by many authors. However, these results generally do not involve the potential function, and important questions still need to be answered. In the case where the random walk is relatively stable, or if one tail of the step distribution is negligible in comparison to the other on average, there has been much less work. Some of these unsettled problems have scarcely been addressed in the last half-century. As revealed in this treatise, the potential function often turns out to play a significant role in their resolution.

Aimed at advanced graduate students specialising in probability theory, this book will also be of interest to researchers and engineers working with random walks and stochastic systems.

By: Kôhei Uchiyama
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 1st ed. 2023
Volume: 2338
Dimensions: Height: 235mm, Width: 155mm,
Weight: 444g
ISBN: 9783031410192
ISBN 10: 303141019X
Series: Lecture Notes in Mathematics
Pages: 276
Publication Date: 29 September 2023
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Kôhei Uchiyama is a professor emeritus of mathematics at the Tokyo Institute of Technology, where he obtained his doctorate in science in 1978. After working at Ryukyu University, Nara Women's University, and Hiroshima University, he moved to the Tokyo Institute of Technology in 1998. While working on various problems concerning statistical mechanics, he has continued interest in Brownian motion and random walks throughout his academic career.