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

The book explores the potential functions of one-dimensional recurrent random walks on the lattice of integers with infinite variance step distribution, providing precise asymptotic results on hitting probabilities, overshoot distributions, Green functions, and absorption probabilities. It is intended for advanced graduate students and researchers in probability theory and stochastic systems.

Format: Paperback / softback
Length: 276 pages
Publication date: 29 September 2023
Publisher: Springer International Publishing AG

This book delves into the potential functions of one-dimensional recurrent random walks on the lattice of integers, characterized by step distributions with infinite variance. The primary objective is to obtain precise estimates of these potential functions, which are then applied to various scenarios, yielding precise asymptotic results. These results encompass 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 holds significant importance in fluctuation theory, particularly when the variance of the step distribution is finite. In this case, the potential function exhibits a simple asymptotic form, facilitating a unified description of recurrent random walks with explicit formulae. However, when the variance is infinite, the potential function exhibits diverse behavior, reflecting the characteristics of the step distribution. In the case of a strictly stable step distribution, significant progress has been made, with many authors establishing remarkable results. However, these results often do not involve the potential function, leaving important questions unanswered. In situations where the random walk is relatively stable or one tail of the step distribution is negligible compared to the other, there has been comparatively less research. This book highlights the crucial role of the potential function in resolving these unsettled problems, which have received limited attention in recent decades.

Dimension: 235 x 155 (mm)
ISBN-13: 9783031410192
Edition number: 1st ed. 2023