On the exact distributional asymptotics for the supremum of a random walk with increments in a class of light-tailed distributions

Research paper by Stan Zachary, Serguei Foss

Indexed on: 29 Oct '06Published on: 29 Oct '06Published in: Mathematics - Probability

Abstract

We study the distribution of the maximum $M$ of a random walk whose increments have a distribution with negative mean and belonging, for some $\gamma>0$, to a subclass of the class $\mathcal{S}_\gamma$--see, for example, Chover, Ney, and Wainger (1973). For this subclass we give a probabilistic derivation of the asymptotic tail distribution of $M$, and show that extreme values of $M$ are in general attained through some single large increment in the random walk near the beginning of its trajectory. We also give some results concerning the spatially local'' asymptotics of the distribution of $M$, the maximum of the stopped random walk for various stopping times, and various bounds.