On a maximum of nearest-neighbor random walk with asymptotically zero drift on lattice of positive half line

TL;DR

This paper analyzes the maximum of a nearest-neighbor random walk with asymptotically zero drift on the positive half line, revealing unique asymptotic behavior distinct from simple random walks.

Contribution

It characterizes the distribution and asymptotics of the maximum of excursions for a specific class of random walks with asymptotically zero drift.

Findings

Distribution of maximum M is characterized.

Asymptotic behavior differs from simple random walks.

Provides new insights into random walks with zero drift.

Abstract

Consider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from $1$ and ending at $0.$ We study the distribution of $M$ and characterize its asymptotics, which is quite different from those of simple random walks.