The maximum on a random time interval of a random walk with long-tailed increments and negative drift

TL;DR

This paper investigates the asymptotic behavior of the maximum of a long-tailed, negatively drifting random walk over a random time interval, extending existing results to general stopping times and providing simplified proofs.

Contribution

It generalizes Asmussen's 1998 result to arbitrary stopping times, simplifies the proof, and offers new converses for the maximum of such random walks.

Findings

Extended asymptotic results to general stopping times

Simplified the proof of existing theorems

Provided new converses for the maximum behavior

Abstract

We study the asymptotics for the maximum on a random time interval of a random walk with a long-tailed distribution of its increments and negative drift. We extend to a general stopping time a result by Asmussen (1998), simplify its proof, and give some converses.