Upper bounds for the maximum of a random walk with negative drift

TL;DR

This paper develops new upper bounds for the maximum of a negatively drifting random walk by truncating summands and splitting the process into finite intervals, with applications to heavy traffic regimes.

Contribution

It introduces a novel approach of splitting the process into random intervals and truncating summands to derive upper bounds for the maximum of the walk.

Findings

Derived upper bounds for the maximum of the random walk.

Validated inequalities in heavy traffic with regularly varying tails.

Reduced the problem to bounds on finite-time maxima.

Abstract

Consider a random walk $S_n=\sum_{i=0}^n X_i$ with negative drift. This paper deals with upper bounds for the maximum $M=\max_{n\ge 1}S_n$ of this random walk in different settings of power moment existences. As it is usual for deriving upper bounds, we truncate summands. Therefore we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby we can reduce the problem of finding upper bounds for $M$ to the problem of finding upper bounds for $M_\tau=\max_{n\le \tau}S_n$. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.