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

Abstract

Consider a random walk S_n = ∑_i=0ⁿX_i with negative drift. This paper deals with upper bounds for the maximum M = max_n≥1S_n of this random walk in different settings of power moment existences. As 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_τ = max_n≤τS_n. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.