Journal of Applied Probability

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

Johannes Kugler and Vitali Wachtel

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Consider a random walk Sn = ∑i=0nXi with negative drift. This paper deals with upper bounds for the maximum M = maxn≥1Sn 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 Mτ = maxn≤τSn. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

Article information

J. Appl. Probab., Volume 50, Number 4 (2013), 1131-1146.

First available in Project Euclid: 10 January 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G52: Stable processes

Limit theorem random walk renewal theorem


Kugler, Johannes; Wachtel, Vitali. Upper bounds for the maximum of a random walk with negative drift. J. Appl. Probab. 50 (2013), no. 4, 1131--1146. doi:10.1239/jap/1389370104.

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