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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Abstract

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

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

Dates
First available in Project Euclid: 10 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1389370104

Digital Object Identifier
doi:10.1239/jap/1389370104

Mathematical Reviews number (MathSciNet)
MR3161378

Zentralblatt MATH identifier
1303.60042

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

Keywords
Limit theorem random walk renewal theorem

Citation

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. https://projecteuclid.org/euclid.jap/1389370104


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