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Upper Bounds for the Maximum of a Random Walk with Negative Drift

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Johannes Kugler  and
Vitali Wachtel
Johannes Kugler*
Affiliation:
University of Munich
Vitali Wachtel*
Affiliation:
University of Munich
*
∗ Postal address: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany.
∗ Postal address: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany.
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Abstract

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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.

Keywords

Limit theoremrandom walkrenewal theorem

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G52: Stable processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1131 - 1146
Copyright
© Applied Probability Trust 

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