### Tail asymptotics for the maximum of perturbed random walk

Victor F. Araman and Peter W. Glynn
Source: Ann. Appl. Probab. Volume 16, Number 3 (2006), 1411-1431.

#### Abstract

Consider a random walk S=(Sn:n≥0) that is “perturbed” by a stationary sequence (ξn:n≥0) to produce the process (Sn+ξn:n≥0). This paper is concerned with computing the distribution of the all-time maximum M=max {Sk+ξk:k≥0} of perturbed random walk with a negative drift. Such a maximum arises in several different applications settings, including production systems, communications networks and insurance risk. Our main results describe asymptotics for ℙ(M>x) as x→∞. The tail asymptotics depend greatly on whether the ξn’s are light-tailed or heavy-tailed. In the light-tailed setting, the tail asymptotic is closely related to the Cramér–Lundberg asymptotic for standard random walk.

First Page:
Primary Subjects: 60K25, 60F17, 68M20, 90F35
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1159804986
Digital Object Identifier: doi:10.1214/105051606000000268
Mathematical Reviews number (MathSciNet): MR2260068
Zentralblatt MATH identifier: 1118.60073

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