The Annals of Applied Probability

Tail asymptotics for the maximum of perturbed random walk

Victor F. Araman and Peter W. Glynn

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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.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 3 (2006), 1411-1431.

Dates
First available in Project Euclid: 2 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1159804986

Digital Object Identifier
doi:10.1214/105051606000000268

Mathematical Reviews number (MathSciNet)
MR2260068

Zentralblatt MATH identifier
1118.60073

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F17: Functional limit theorems; invariance principles 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90F35

Keywords
Perturbed random walk Cramér–Lundberg approximation tail asymptotics coupling heavy tails

Citation

Araman, Victor F.; Glynn, Peter W. Tail asymptotics for the maximum of perturbed random walk. Ann. Appl. Probab. 16 (2006), no. 3, 1411--1431. doi:10.1214/105051606000000268. https://projecteuclid.org/euclid.aoap/1159804986


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References

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The Annals of Applied Probability

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