Tail asymptotics for the maximum of perturbed random walk

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

2006-10-09

arxiv.org/abs/math/0610271v1

Annals of Applied Probability 2006, Vol. 16, No. 3, 1411-1431

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000268 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000268

Consider a random walk $S=(S_n:n\geq 0)$ that is ``perturbed'' by a stationary sequence $(\xi_n:n\geq 0)$ to produce the process $(S_n+\xi_n:n\geq0)$. This paper is concerned with computing the distribution of the all-time maximum $M_{\infty}=\max \{S_k+\xi_k:k\geq0\}$ 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 $\mathbb{P}(M_{\infty}>x)$ as $x\to\infty$. The tail asymptotics depend greatly on whether the $\xi_n$'s are light-tailed or heavy-tailed. In the light-tailed setting, the tail asymptotic is closely related to the Cram\'{e}r--Lundberg asymptotic for standard random walk.

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