Tail asymptotics for the maximum of perturbed random walk

Abstract

Consider a random walk $S=(S_n:n\ge 0)$ that is ``perturbed'' by a stationary sequence $({\xi }_n:n\ge 0)$ to produce the process $(S_n+{\xi }_n:n\ge 0)$. This paper is concerned with computing the distribution of the all-time maximum $M_{\infty }=\max \{S_k+{\xi }_k:k\ge 0\}$ 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 $\mathbb{P}(M_{\infty }>x)$ 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 Cramér--Lundberg asymptotic for standard random walk.

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