Tail Asymptotics for the Maximum of Perturbed Random Walk

V. Araman and P. W. Glynn

Annals of Applied Probability, Vol. 16, 1411-1431 (2006)

Consider a random walk S = (Sn:n≥0) that is “perturbed” by a stationary sequence (ξn:n≥0) to produce the process (Snn:n≥0). This paper is concerned with computing the distribution of the all-time maximum M=max{Skk: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.