Tail Asymptotics for the Maximum of Perturbed Random WalkV. Araman and P. W. Glynn Annals of Applied Probability, Vol. 16, 14111431 (2006) Consider a random walk S = (S_{n}:n≥0) that is “perturbed” by a stationary sequence (ξ_{n}:n≥0) to produce the process (S_{n}+ξ_{n}:n≥0). This paper is concerned with computing the distribution of the alltime maximum M_{∞}=max{S_{k}+ξ_{k}: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 ξ_{n}’s are lighttailed or heavytailed. In the lighttailed setting, the tail asymptotic is closely related to the Cramér– Lundberg asymptotic for standard random walk.
