Tail asymptotics for the maximum of perturbed random walk

Abstract

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 all-time 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 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.