Tail Asymptotics for the Maximum of Perturbed Random WalkV. 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 (Sn+ξn:n≥0). This paper is concerned with computing the distribution of the all-time maximum M∞=max{Sk+ξk: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.
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