Tail asymptotics for the maximum of perturbed random walk

• 成果类型: Article
• 署名作者: Araman, Victor F.; Glynn, Peter W.
• 署名单位: New York University; Stanford University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051606000000268
• 发表日期: 2006
• 页码: 1411-1431
• 关键词: ruin probabilities; inventory model; renewal theory; approximations

摘要

Consider a random walk S=(S-n:n >= 0) that is perturbed by a stationary sequence (xi(n): n >= 0) to produce the process (S-n+xi(n):n >= 0). This paper is concerned with computing the distribution of the all-time maximum M-infinity= max{S-k+xi(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-infinity > x) as x ->infinity. 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 Cramer-Lundberg asymptotic for standard random walk.