Curve crossing for random walks reflected at their maximum

• 成果类型: Article
• 署名作者: Doney, Ron; Maller, Ross
• 署名单位: University of Manchester; Australian National University; Australian National University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000953
• 发表日期: 2007
• 页码: 1351-1373
• 关键词: power-law boundaries; Levy processes; passage times; large numbers; moments; THEOREMS; QUEUE

摘要

Let R-n = max(0 <= j <= n) S-j - S-n be a random walk S-n reflected in its maximum. Except in the trivial case when P(X >= 0) = 1, R-n will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of R-n above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of S-n is necessary for passage of R-n above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of R-n above linear and square root boundaries.