RANDOM WALKS IN CONES

• 成果类型: Article
• 署名作者: Denisov, Denis; Wachtel, Vitali
• 署名单位: University of Manchester; University of Munich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP867
• 发表日期: 2015
• 页码: 992-1044
• 关键词: CENTRAL LIMIT-THEOREMS; brownian-motion; exit times; potential-theory; weyl chambers; small steps; stay; sums

摘要

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.