SELF-SIMILAR SCALING LIMITS OF MARKOV CHAINS ON THE POSITIVE INTEGERS

• 成果类型: Article
• 署名作者: Bertoin, Jean; Kortchemski, Igor
• 署名单位: University of Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1157
• 发表日期: 2016
• 页码: 2556-2595
• 关键词: recurrent extensions; excursions

摘要

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If X-n is such a Markov chain started at n, we establish a limit theorem for 1/n X-n appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by X-n to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which X-n exhibits different behavior. The present results extend those of Haas and Miermont [Bernoulli 17 (2011) 1217-1247] who focused on the case of nonincreasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive and exchangeable coalescence-fragmentation processes.