Conditioned local limit theorems for random walks defined on finite Markov chains

• 成果类型: Article
• 署名作者: Grama, Ion; Lauvergnat, Ronan; Le Page, Emile
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00948-8
• 发表日期: 2020
• 页码: 669-735
• 关键词: ASYMPTOTIC-BEHAVIOR; 1st-passage times; random-variables; potential-theory; distributions; PRODUCTS

摘要

Let (Xn) n0 be a Markov chain with values in a finite state space X starting at X0 = x. X and let f be a real function defined on X. Set Sn = nk =1 f ( Xk), n 1. For any y. R denote by ty the first time when y + Sn becomes non-positive. We study the asymptotic behaviour of the probability Px y + Sn. [z, z + a], ty > n as n.+8. We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order n3/2 and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability Px ty = n as n ->+infinity.