LIMIT THEOREMS FOR MARKOV WALKS CONDITIONED TO STAY POSITIVE UNDER A SPECTRAL GAP ASSUMPTION

• 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: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1197
• 发表日期: 2018
• 页码: 1807-1877
• 关键词: ordered random-walks; affine random-walks; asymptotic-behavior; potential-theory; 1st-passage times; conical domains; stable laws; distributions; CONVERGENCE; chains

摘要

Consider a Markov chain (X-n)(n >= 0) with values in the state space X. Let f be a real function on X and set S-n = Sigma(n)(i=1) f(X-i), n >= 1. Let P-x be the probability measure generated by the Markov chain starting at X-0 = x. For a starting point y is an element of R, denote by tau(y) the first moment when the Markov walk (y + S-n)(n >= 1) becomes nonpositive. Under the condition that S-n has zero drift, we find the asymptotics of the probability P-x (tau(y) > n) and of the conditional law P-x(y + S-n <=.root n vertical bar tau(y) > n) as n -> +infinity.