Central limit theorems for iterated random Lipschitz mappings

• 成果类型: Article
• 署名作者: Hennion, H; Hervé, L
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Rennes; Universite de Rennes
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000467
• 发表日期: 2004
• 页码: 1934-1984
• 关键词: independent random matrices; characteristic exponent; PRODUCTS

摘要

Let M be a noncompact metric space in which every closed ball is compact, and let G be a semigroup of Lipschitz mappings of M. Denote by (Y-n)(ngreater than or equal to1) a sequence of independent G-valued, identically distributed random variables (r.v.'s), and by Z an M-valued r.v. which is independent of the r.v. Y-n, n greater than or equal to 1. We consider the Markov chain (Z(n))(ngreater than or equal to0) with state space M which is defined recursively by Z(0) = Z and Z(n+1) = Y(n+1)Z(n) for n > 0. Let be a real-valued function on G x M. The aim of this paper is to prove central limit theorems for the sequence of r.v.'s (xi(Y-n, Z(n-1)))(ngreater than or equal to1). The main hypothesis is a condition of contraction in the mean for the action on M of the mappings Y-n; we use a spectral method based on a quasi-compactness property of the transition probability of the chain mentioned above, and on a special perturbation theorem.