Central limit theorems for iterated random Lipschitz mappings

Abstract

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)_n≥1 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≥1. We consider the Markov chain (Z_n)_n≥0 with state space M which is defined recursively by Z₀=Z and Z_n+1=Y_n+1Z_n for n≥0. Let ξ be a real-valued function on G×M. The aim of this paper is to prove central limit theorems for the sequence of r.v.’s (ξ(Y_n,Z_n−1))_n≥1. 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.