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 (Yn)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. Yn, n≥1. We consider the Markov chain (Zn)n≥0 with state space M which is defined recursively by Z0=Z and Zn+1=Yn+1Zn 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(Yn,Zn−1))n≥1. The main hypothesis is a condition of contraction in the mean for the action on M of the mappings Yn; 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.
"Central limit theorems for iterated random Lipschitz mappings." Ann. Probab. 32 (3) 1934 - 1984, July 2004. https://doi.org/10.1214/009117904000000469