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July 2004 Central limit theorems for iterated random Lipschitz mappings
Hubert Hennion, Loïc Hervé
Ann. Probab. 32(3): 1934-1984 (July 2004). DOI: 10.1214/009117904000000469


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)n1 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, n1. We consider the Markov chain (Zn)n0 with state space M which is defined recursively by Z0=Z and Zn+1=Yn+1Zn for n0. 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,Zn1))n1. 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.


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Hubert Hennion. Loïc Hervé. "Central limit theorems for iterated random Lipschitz mappings." Ann. Probab. 32 (3) 1934 - 1984, July 2004.


Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1062.60017
MathSciNet: MR2073182
Digital Object Identifier: 10.1214/009117904000000469

Primary: 60F05 , 60J05

Keywords: central limit theorems , iterated function systems , Markov chains , Spectral method

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • July 2004
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