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Medientyp:
E-Artikel
Titel:
Central Limit Theorems for Iterated Random Lipschitz Mappings
Beteiligte:
Hennion, Hubert;
Hervé, Loïc
Erschienen:
Institute of Mathematical Statistics, 2004
Erschienen in:The Annals of Probability
Sprache:
Englisch
ISSN:
0091-1798
Entstehung:
Anmerkungen:
Beschreibung:
<p> 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<sub>n</sub>)<sub>n≥ 1</sub>a sequence of independent G-valued, identically distributed random variables (r.v.'s), and by Z and M-valued r.v. which is independent of the r.v. Y<sub>n</sub>, n ≥ 1. We consider the Markov chain (Z<sub>n</sub>)<sub>n≥ 0</sub>with state space M which is defined recursively by Z<sub>0</sub>=Z and Z<sub>n+1</sub>=Y<sub>n+1</sub>Z<sub>n</sub>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<sub>n</sub>,Z<sub>n-1</sub>))<sub>n≥ 1</sub>. The main hypothesis is a condition of contraction in the mean for the action on M of the mappings Y<sub>n</sub>; 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. </p>