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Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

Published online by Cambridge University Press:  15 May 2003

Jean-Pierre Conze  and
Albert Raugi
Jean-Pierre Conze
Affiliation:
IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France; conze@univ-rennes1.fr.
Albert Raugi
Affiliation:
IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France; conze@univ-rennes1.fr.
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Abstract

We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet" condition and apply it to a class of transition operators. This gives the convergence of the series ∑k≥0krPkƒ, $r \in \mathbb{N}$, under some regularity assumptions and implies the central limit theorem with a rate in $n^{- \frac{1}{2} }$ for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

Keywords

Transfer operatorconvergence of iteratesMarkov chainsrate in the TCL for dynamical systemsBorel-Cantelli propertynon uniformly hyperbolic map.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 115 - 146
Copyright
© EDP Sciences, SMAI, 2003

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