ESAIM: Probability and Statistics

Table of Contents - May 2003 - Volume 7  

Research Article

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

Conze, Jean-Pierrea1 and Raugi, Alberta1

a1 IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France; conze@univ-rennes1.fr.

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.

(Received January 28 2002)

(Online publication May 15 2003)

Key Words:

  • Transfer operator;
  • convergence of iterates;
  • Markov chains;
  • rate in the TCL for dynamical systems;
  • Borel-Cantelli property;
  • non uniformly hyperbolic map.

Mathematics Subject Classification:

  • 60J10;
  • 37A05;
  • 37A25
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