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

Jean-Pierre Conze; Albert Raugi

ESAIM: Probability and Statistics (2010)

  • Volume: 7, page 115-146
  • ISSN: 1292-8100

Abstract

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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 , under some regularity assumptions and implies the central limit theorem with a rate in n - 1 2 for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

How to cite

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Conze, Jean-Pierre, and Raugi, Albert. "Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains." ESAIM: Probability and Statistics 7 (2010): 115-146. <http://eudml.org/doc/104299>.

@article{Conze2010,
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. },
author = {Conze, Jean-Pierre, Raugi, Albert},
journal = {ESAIM: Probability and Statistics},
keywords = {Transfer operator; convergence of iterates; Markov chains; rate in the TCL for dynamical systems; Borel-Cantelli property; non uniformly hyperbolic map.; transfer operator; rate in the TCL for dynamical systems; non uniformly hyperbolic map},
language = {eng},
month = {3},
pages = {115-146},
publisher = {EDP Sciences},
title = {Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains},
url = {http://eudml.org/doc/104299},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Conze, Jean-Pierre
AU - Raugi, Albert
TI - Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 115
EP - 146
AB - 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.
LA - eng
KW - Transfer operator; convergence of iterates; Markov chains; rate in the TCL for dynamical systems; Borel-Cantelli property; non uniformly hyperbolic map.; transfer operator; rate in the TCL for dynamical systems; non uniformly hyperbolic map
UR - http://eudml.org/doc/104299
ER -

References

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