941 articles – 1212 references  [version française]
 HAL: hal-00008462, version 1
 Available versions: v1 (2005-09-06) v2 (2006-10-17)
 Limit theorems for geometrically ergodic Markov chains
 (2005-09-06)
 Let $(E,\cE)$ be a countably generated state space, let $(X_n)_n$ be an aperiodic and $\psi$-irreducible $V$-geometrically ergodic Markov chain on $E$, with $V : E\r [1,+\infty[$ and $\psi$ a $\sigma$-finite positive measure on $E$. Let $\pi$ be the $P$-invariant distribution, and let $\xi : E\r\R$ measurable and dominated by $\sqrt V$. Then $\sigma^2 = \lim_n n^{-1}\E_x[(S_n)^2]$ exists for any $x\in E$ (and does not depend on $x$), and if $\sigma^2 >0$, then $n^{-1}[\xi(X_1)+\ldots+\xi(X_n)-n\pi(\xi)]$ converges in distribution to the normal distribution ${\cal N}(0,\sigma^2)$. In this work we prove that, for any initial distribution $\mu_0$ satisfying $\mu_{_0}(V) < +\infty$ and under the condition $\sigma^2>0$, - If $\xi$ is dominated by $V^\alpha$ with $\alpha<\frac{1}{4}$, then the rate of convergence in the c.l.t is $O(\frac{1}{\sqrt n})$. - If $\xi$ is dominated by $V^\alpha$ with $\alpha<\frac{1}{2}$, then $(\xi(X_n))_n$ satisfies a local limit theorem and a renewal theorem under a usual non-arithmeticity assumption.
 1: Institut de Recherche Mathématique de Rennes (IRMAR) CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
 Subject : Mathematics/Probability
 Keyword(s): Markov chains : geometric ergodicity : central limit and renewal theorems : spectral method
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 hal-00008462, version 1 http://hal.archives-ouvertes.fr/hal-00008462 oai:hal.archives-ouvertes.fr:hal-00008462 From: Loïc Hervé <> Submitted on: Tuesday, 6 September 2005 08:11:19 Updated on: Tuesday, 6 September 2005 09:27:14