Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques

Abstract

Let $Q$ be a transition probability on a measurable space $E$ which admits an invariant probability measure, let $(X_n{)}_n$ be a Markov chain associated to $Q$, and let $\xi$ be a real-valued measurable function on $E$, and $S_n={\sum }_{k=1}^n\xi (X_k)$. Under functional hypotheses on the action of $Q$ and the Fourier kernels $Q(t)$, we investigate the rate of convergence in the central limit theorem for the sequence $(\frac{S_n}{\sqrt{n}}{)}_n$. According to the hypotheses, we prove that the rate is, either $\mathrm{O}(n^{-\tau /2})$ for all $\tau <1$, or $\mathrm{O}(n^{-1/2})$. We apply the spectral Nagaev's method which is improved by using a perturbation theorem of Keller and Liverani, and a majoration of $|\mathbb{E}[\mathrm{e}^{\mat hrm{i}t{S_n}/{\sqrt{n}}}]-\mathrm{e}^{{-t^2}/{2}}|$ obtained by a method of martingale difference reduction. When $E$ is not compact or $\xi$ is not bounded, the conditions required here on $Q(t)$ (in substance, some moment conditions on $\xi$) are weaker than the ones usually imposed when the standard perturbation theorem is used in the spectral method. For example, in the case of $V$-geometric ergodic chains or Lipschitz iterative models, the rate of convergence in the c.l.t. is $\mathrm{O}(n^{-1/2})$ under a third moment condition on $\xi$.

Published in at http://dx.doi.org/10.1214/07-AIHP101 the Annales de l'Institut Henri Poincaré - Probabilités et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• On mean central limit theorems for stationary sequences

Cited by in corpus (1)

• A uniform Berry--Esseen theorem on $M$-estimators for geometrically ergodic Markov chains