Vitesse de Convergence dans le Th\'eor\`eme Limite Central pour Cha\^ines de Markov de Probabilit\'e de Transition Quasi-Compacte

Abstract

Let $Q$ be a transition probability on a measurable space $E$, 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 its 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 $O(n^{-\frac{\tau}{2}})$ for all $\tau<1$, or $O(n^{-{1/2}})$. We apply the spectral method of Nagaev which is improved by using a perturbation theorem of Keller and Liverani and a method of martingale difference reduction. When $E$ is not compact or $\xi$ is not bounded, the conditions required here are weaker than the ones usually imposed when the standard perturbation theorem is used. For example, in the case of $V$-geometric ergodic chains or Lipschitz iterative models, the rate of convergence in the c.l.t is $O(n^{-{1/2}})$ under a third moment condition on $\xi$.