Vitesse de Convergence dans le Théorème Limite Central pour Chaînes de Markov de Probabilité 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 $\xi$.

16 pages