Vitesse de convergence dans le th\'{e}or\`{e}me limite central pour des cha\^{i}nes 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 $\xi$.