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 (Xn)n be a Markov chain associated to Q, and let ξ be a real-valued measurable function on E, and $S_n=\sum^n_{k=1}ξ(X_k)$. 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 O(n−τ/2) for all τ<1, or 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 $|\mathbb{E}[\mathrm{e}^{\mathrm{i}t{S_{n}}/{\sqrt{n}}}]-\mathrm{e}^{{-t^{2}}/{2}}|$ obtained by a method of martingale difference reduction. When E is not compact or ξ is not bounded, the conditions required here on Q(t) (in substance, some moment conditions on ξ) 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 O(n−1/2) under a third moment condition on ξ.