Rate of convergence in the central limit theorem for strongly ergodic Markov chains

International audienceLet 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 = Sigma(n)(k=1)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 (S-n/root n)(n). According to the hypotheses, we prove that the rate is, either O(n(-tau/2)) for all tau < 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 vertical bar E[e(itSn/root n)] - e(-t2/2)vertical bar obtained by a method of martingale difference reduction. When E is not compact or is not bounded, the conditions required here on Q (t) (in substance, some moment conditions on) 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.1.t. is O(n(-1/2)) under a third moment condition on