2006/09/26 by Loïc Hervé, Hervé, Loïc
Mathematics · #60J05 - 60F05 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F05 #msc:60J05
paper · pdf · doi:10.48550/arxiv.math/0609720
16 pages
arxiv created 2006/09/26 · arxiv updated 2009/12/01
Let Q be a transition probability on a measurable space E, let (X_n)_n be a Markov chain associated to Q, and let ξ be a real-valued measurable function on E, and S_n = ∑_k=1n ξ(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 ((S_n)/(√ n))_n. According to the hypotheses, we prove that the rate is, either O(n^-\fracτ2) for all τ<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 ξ 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 ξ.