On the rate of convergence in the central limit theorem for Markov-chains

Abstract

All results concerning the accuracy of the normal approximation for sums of not necessarily independent random variables assume Doeblin's condition or the condition of (p-mixing (see e.g. [1, 3, 5, 7, 9]). Both assumptions mean in some sense that the random variables are "asymptotically independent", and they are rarely fulfilled for Markov-chains. Using Doeblin's condition or the condition of (p-mixing the rate of convergence to the normal distribution obtained in some of the papers cited above is of order n -1/2. The authors do not know of any results on the accuracy of the normal approximation holding without such conditions. In this paper we prove under weak moment conditions that for Markov-chains the normal approximation is of order n -" for each c~ < 1/4.