The Berry - Esseen bound for general Markov chains

The subject of this communication is the estimation of the proximity of the normal approximation for the distribution of sums of random variables defined on the Markov chain. More exactly, our aim is to extend the Berry - Esseen bound to non -uniformly ergodic Markov chains. Bolthausen (1980) obtained the bound $O(n^{1/2})$ in CLT for countable Markov chains, where $n$ is number of summands. In 1982 he extended this bound to the chains with the general phase space, by using Nummelin's (1978) splitting technique. Strictly speaking, it is not the Berry - Esseen bound in full measure since it does not depend explicitly on ergodic properties of the Markov chain and does not contain an absolute constant. In brief Bolthausen's bound takes into account only the dependence of $n$. By contrast, our bound includes some parameters related to ergodic properties of the Markov chain, along with an absolute constant. We assume that the Markov chain under consideration satisfies the conditions which are used by Athreya and Ney (1978). Remark that our bound is valid for any initial distribution, whereas Bolthausen confines himself to the case of the stationary Markov chain. It should be noted that in contrast with Bolthausen we do not use the splitting technique, our approach is completely different.