THE CENTRAL LIMIT THEOREM FOR STATIONARY MARKOV CHAINS UNDER INVARIANT SPLITTINGS

Abstract

The central limit theorem (CLT) for stationary ergodic Markov chains is investigated. We give a short survey of related results on the CLT for general (not necessarily Harris recurrent) chains and formulate a new sufficient condition for its validity. Furthermore, Markov operators are considered which admit invariant orthogonal splittings of the space of square-integrable functions. We show how conditions for the CLT can be improved if this additional structure is taken into account. Finally we give examples of this situation, namely endomorphisms of compact Abelian groups and random walks on compact homogeneous spaces.