The central limit theorem for Markov chains started at a point

The aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic Markov chain on a general state space, with respect to the law of the chain started at a point. No irreducibility assumption nor mixing conditions are imposed; the only assumption bears on the growth of the L²-norms of the ergodic sums for the function generating the additive functional, which must be 0 (nα) with α < 1/2. The result holds almost surely with respect to the invariant probability of the chain.