A regularity condition and a limit theorem for Harris ergodic Markov chains

Let (Xn)n⩾0 be a Harris ergodic Markov chain and f be a real function on its state space. Consider the block sums ζ(i) for |f|, i⩾1, between consecutive visits to the atom given by the splitting technique of Nummelin. A regularity condition on the invariant probability measure π and a drift property are introduced and proven to characterize the finiteness of the third moment of ζ(i). This is applied to obtain versions of an almost sure invariance principle for the partial sums of (f(Xn)), which is moreover given in the general case, due to Philipp and Stout for the countable state space case and to Csáki and Csörgő when the chain is strongly aperiodic. Conditions on the strong mixing coefficients are considered. A drift property equivalent to the finiteness of the second moment of ζ(i) is also given and applied to the functional central limit theorem.