Moderate deviations for Markov chains with atom

We obtain in this paper moderate deviations for functional empirical processes of general state space valued Markov chains with atom under weak conditions: a tail condition on the first time of return to the atom, and usual conditions on the class of functions. Our proofs rely on the regeneration method and sharp conditions issued of moderate deviations of independent random variables. We prove our result in the nonseparable case for additive and unbounded functionals of Markov chains, extending the work of de Acosta and Chen (J. Theoret. Probab. (1998) 75–110) and Wu (Ann. Probab. (1995) 420–445). One may regard it as the analog for the Markov chains of the beautiful characterization of moderate deviations for i.i.d. case of Ledoux 1992. Some applications to Markov chains with a countable state space are considered.