Limit theorems for stationary Markov processes with L2-spectral gap

Let (Xt, Yt)t∈T be a discrete or continuous-time Markov process with state space X × R where X is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. (Xt, Yt)t∈T is assumed to be a Markov additive process. In particular, this implies that the first component (Xt)t∈T is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process (Yt)t∈T is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have supt∈(0,1]∩T Eπ,0[|Yt|] < ∞ with the expected order α with respect to the independent case (up to some ε > 0 for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process (Xt)t∈T has an invariant probability distribution π, is stationary and has the L(π)-spectral gap property (that is, (Xt)t∈N is ρ-mixing in the discrete-time case). The case where (Xt)t∈T is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M -estimators associated with ρ-mixing Markov chains. subject classification : 60J05, 60F05, 60J25, 60J55, 37A30, 62M05