Limit theorems for stationary Markov processes with L2-spectral gap

Let $(X_t, Y_t)_{t\in \mathbb{T}}$ be a discrete or continuous-time Markov process with state space $\mathbb{X} \times \mathbb{R}^d$ where $\mathbb{X}$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{t\in \mathbb{T}}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{t\in \mathbb{T}}$ 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 $(Y_t)_{t\in \mathbb{T}}$ 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 $\sup\{ t\in(0,1]\cap \mathbb{T} : \mathbb{E}{\pi,0}[|Y_t| ^{\alpha}] 0$ 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 $(X_t)_{t\in \mathbb{T}}$ has an invariant probability distribution $\pi$, is stationary and has the $\mathbb{L}^2(\pi)$-spectral gap property (that is, $(X_t)_{t\in \mathbb{N}}$ is $\rho$-mixing in the discrete-time case). The case where $(X_t)_{t\in \mathbb{T}}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $\rho$-mixing Markov chains.