# Limit theorems for stationary Markov processes with L2-spectral gap

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1 IRMAR - Institut de Recherche Mathématique de Rennes

Abstract : 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 $\sup\{ t\in0,1\cap \mathbb{T} : \mathbb{E}{\pi,0}|Y t| ^{\alpha} < 1\}$ with the expected order with respect to the independent case up to some $\varepsilon > 0$ 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 $ ho$-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 $ ho$-mixing Markov chains.

Keywords : Markov additive process central limit theorems Berry-Esseen bound Edgeworth expansion spectral method $ ho$-mixing M-estimator

Author: ** Déborah Ferré - Loïc Hervé - James Ledoux - **

Source: https://hal.archives-ouvertes.fr/