Limit theorems for stationary Markov processes with L2-spectral gap

TL;DR

This paper establishes classical limit theorems for additive functionals of stationary Markov processes with an L2-spectral gap, including CLT, local limit theorem, Berry-Esseen, and Edgeworth expansions, under specific moment and nonlattice conditions.

Contribution

It extends limit theorems to Markov additive processes with spectral gap assumptions, providing new results for stationary Markov processes with additive functionals.

Findings

Proves CLT, local limit theorem, Berry-Esseen, and Edgeworth expansions for Markov additive processes.

Derives a Berry-Esseen bound for M-estimators in $ ho$-mixing Markov chains.

Shows results under spectral gap and moment conditions, with brief discussion on non-stationary cases.

Abstract

Let $(X_t, Y_t)_{t\in T}$ be a discrete or continuous-time Markov process with state space $X \times R^d$ where $X$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{t\in T}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{t\in T}$ 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 $(Y_t)_{t\in T}$ 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 T : 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 $(X_t)_{t\in T}$ has an invariant probability distribution $\pi$, is stationary and has the $L^2(\pi)$-spectral gap property (that is, $(X_t)t\in N}$ is $\rho$-mixing in the discrete-time case). The case where $(X_t)_{t\in T}$ 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.