A uniform Berry–Esseen theorem on $M$-estimators for geometrically ergodic Markov chains

Abstract

Let {$X_n$}$_{n≥0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_n(α) := n^{−1}∑_{k=1}^nF(α, X_{k−1}, X_k), α \in A \subset \mathbb {R}$. Consider an $M$ estimator $\widehat{α}_n$, that is, a measurable function of the observations satisfying $M_{n}(\widehat{\alpha}_{n})\leq\min_{\alpha\in\mathcal{A}}M_{n}(\alpha)+c_{n}$ with {$c_n$}$_{n≥1}$ some sequence of real numbers going to zero. Under some standard regularity and moment assumptions, close to those of the i.i.d. case, the estimator $\widehat{α}_n$ satisfies a Berry–Esseen theorem uniformly with respect to the underlying probability distribution of the Markov chain.