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

Let $\{X_{n}\}_{n\geq 0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_{n}(\alpha):=n^{-1}\sum^{n}_{k=1} F(\alpha, X_{k-1}, X_{k}), \alpha \in \mathcal A \subset \mathbb R$. Consider an $M$-estimator $\widehat{\alpha}_{n}$, that is as a measurable function of the observations satisfying $M_{n} (v)\leq min_{\alpha \in\mathcal A} M_{n}(\alpha)+ c_{n}$ with $\{c_{n}\}_{n\geq 1}$ 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 $\widehat{\alpha}_{n}$ satisfies a Berry-Esseen theorem uniformly with respect to the underlying probability distribution of the Markov chain.