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

Let $\{X_n\}_{n\ge0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_n(α):=n^{-1}\sum_{k=1}^nF(α,X_{k-1},X_k)$, $α\in\mathcal{A}\subset \mathbb {R}$. Consider an $M$ estimator $\hatα_n$, that is, a measurable function of the observations satisfying $M_n(\hatα_n)\leq \min_{α\in\mathcal{A}}M_n(α)+c_n$ with $\{c_n\}_{n\geq1}$ 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 $\hatα_n$ satisfies a Berry--Esseen theorem uniformly with respect to the underlying probability distribution of the Markov chain.