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

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1 IRMAR - Institut de Recherche Mathématique de Rennes

Abstract : 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.

Keywords : Spectral method

Author: ** Loïc Hervé - James Ledoux - Valentin Patilea - **

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