Bernoulli

  • Bernoulli
  • Volume 18, Number 2 (2012), 703-734.

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

Loïc Hervé, James Ledoux, and Valentin Patilea

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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.

Article information

Source
Bernoulli, Volume 18, Number 2 (2012), 703-734.

Dates
First available in Project Euclid: 16 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1334580730

Digital Object Identifier
doi:10.3150/10-BEJ347

Mathematical Reviews number (MathSciNet)
MR2922467

Zentralblatt MATH identifier
1279.60089

Keywords
asymptotic properties of estimators Markov chains weak spectral method

Citation

Hervé, Loïc; Ledoux, James; Patilea, Valentin. A uniform Berry–Esseen theorem on $M$-estimators for geometrically ergodic Markov chains. Bernoulli 18 (2012), no. 2, 703--734. doi:10.3150/10-BEJ347. https://projecteuclid.org/euclid.bj/1334580730


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