Parametric first-order Edgeworth expansion for Markov additive functionals. Application to $M$-estimations

Ferré, D.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 781-808

We give a spectral approach to prove a parametric first-order Edgeworth expansion for bivariate additive functionals of strongly ergodic Markov chains. In particular, given any $V$-geometrically ergodic Markov chain ${\left(X_n\right)}_{n\in \mathbb{N}}$ whose distribution depends on a parameter $\theta$, we prove that $\{{\xi }_p(X_{n-1},X_n);p\in 𝒫,n\ge 1\}$ satisfies a uniform (in $(\theta ,p)$) first-order Edgeworth expansion provided that $\{{\xi }_p(·,·);p\in 𝒫\}$ satisfies some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence ${\left(X_n\right)}_{n\in \mathbb{N}}$ need not be stationary. This result is applied to $M$-estimators of Markov chains and in particular of $V$-geometrically ergodic Markov chains. The $M$-estimators of some autoregressive processes are studied.

Grâce à une approche spectrale, nous donnons des conditions assurant la validité du développement d’Edgeworth d’ordre 1 paramétrique, dans le cadre général des fonctionnelles bivariées et additives de chaînes de Markov fortement ergodiques. En particulier, soit ${\left(X_n\right)}_{n\in \mathbb{N}}$ une chaîne de Markov $V$-géométriquement ergodique dont la loi dépend d’un paramètre $\theta$. Nous montrons alors que $\{{\xi }_p(X_{n-1},X_n);p\in 𝒫,n\ge 1\}$ satisfait un développement d’Edgeworth d’ordre 1 uniforme (en $(\theta ,p)$) si $\{{\xi }_p(·,·);p\in 𝒫\}$ satisfait une condition de type non-lattice ainsi qu’une condition quasi-optimale de moment-domination. De plus, ce résultat est établi dans le cas où les données ${\left(X_n\right)}_{n\in \mathbb{N}}$ ne sont pas nécessairement stationnaires. Ce résultat est appliqué en particulier aux $M$-estimateurs associés à des chaînes de Markov $V$-géométriquement ergodiques. Les $M$-estimateurs de processus autorégressifs sont étudiés.

Classification : 60F05, 60J05, 62F12, 62M05
Keywords: edgeworth expansion, $V$-geometrically ergodic Markov chain, non-arithmeticity condition, perturbation operator

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