Parametric first-order Edgeworth expansion for Markov additive functionals. Application to M-estimations
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 781-808.

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 (X n ) n∈ℕ une chaîne de Markov V-géométriquement ergodique dont la loi dépend d’un paramètre θ. Nous montrons alors que {ξ p (X n-1 ,X n );p∈𝒫,n≥1} satisfait un développement d’Edgeworth d’ordre 1 uniforme (en (θ,p)) si {ξ p (·,·);p∈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 (X n ) n∈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.

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 (X n ) n∈ℕ whose distribution depends on a parameter θ, we prove that {ξ p (X n-1 ,X n );p∈𝒫,n≥1} satisfies a uniform (in (θ,p)) first-order Edgeworth expansion provided that {ξ p (·,·);p∈𝒫} satisfies some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence (X n ) n∈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.

DOI : 10.1214/13-AIHP592
Classification : 60F05, 60J05, 62F12, 62M05
Mots-clés : edgeworth expansion, $V$-geometrically ergodic Markov chain, non-arithmeticity condition, perturbation operator 
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     title = {Parametric first-order {Edgeworth} expansion for {Markov} additive functionals. {Application} to $M$-estimations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Ferré, D. Parametric first-order Edgeworth expansion for Markov additive functionals. Application to $M$-estimations. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 781-808. doi : 10.1214/13-AIHP592. http://www.numdam.org/articles/10.1214/13-AIHP592/

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