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 une chaîne de Markov -géométriquement ergodique dont la loi dépend d’un paramètre . Nous montrons alors que satisfait un développement d’Edgeworth d’ordre 1 uniforme (en ) si 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 ne sont pas nécessairement stationnaires. Ce résultat est appliqué en particulier aux -estimateurs associés à des chaînes de Markov -géométriquement ergodiques. Les -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 -geometrically ergodic Markov chain whose distribution depends on a parameter , we prove that satisfies a uniform (in ) first-order Edgeworth expansion provided that satisfies some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence need not be stationary. This result is applied to -estimators of Markov chains and in particular of -geometrically ergodic Markov chains. The -estimators of some autoregressive processes are studied.
Mots clés : edgeworth expansion, $V$-geometrically ergodic Markov chain, non-arithmeticity condition, perturbation operator
@article{AIHPB_2015__51_2_781_0, author = {Ferr\'e, D.}, 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}, pages = {781--808}, publisher = {Gauthier-Villars}, volume = {51}, number = {2}, year = {2015}, doi = {10.1214/13-AIHP592}, mrnumber = {3335025}, zbl = {1322.60158}, language = {en}, url = {http://www.numdam.org/articles/10.1214/13-AIHP592/} }
TY - JOUR AU - Ferré, D. TI - Parametric first-order Edgeworth expansion for Markov additive functionals. Application to $M$-estimations JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 781 EP - 808 VL - 51 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/13-AIHP592/ DO - 10.1214/13-AIHP592 LA - en ID - AIHPB_2015__51_2_781_0 ER -
%0 Journal Article %A Ferré, D. %T Parametric first-order Edgeworth expansion for Markov additive functionals. Application to $M$-estimations %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 781-808 %V 51 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/13-AIHP592/ %R 10.1214/13-AIHP592 %G en %F AIHPB_2015__51_2_781_0
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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