Comparison between criteria leading to the weak invariance principle

Durieu, Olivier; Volný, Dalibor

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 324-340.

Le but de cet article est de comparer différents critères conduisant au théoreme limite centrale et au principe d'invariance faible. Ces critères sont la décomposition martingale-cobord développée par Gordin dans Dokl. Akad. Nauk SSSR 188 (1969), le critère projectif introduit par Dedecker dans Probab. Theory Related Fields 110 (1998), par la suite amélioré par Dedecker et Rio dans Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) et la condition introduite par Maxwell et Woodroofe dans Ann. Probab. 28 (2000), plus tard améliorée par Peligrad et Utev dans Ann. Probab. 33 (2005). On montre que dans tout système dynamique ergodique d’entropie strictement positive, si l’on considère deux de ces critères, on peut trouver une fonction dans ${𝕃}^2$ vérifiant le premier mais pas le deuxième.

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in ${𝕃}^2$ satisfying the first but not the second.

Classification : 60F05, 60F17, 60G10, 28D05, 60G42
Mots clés : stationary process, central limit theorem, weak invariance principle, martingale approximation, projective criterion

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