Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Comparison between criteria leading to the weak invariance principle

Olivier Durieu and Dalibor Volný

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Abstract

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 $\mathbb{L}^{2}$ satisfying the first but not the second.

Résumé

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 $\mathbb{L}^{2}$ vérifiant le premier mais pas le deuxième.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 44, Number 2 (2008), 324-340.

Dates
First available in Project Euclid: 11 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1207948222

Digital Object Identifier
doi:10.1214/07-AIHP123

Mathematical Reviews number (MathSciNet)
MR2446326

Zentralblatt MATH identifier
1182.60010

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G10: Stationary processes 28D05: Measure-preserving transformations 60G42: Martingales with discrete parameter

Keywords
Stationary process Central limit theorem Weak invariance principle Martingale approximation Projective criterion

Citation

Durieu, Olivier; Volný, Dalibor. Comparison between criteria leading to the weak invariance principle. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 2, 324--340. doi:10.1214/07-AIHP123. https://projecteuclid.org/euclid.aihp/1207948222.


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