Comparison between criteria leading to the weak invariance principle

Olivier Durieu; Dalibor Volný

doi:10.1214/07-AIHP123

Comparison between criteria leading to the weak invariance principle

Probability 2008-12-18 v1

Authors: Olivier Durieu , 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. Poincar\'{e} 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.

Keywords

weak invariance principle central limit theorem martingale-coboundary decomposition projective criterion ergodic dynamical system positive entropy L^2 function

Cite

@article{arxiv.0805.3450,
  title  = {Comparison between criteria leading to the weak invariance principle},
  author = {Olivier Durieu and Dalibor Volný},
  journal= {arXiv preprint arXiv:0805.3450},
  year   = {2008}
}

Comments

Published in at http://dx.doi.org/10.1214/07-AIHP123 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Comparison between criteria leading to the weak invariance principle

Abstract

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