Comparison between criteria leading to the weak invariance principle

Details Comparison between criteria leading to the weak invariance principle Comparison between criteria leading to the weak invariance principle

2008-05-22

arxiv.org/abs/0805.3450v1

Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques 2008, Vol. 44, No. 2, 324-340

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AIHP123 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiqu

Scientific paper

10.1214/07-AIHP123

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.

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