A quenched weak invariance principle
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, pp. 872-898.

In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and nonirreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, an approach which has interest in itself.

Dans cet article, nous étudions le théorème central limite conditionnel presque sûr, ainsi que sa forme fonctionnelle, pour des suites stationnaires de variables aléatoires réelles satisfaisant une condition de type projectif. Nous donnons des applications de ces résultats aux processus fortement mélangeants ainsi qu'à des chaînes de Markov nonirréductibles. Les preuves sont essentiellement basées sur une approximation normale de suites doublement indexées de variables aléatoires de type martingale.

DOI: 10.1214/13-AIHP553
Classification: 60F05,  60F17,  60J05
Keywords: quenched central limit theorem, weak invariance principle, strong mixing, Markov chains
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Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda. A quenched weak invariance principle. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, pp. 872-898. doi : 10.1214/13-AIHP553. http://www.numdam.org/articles/10.1214/13-AIHP553/`

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