Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A quenched weak invariance principle

Jérôme Dedecker, Florence Merlevède, and Magda Peligrad

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 3 (2014), 872-898.

Dates
First available in Project Euclid: 20 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1403277001

Digital Object Identifier
doi:10.1214/13-AIHP553

Mathematical Reviews number (MathSciNet)
MR3224292

Zentralblatt MATH identifier
1304.60031

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60J05: Discrete-time Markov processes on general state spaces

Keywords
Quenched central limit theorem Weak invariance principle Strong mixing Markov chains

Citation

Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda. A quenched weak invariance principle. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 3, 872--898. doi:10.1214/13-AIHP553. https://projecteuclid.org/euclid.aihp/1403277001


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