An improvement of the mixing rates in a counter-example to the weak invariance principle

Giraudo, Davide

doi:10.1016/j.crma.2015.07.013

Probability theory

An improvement of the mixing rates in a counter-example to the weak invariance principle

Présenté par : Jean-François Le Gall

Giraudo, Davide ¹

¹ Université de Rouen, LMRS, Avenue de l'Université, BP 12 76801 Saint-Étienne-du-Rouvray cedex, France

Comptes Rendus. Mathématique, Tome 353 (2015) no. 10, pp. 953-958.

Résumé
Abstract

Dans [1], les auteurs ont fourni un exemple de processus strictement stationnaire β-mélangeant vérifiant le théorème limite central, mais pas le principe d'invariance faible. Pour tout $q < 1 / 2$ , le processus peut être construit avec des taux de mélange de l'ordre de $N^{- q}$ . L'objectif de cette note est de montrer que la même construction peut fournir des taux de mélange de l'ordre de $N^{- q}$ pour un $q < 1$ donné.

In [1], the authors gave an example of absolutely regular strictly stationary process that satisfies the central limit theorem, but not the weak invariance principle. For each $q < / 1 / 2$ , the process can be constructed with mixing rates of order $N^{- q}$ . The goal of this note is to show that actually the same construction can give mixing rates of order $N^{- q}$ for a given $q < 1$ .

Reçu le : 2015-02-09
Accepté le : 2015-07-27
Publié le : 2015-10-01

DOI : 10.1016/j.crma.2015.07.013

Affiliations des auteurs :

Giraudo, Davide ¹

¹ Université de Rouen, LMRS, Avenue de l'Université, BP 12 76801 Saint-Étienne-du-Rouvray cedex, France

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Giraudo, Davide. An improvement of the mixing rates in a counter-example to the weak invariance principle. Comptes Rendus. Mathématique, Tome 353 (2015) no. 10, pp. 953-958. doi : 10.1016/j.crma.2015.07.013. http://www.numdam.org/articles/10.1016/j.crma.2015.07.013/

Bibliographie
Cité par

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[2] Giraudo, D.; Volný, D. A counter example to central limit theorem in Hilbert spaces under a strong mixing condition, Electron. Commun. Probab., Volume 19 (2014) (MR 3254741)

[3] Rosenthal, H.P. On the subspaces of $L^{p} (p > 2)$ spanned by sequences of independent random variables, Israel J. Math., Volume 8 (1970), pp. 273-303 MR 0271721 (42 #6602)

[4] Volkonskiĭand, V.A.; Rozanov, Yu.A. Some limit theorems for random functions. I, Teor. Veroâtn. Ee Primen., Volume 4 (1959), pp. 186-207 MR 0105741 (21 #4477)

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