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

Davide Giraudo

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

Jean-François Le Gall

Davide Giraudo ¹

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

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

Abstract (Original language)
Résumé

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 $q < 1$ .

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.

Article information
Cite this article

Received: 2015-02-09
Accepted: 2015-07-27
Published online: 2015-10-01

DOI: 10.1016/j.crma.2015.07.013

Author's affiliations:

Davide Giraudo ¹

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

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

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References
Cited by

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

[3] H.P. Rosenthal 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] V.A. Volkonskiĭand; Yu.A. Rozanov 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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