Probability theory
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.

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 Nq. The goal of this note is to show that actually the same construction can give mixing rates of order Nq 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 Nq. L'objectif de cette note est de montrer que la même construction peut fournir des taux de mélange de l'ordre de Nq pour un q<1 donné.

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DOI: 10.1016/j.crma.2015.07.013
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, Volume 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/

[1] Giraudo, D.; Volný, D. A strictly stationary β-mixing process satisfying the central limit theorem but not the weak invariance principle, Stoch. Process. Appl., Volume 124 (2014) no. 11, pp. 3769-3781 (MR 3249354)

[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 Lp(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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