A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion
Aazizi, Soufiane1
1 Department of Mathematics, Faculty of Sciences Semlalia Cadi Ayyad University, B.P. 2390 Marrakesh, Morocco
Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 2, pp. 141-150.

Nous donnons une technique simple pour calculer les limites Berry–Esséen pour la variation quadratique du mouvement Brownien subfractional (subfBm). Notre approche a deux ingrédients principaux  : (i) majorer la covariance des variations quadratiques de subfBm par la covariance de la variation quadratique du mouvement Brownien fractionnaire (FBM)  ; et (ii) utiliser les résultats existants sur fBm dans [1, 2, 4]. En conséquence, nous obtenons une simple et directe preuve pour calculer le taux de convergence des variations quadratiques de subfBm. En outre, nous améliorons aussi ce taux de convergence pour obtenir ceux du mouvement Brownien fractionnaire dans [2].

We give a simple technic to derive the Berry–Esséen bounds for the quadratic variation of the subfractional Brownian motion (subfBm). Our approach has two main ingredients: (i) bounding from above the covariance of quadratic variation of subfBm by the covariance of the quadratic variation of fractional Brownian motion (fBm); and (ii) using the existing results on fBm in [1, 2, 4]. As a result, we obtain simple and direct proof to derive the rate of convergence of quadratic variation of subfBm. In addition, we also improve this rate of convergence to meet the one of fractional Brownian motion in [2].

DOI : 10.5802/ambp.358
Mots-clés : Fractional Brownian motion, Malliavin calculus, Kolmogorov distance, Subfractional Brownian motion, Stein method, Quadratic variation.
Aazizi, Soufiane 1

1 Department of Mathematics, Faculty of Sciences Semlalia Cadi Ayyad University, B.P. 2390 Marrakesh, Morocco 
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     title = {A {Simple} {Proof} of {Berry{\textendash}Ess\'een} {Bounds} for the {Quadratic} {Variation} of the {Subfractional} {Brownian} {Motion}},
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Aazizi, Soufiane. A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion. Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 2, pp. 141-150. doi : 10.5802/ambp.358. https://www.numdam.org/articles/10.5802/ambp.358/

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