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.

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
@article{AMBP_2016__23_2_141_0,
     author = {Aazizi, Soufiane},
     title = {A {Simple} {Proof} of {Berry{\textendash}Ess\'een} {Bounds} for the {Quadratic} {Variation} of the {Subfractional} {Brownian} {Motion}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {141--150},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {23},
     number = {2},
     year = {2016},
     doi = {10.5802/ambp.358},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.358/}
}
TY  - JOUR
AU  - Aazizi, Soufiane
TI  - A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion
JO  - Annales mathématiques Blaise Pascal
PY  - 2016
SP  - 141
EP  - 150
VL  - 23
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.358/
DO  - 10.5802/ambp.358
LA  - en
ID  - AMBP_2016__23_2_141_0
ER  - 
%0 Journal Article
%A Aazizi, Soufiane
%T A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion
%J Annales mathématiques Blaise Pascal
%D 2016
%P 141-150
%V 23
%N 2
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.358/
%R 10.5802/ambp.358
%G en
%F AMBP_2016__23_2_141_0
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. http://www.numdam.org/articles/10.5802/ambp.358/

[1] Breton, Jean-Christophe; Nourdin, Ivan Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion, Electron. Commun. Probab., Volume 13 (2008), pp. 482-493 | DOI

[2] Nourdin, Ivan Lectures on Gaussian approximations with Malliavin calculus, Séminaire de Probabilités XLV (Lecture Notes in Mathematics), Volume 2078, Springer, 2013, pp. 3-89

[3] Nourdin, Ivan; Nualart, David; Tudor, Ciprian A. Central and non-central limit theorems for weighted power variations of fractional Brownian motion, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 46 (2010) no. 4, pp. 1055-1079 | DOI

[4] Nourdin, Ivan; Peccati, Giovanni Stein’s method on Wiener chaos, Probab. Theory Relat. Fields, Volume 145 (2009) no. 1-2, pp. 75-118 | DOI

[5] Nualart, David The Malliavin calculus and related topics, Probability and Its Applications, Springer, 2006, xiv+382 pages

[6] Tudor, Constantin Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion, J. Math. Anal. Appl., Volume 375 (2011) no. 2, pp. 667-676 | DOI

Cité par Sources :