A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales

Songqi Wu; Xiaohui Ma; Hailin Sang; Xiequan Fan

doi:10.5802/crmath.81

Probabilités

A Berry–Esseen bound of order

\frac{1}{\sqrt{n}}

for martingales
[Une borne de Berry–Esseen d’ordre

\frac{1}{\sqrt{n}}

pour les martingales]

Songqi Wu ¹ ; Xiaohui Ma ¹ ; Hailin Sang ² ; Xiequan Fan ¹

¹ Center for Applied Mathematics, Tianjin University, Tianjin, China
² Department of Mathematics, The University of Mississippi, University, MS 38677, USA

Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 701-712.

Résumé
Abstract

Renz [13] a établi un taux de convergence $1 / \sqrt{n}$ dans le théorème de la limite centrale pour les martingales avec certaines conditions restrictives. Dans le présent article, une modification des méthodes, développées par Bolthausen [2] et Grama et Haeusler [6], est appliquée pour obtenir le même taux de convergence pour une classe de martingales plus générales. Une application aux processus linéaires est discutée.

Renz [13] has established a rate of convergence $1 / \sqrt{n}$ in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen [2] and Grama and Haeusler [6], is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.

Reçu le : 2020-01-23
Accepté le : 2020-06-02
Publié le : 2020-10-02

DOI : 10.5802/crmath.81

Affiliations des auteurs :

Songqi Wu ¹ ; Xiaohui Ma ¹ ; Hailin Sang ² ; Xiequan Fan ¹

¹ Center for Applied Mathematics, Tianjin University, Tianjin, China
² Department of Mathematics, The University of Mississippi, University, MS 38677, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Songqi Wu and Xiaohui Ma and Hailin Sang and Xiequan Fan},
     title = {A {Berry{\textendash}Esseen} bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {701--712},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {6},
     year = {2020},
     doi = {10.5802/crmath.81},
     language = {en},
}

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Songqi Wu; Xiaohui Ma; Hailin Sang; Xiequan Fan. A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 701-712. doi : 10.5802/crmath.81. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.81/

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