Résumé
Nous étendons la méthode de démonstration du théorème de Berry-Esseen proposée par Bergström aux suites de variables aléatoires faiblement dépendantes. En particulier, nous montrons que, pour les suites stationnaires de variables aléatoires réelles bornées, la vitesse de convergence dans le théorème limite central en distance de Lévy est de l'ordre den −1/2 dès que la suite (θ p)p>0 des coefficients de mélange uniforme satisfait la condition Σ p>0 pθ p <∞
Abstract
We extend the method of Bergström for the rates of convergence in the central limit theorem to weakly dependent sequences. In particular, we prove that, for stationary and uniformly mixing sequences of real-valued and bounded random variables, the rate of convergence in the central limit theorem is of the order ofn −1/2 as soon as the sequence (θ p)p>0 of uniform mixing coefficients satisfies Σ p>0 pθ p <∞.
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Rio, E. Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. Probab. Th. Rel. Fields 104, 255–282 (1996). https://doi.org/10.1007/BF01247840
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DOI: https://doi.org/10.1007/BF01247840