Baum–Katz type theorems for martingale arrays

Shunli Hao; Quansheng Liu

doi:10.1016/j.crma.2011.12.006

Probability Theory

Baum–Katz type theorems for martingale arrays

Presented by: the Editorial Board

Shunli Hao ^1,²; Quansheng Liu ^1,²

¹ Université de Bretagne-Sud, UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, campus de Tohannic, BP 573, 56017 Vannes, France
² Université Européenne de Bretagne, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 91-96.

Abstract
Résumé

We show convergence rates in the law of large numbers for martingale arrays. The results extend the classical theorems of Baum and Katz (1965) [2] for sums of independent and identically distributed (i.i.d.) random variables. They improve a result of Ghosal and Chandra (1998) [6] for martingale arrays, and generalize a result of Alsmeyer (1990) [1] for a single martingale. As an application, we obtain a new theorem about the convergence rate of Cesàro summation of identically distributed random variables.

Nous montrons la vitesse de convergence dans la loi des grand nombres pour un tableau de martingales. Les résultats étendent les théorèmes classiques de Baum et Katz (1965) [2] pour les sommes de variables aléatoires indépendantes et identiquement distribuées (i.i.d.). Ils améliorent un résultat de Ghosal et Chandra (1998) [6] pour des tableaux de martingales, et généralisent un résultat dʼAlsmeyer (1990) [1] pour une seule martingale. Comme application, nous obtenons un théorème nouveau concernant la vitesse de convergence pour des sommes de Cesàro de variables aléatoires identiquement distribuées.

Received: 2011-11-14
Accepted: 2011-12-08
Published online: 2011-12-27

DOI: 10.1016/j.crma.2011.12.006

Author's affiliations:

Shunli Hao ^{1, 2}; Quansheng Liu ^{1, 2}

¹ Université de Bretagne-Sud, UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, campus de Tohannic, BP 573, 56017 Vannes, France
² Université Européenne de Bretagne, France

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     author = {Shunli Hao and Quansheng Liu},
     title = {Baum{\textendash}Katz type theorems for martingale arrays},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {91--96},
     publisher = {Elsevier},
     volume = {350},
     number = {1-2},
     year = {2012},
     doi = {10.1016/j.crma.2011.12.006},
     language = {en},
}

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Shunli Hao; Quansheng Liu. Baum–Katz type theorems for martingale arrays. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 91-96. doi : 10.1016/j.crma.2011.12.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.12.006/

References
Cited by

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