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Convergence to infinitely divisible distributions with finite variance for someweakly dependent sequences

Published online by Cambridge University Press:  15 November 2005

Jérôme Dedecker  and
Sana Louhichi
Jérôme Dedecker
Affiliation:
Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, Site Chevaleret, 13 rue Clisson, 75013 Paris, France; dedecker@ccr.jussieu.fr
Sana Louhichi
Affiliation:
Laboratoire de Probabilités, Statistique et modélisation, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France; Sana.Louhichi@math.u-psud.fr
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Abstract

We continue the investigation started in a previous paper, on weak convergence to infinitely divisible distributions with finite variance. In the present paper, we study this problem for some weakly dependent random variables, including in particular associated sequences. We obtain minimal conditions expressed in terms of individual random variables. As in the i.i.d. case, we describe the convergence to the Gaussian and the purely non-Gaussian parts of the infinitely divisible limit. We also discuss the rate of Poisson convergence and emphasize the special case of Bernoulli random variables. The proofs are mainly based on Lindeberg's method.

Keywords

Infinitely divisible distributionsLévy processesweak dependenceassociationbinary random variablesnumber of exceedances.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 38 - 73
Copyright
© EDP Sciences, SMAI, 2005

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