Document ESAIM: P&S, February 2005, Vol. 9, pp. 38-73
DOI: 10.1051/ps:2005003
Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences
Jérôme Dedecker1 and Sana Louhichi21 Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, Site Chevaleret, 13 rue Clisson, 75013 Paris, France; dedecker@ccr.jussieu.fr
2 Laboratoire de Probabilités, Statistique et modélisation, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France; Sana.Louhichi@math.u-psud.fr
(Received July 3, 2003.)
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.
Mathematics Subject Classification. 60E07, 60F05
Key words: Infinitely divisible distributions, Lévy processes, weak dependence, association, binary random variables, number of exceedances.
© EDP Sciences, SMAI 2005