Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences

¹ 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: 3 July 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.