Research Article

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

Dedecker, Jérôme^a1 and Louhichi, Sana^a2

^a1 Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, Site Chevaleret, 13 rue Clisson, 75013 Paris, France; dedecker@ccr.jussieu.fr

^a2 Laboratoire de Probabilités, Statistique et modélisation, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France; Sana.Louhichi@math.u-psud.fr

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.

(Received July 3 2003)

(Online publication November 15 2005)

Key Words:

• Infinitely divisible distributions;
• Lévy processes;
• weak dependence;
• association;
• binary random variables;
• number of exceedances.

Mathematics Subject Classification:

• 60E07;
• 60F05