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Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences. (English) Zbl 1136.60308
Summary: We continue the investigation started in a previous paper [Stochastic Processes Appl. 115, No. 5, 737–768 (2005; Zbl 1070.60033)], 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.
60E07Infinitely divisible distributions; stable distributions
60F05Central limit and other weak theorems