Conditional convergence to infinitely divisible distributions with finite variance

We obtain new conditions for partial sums of an array with stationary rows to converge to a mixture of infinitely divisible distributions with finite variance. More precisely, we show that these conditions are necessary and sufficient to obtain conditional convergence. If the underlying σ -algebras are nested, conditional convergence implies stable convergence in the sense of Rényi. From this general result we derive new criteria expressed in terms of conditional expectations, which can be checked for many processes such as m-conditionally centered arrays or mixing arrays. When it is relevant, we establish the weak convergence of partial sum processes to a mixture of Lévy processes in the space of cadlag functions equipped with Skorohodʹs topology. The cases of Wiener processes, Poisson processes and Bernoulli distributed variables are studied in detail.