Weak Limits for Multivariate Random Sums

Let {Xi, i⩾1} be a sequence of i.i.d. random vectors inRd, and letνp, 0<p<1, be a positive, integer valued random variable, independent ofXis. Theν-stable distributions are the weak limits of properly normalized random sums ∑νpi=1 Xiasνp→P∞ andpνp→dν. We study the properties ofν-stable laws through their representation via stable laws. In particular, we estimate their tail probabilities and provide conditions for finiteness of their moments.