Weak Limits for Multivariate Random Sums

Let {Xi, i[greater-or-equal, slanted]1} be a sequence of i.i.d. random vectors inRd, and let[nu]p, 0<p<1, be a positive, integer valued random variable, independent ofXis. The[nu]-stable distributions are the weak limits of properly normalized random sums [summation operator][nu]pi=1 Xias[nu]p[infinity] andp[nu]p[nu]. We study the properties of[nu]-stable laws through their representation via stable laws. In particular, we estimate their tail probabilities and provide conditions for finiteness of their moments.