Weak limits for multivariate random sums

Let {Xi, i greater than or equal to 1} be a sequence of i.i.d. random vecto rs in R-d, and let nu(p), 0 < p < 1, be a positive, integer valued random v ariable, independent of Xis. The v-stable distributions are the weak limits of properly normalized random sums Sigma(t=1)(nu p), X-i, as nu(p) (P) und er right arrow infinity and p nu(P) (d) under right arrow 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. (C) 1998 Academic Press.