Random sums of independent random vectors attracted by (semi)-stable hemigroups

Let (Xn) be a sequence of independent not necessarily identically distributed random vectors belonging to the domain of attraction of a stable or semistable hemigroup, i.e. for an increasing sampling sequence (kn) such that kn+1/kn → c ≥ 1 and linear operators An, the normalized sums [wzór] converge in distribution uniformly on compact subsets of {0 ≤ s < t} to some full probability μs,t. Suppose that (Tn) is a sequence of positive integer valued random variables such that Tn/kn converges in probability to some positive random variable, where we do not assume (Xn) and (Tn) to be independent. Then weak limit theorems of random sums, where the sampling sequence (kn) is replaced by random sample sizes (Tn), are presented.