LIMIT THEOREMS FOR PRODUCTS OF SUMS OF INDEPENDENT RANDOM VARIABLES∗ BY

Abstract. Let {Xn, n 1} be a sequence of independent random variables with finite second moments and {Nn, n 1} be a sequence of positive integer-valued random variables. Write Sn = ∑n k=1(Xk−EXk), n 1, and let N be a standard normal random variable. In the paper the convergences ` n∏ k=1 (Sk/ak + 1) ´γn D− → eN and `Nn∏ k=1 (Sk/ak + 1) ´γn D− → eN are considered for some sequences {an} and {γn} of positive integer num-bers such that Sn + an 0 a.e. The case when γn are random variables is also considered. The main results generalize the main theorems presented by Pang et al. [3]