The convergence of distribution functions of random sums of independent random variables with finite variances

Abstract

Let (Xnk) be a double sequence of random variables with finite variances. Let (Zn) be a sequence of positive integral-valued random variables such that for each n, Zn, Xn1, Xn2,... are independent. In this study, we give necessary conditions and sufficient conditions for the sequence of distribution functions of the sums Xn1+Xn2+...+XnZn- (Zn)Sigma mu nk(k=1) to be weakly convergent, where mu nk is the expectation of Xnk.