Let $\{X_n,n\ge 1\}$ be a sequence of independent random variables such that $E{X}_n=a_n$, $E{(X_n-a_n)}^2={\sigma }_n^2$, $n\ge 1$. Let $\{N_n,n\ge 1\}$ be a sequence od positive integer-valued random variables. Let us put $S_{N_n}={\sum }_{k=1}^{N_n}{X}_k$, $L_n={\sum }_{k=1}^n{a}_k$, $s_n^2={\sum }_{k=1}^n{\sigma }_k^2$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\{(S_{N_n}-L_n)/s_n,n\ge 1\}$, as $n\to \infty$. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).