Weak compactness of random sums of independent random variables

The shift-compactness of random sums $S_{N_n}^{(n)}$, $S_k^{(n)}=X_{n,1}+\cdots +X_{n,k}$, of independent random variables is investigated under the assumptions that in each sum the summands and their number $N_n$ are independent and that the summands satisfy the condition of uniform asymptotic negligibility in the form
$$ \max_{1\le k\le N_n}\mathsf{P}\{|X_{n,k}|\ge\varepsilon\}\to0 $$
in probability for each $\varepsilon>0$. Some necessary and sufficient conditions are given for the weak compactness of random sums $S_{N_n}^{(n)}-A_n$, and the form of centering constants $A_n$ is described.