Asymptotic distribution of sequences with random indices

A general form is determined for the limit distribution function of a sequence of random vectors with random indices $(S_1(N_n^{(1)},\dots,S_r(N_n^{(r)}))$ in the case when sequences $(S_1(n),\dots,S_r(n))$ and $(N_n^{(1)},\dots,N_n^{(r)})$ for appropriate normalization have a nonsingular joint limit distribution.