On the transfer theorems for observed and unobserved random variables

We characterize the possible weak limits of [Formula} for a sequence {єn,n ≥ 1} of independent random variables [Formula] and a non-random variables (P[єn Є {0,1}] = 1 for n ≥ 1) and a non-random normalizing sequence {kn,n ≥ 1} of positive reals. We consider two cases: when {Xn, n ≥ 1} and {єn, n ≥ 1} are independent or dependent. In the first case we obtain results generalizing transfer theorems, whereas in the other case, only a partial characterization was possible.