A necessary condition is given for the convergence of distributions of the sums of a random number of independent random variables. This is made on the basis of a theorem which gives sufficient conditions for the convergence of distributions of randomly stopped stochastic processes. The random indices are supposed to be independent of the sequence of summands.
"Limit Theorems for the Distributions of the Sums of a Random Number of Random Variables." Ann. Math. Statist. 43 (6) 1902 - 1913, December, 1972. https://doi.org/10.1214/aoms/1177690861