On weak and Moments Convergence of Randomly Indexed Sums

Let {X_n, n ? 1) be a sequence of independent random variables such that EX_n = a_n, E(X_n ? a_n)² = σ, n ? 1. Let {N_n, n ? 1} be a sequence of positive integer-valued random variables. Let us put In this paper we present necessary and sufficient conditions for weak and moments convergence of the sequence {(S-L_n)/s_n, n ? 1}, as n → ∞. Hermite polinomial type limit theorems are also considered. The obtained results extend the main theorem of M. Finkelstein and H. G. Tucker (1989).