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 2 , 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   N   n‐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).