On weak and Moments Convergence of Randomly Indexed Sums

Abstract

Abstract 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).

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