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On weak and Moments Convergence of Randomly Indexed Sums

Andrzej Krajka, Z. Rychlik

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Abstract

Abstract Let {Xn, n ⩾ 1) be a sequence of independent random variables such that EXn = an, E(Xn − an)2 = σ , n ⩾ 1. Let {Nn, 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 ‐Ln)/sn, 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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What this paper is about

Abstract Let {Xn, n ⩾ 1) be a sequence of independent random variables such that EXn = an, E(Xn − an)2 = σ , n ⩾ 1. Let {Nn, 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 ‐Ln)/sn, 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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Available abstract

Abstract Let {Xn, n ⩾ 1) be a sequence of independent random variables such that EXn = an, E(Xn − an)2 = σ , n ⩾ 1. Let {Nn, 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 ‐Ln)/sn, 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).

Key concepts: Mathematics, Sequence (biology), Combinatorics, Hermite polynomials, Limit of a sequence, Integer (computer science), Type (biology), Random variable

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