Premium
On weak and Moments Convergence of Randomly Indexed Sums
Author(s) -
Krajka A.,
Rychlik Z.
Publication year - 1992
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19921570121
Subject(s) - mathematics , sequence (biology) , combinatorics , hermite polynomials , limit of a sequence , integer (computer science) , type (biology) , random variable , weak convergence , moment (physics) , convergence (economics) , limit (mathematics) , discrete mathematics , pure mathematics , mathematical analysis , statistics , ecology , genetics , physics , computer security , classical mechanics , computer science , economics , asset (computer security) , biology , programming language , economic growth
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 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).