Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions

Abstract

ABSTRACT In this work, suppose that {X n ; n ≥ 1}is a sequence of asymptotically negatively associated random variables and {a ni ; 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers such that ∑ i = 1 n | a n i | q = O ( n ) for some q > max { α p − 1 α − 1 / 2 , 2 } with αp > 1 and α > 1 2 . Let l (x) > 0 be a slowly varying function at infinity. We establish some equivalent conditions of the complete convergence for weighted sums of this form ∑ n = 1 ∞ n α p − 2 l ( n ) P ( max 1 ≤ j ≤ n | ∑ i = 1 j a n i X i | > ε n α ) < ∞   for   all   ε   >   0 . As applications, some strong laws of large numbers for weighted sums of asymptotically negatively associated random variables are also obtained.