On the strong convergence for weighted sums of negatively associated random variables

Let {Xn,n≥1} be a sequence of identically distributed negatively associated random variables and let {ani,1≤i≤n,n≥1} be an array of constants satisfying ∑i=1n|ani|α=O(n) for some 1<α≤2. In this paper, we give moment conditions for complete convergence of the form ∑n=1∞n−1P(max1≤m≤n|∑i=1maniXi|>ϵbn)<∞,∀ϵ>0, where bn=n1/α(logn)1/γ and α>γ>0. Our moment conditions are almost optimal. As a corollary, a strong law of large numbers for weighted sums is obtained.