Complete Convergence for Negatively Dependent Random Variables

Let {Xn, n ≥ 1} be a sequence of i.i.d., real random variables. Hsu and Rabbins [5] proved that if E(X) = 0 and E(X) < ∞, then the sequence 1 n ∑n i=1 Xi converges to 0 completely. (i.e., the series ∑∞ n=1 P [|Sn| > nε] < ∞, converges for every ε > 0). Now let {Xn, n ≥ 1} be a sequence of negatively dependent real random variables. In this paper, we proved the complete convergence of the sequence 1 n ∑n i=1 Xi, via. exponential bounds. In addition if {Xnk, 1 ≤ k ≤ n, n ≥ 1} is an array of rowwise pairwise negatively dependent random variables, we proved complete convergence of the sequence { 1 nα ∑n k=1 Xnk, n ≥ 1} where α > 0. To prove these theorems we need to the following definitions and lemmas.