Complete convergence for negatively dependent random variables

Let {Xn, n ≥ 1} be a sequence of independent and identically random variables. In 1947 Hsu and Rabbins proved that if E[X] = 0 and E[X2] < ∞, then 1 n ∑n k=1Xk converges to 0 completely. Recently, the strong convergence of weighted sums for the case of independent random variables has been discussed by Wu (1999), Hu and et. (2000, 2003) proved the complete convergence theorem for arrays of independent random variables and Amini and Bozorgnia (2003) studied complete convergence of the sequence 1 n ∑n k=1Xk, via. exponential bounds in the case of negatively dependent and identically random variables. Zarei (2006) extended some results of Chow (1966) for negatively dependent and identically distributed random variables. In this paper, we study complete convergence of weighted sums Tn = ∑n k=1 ankXk where {Xn, n ≥ 1} is a sequence of negatively dependent random variables and ank, n ≥ 1, k ≥ 1 is an array of real numbers where ank = 0 if k > n, |ank| ≤ CAn for An = ∑∞ k=1 a 2 nk and some 0 < C <∞. In fact we omit the condition identically distributed random variables. The material in this note is closely related to Zarei (2006) and chow(1966). In this paper we define X+ = max{0, X} and X− = max{0,−X} and IA denotes indicator function A. To prove the main result we need to the following definitions and lemmas.