In this paper, we generalize a theorem of Shao [12] by assuming that {Xn} is a sequence of linear negatively dependent random variables. Also, we extend some theorems of Chao [6] and Thrum [14]. It is shown by an elementary method that for linear negatively dependent identically random variables with finite p –th absolute moment (p³2) p the weighted sums 1/An åni=1 aniXi converge to zero as where
An=n1/p and (åni=1 a2ni) {an,i} a is an array of real numbers. Moreover, we prove the almost sure convergence for weighted sums åni=1 aniX, n³1, when {Xi, i ³ 1} i is a sequence of pairwise negative quadrant dependence stochastically bounded random variables under some suitable conditions on an,i.