In this paper, we generalize a theorem of Shao [12] by assuming that 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 -th absolute moment the weighted sums converge to zero as where and is an array of real numbers. Moreover, we prove the almost sure convergence for weighted sums , when is a sequence of pairwise negative quadrant dependence stochastically bounded random variables under some suitable conditions on .