Let {$X_n,;n{ge}1$} be a sequence of negatively associated random variables which are dominated randomly by another random variable. We discuss the limit properties of weighted sums ${Sigma}^n_{i=1}a_{ni}X_i$ under some appropriate conditions, where {$a_{ni},;1{le};i;{le};n,;n;{ge};1$} is an array of constants. As corollary, the results of Bai and Cheng (2000) and Sung (2001) are extended from the i.i.d. case to not necessarily identically distributed negatively associated setting. The corresponding results of Chow and Lai (1973) also are extended.