Let {$V_{nk},;k;{geq};1,;{geq};1$} be an array of rowwise independent random elements which are stochastically dominated by a random variable X with $E|X|^{frac{alpha}{gamma}+{ heta}}log^{ ho}(|X|);<;{infty}$ for some ${ ho};>;0,;{alpha};>;0,;{gamma};>;0,;{ heta};>;0$ such that ${ heta}+{alpha}/{gamma}<2$. Let {$a_{nk},k{geq}1,n{geq}1$) be an array of suitable constants. A complete convergence result is obtained for the weighted sums of the form $sum{^infty_k_=_1};a_{nk}V_{nk}$.