Let $\{a_{ni},1\leq i\leq n,n\geq1\}$ be an array of real numbers and $\{X_n,n\geq1\}$ be a sequence of random variables satisfying the Rosenthal type inequality, which is stochastically dominated by a random variable $X$. Under mild conditions, we present some results on complete convergence for weighted sums $\sum^n_{i=1}a_{ni}X_i$ of random variables satisfying the Rosenthal type inequality. The results obtained in the paper generalize some known ones in the literatures.