Let {$X_i,-{infty}$ < 1 < $infty$} be a doubly infinite sequence of identically distributed and negatively associated random variables with mean zero and finite variance and {$a_i,;-{infty}$ < i < ${infty}$} be an absolutely summable sequence of real numbers. Define a moving average process as $Y_n={sum}_{i=-infty}^{infty}a_{i+n}X_i$, n $geq$ 1 and $S_n=Y_1+{cdots}+Y_n$. In this paper we prove that E|$X_1$|$^rh$($|X_1|^p$) < $infty$ implies ${sum}_{n=1}^{infty}n^{r/p-2-q/p}h(n)E{max_{1{leq}k{leq}n}|S_k|-{epsilon}n^{1/p}}{_+^q}<{infty}$ and ${sum}_{n=1}^{infty}n^{r/p-2}h(n)E{sup_{k{leq}n}|k^{-1/p}S_k|-{epsilon}}{_+^q}<{infty}$ for all ${epsilon}$ > 0 and all q > 0, where h(x) > 0 (x > 0) is a slowly varying function, 1 ${leq}$ p < 2 and r > 1 + p/2.