Let $\{X_j: j \in \mathbb{N}\}$ be a sequence of associated random variables with zero mean and let $r > 2$. We give two conditions--on the moments and on the covariance structure of the process--which guarantee that $\sup_{m \in \mathbb{N} \cup \{0\}} E| \sum^{m+n}_{j=m+1} X_j|^r = O(n^{r/2})$ holds. Examples show that neither condition can be weakened.