Inequalities for the moments of sums of associated multi-indexed variables

Exact upper bounds are obtained for absolute moments of order $r>2$ for finite sums of associated random variables forming a centered field on $\mathbf{N}^d$ or a countable set $T$. These estimates have the form $O(|V|^\tau)$ where $|V|$ is the number of summands. It is shown how the dependence of the summands and existence of their moments determine $\tau$.