Rate of convergence in the central limit theorem for fields of associated random variables

The rate of convergence of standardized sums $S(V)=\sum_{j\in V}X_j$ to the normal law is established for a field $\{X_j,j\in\mathbf Z^d\}$ of associated random variables and arbitrarily increasing finite sets $V\subset\mathbf Z^d$. An exponential type of decay is assumed for the Cox–Grimmet coefficient $u(\,\cdot\,)$ as well as $\sup_j\mathbf E|X_j|^s<\infty$ for some $s>2$.