Abstract: In this paper we establish a Berry-Esseen inequality and an invarianoe prinoipe for associated random fields \left\X_\mathrmn(N), n \in \Lambda(K(N))\right\ under the assumptions i) E X_n^2(N) \geqslant 0 >0 and E X_n(N)=0 for every n, N ii) \lim _A \rightarrow \infty \sup _N \sup _n \in \Lambda(K(N)) \int_\mid(| >| >\Delta x^2 d P\left(X_n(N)< x\right)=0 where \K(N)\ is a sequence in z^d with \|\Lambda(K(N)) \|\rightarrow \infty. iii) \sum_j: 1 / n=f \mid>r \operatornameCov\left(X_j(N), X_n(N)\right) \leqslant u(r) for r \geqslant 0 where u(r) \rightarrow 0 as \boldsymbolr \rightarrow \infty. The results obtained improve the theorem of Cox and Grimment (1984).