Asymptotic normality of random fields of positively or negatively associated processes

Consider a random field of real-valued random variables with finite second moment and subject to covariance invariance and finite susceptibility. Under the basic assumption of positive or negative association, asymptotic normality is established. More specifically, it is shown that the joint distribution of suitably normalized and centered at expectation sums of random variables, over any finite number of appropriately selected rectangles, is asymptotically normal. The mean vector of the limiting distribution is zero and the covariance matrix is a specified diagonal matrix.