On the extremal behavior of a nonstationary normal random field

Let X={_Xn}n?1$\mathbf{X}=\{X_{\mathbf{n}}{\}}_{\mathbf{n}?\mathbf{1}}$ be a nonstationary random field satisfying a long range weak dependence for each coordinate at a time and a local dependence condition that avoids clustering of exceedances of high values. For these random fields, the probability of no exceedances of high values can be approximated by exp(−τ)$\exp (-\tau )$, where τ$\tau$ is the limiting mean number of exceedances. We present a class of nonstationary normal random fields for which this result can be applied.