The asymptotic distribution of the maxima of a Gaussian random field on a lattice

Under mild conditions on the covariance function of a stationary Gaussian process, the maxima behaves asymptotically the same as the maxima of independent, identically distributed Gaussian random variables. In order to achieve extremal clustering, Hsing et al. (Ann Appl Probab 6:671–686, 1996) considered a triangular array of Gaussian sequences in which the correlation between “neighboring” observations approaches 1 at a certain rate. Using analogues of the conditions of Hsing et al., which allows for strong local dependence among variables but asymptotic independence, it is possible to show that two-dimensional Gaussian random fields also exhibit extremal clustering in the limit. A closed form expression for the extremal index governing the clustering will be provided. The results apply to Gaussian random fields in which the spatial domain is rescaled.