Asymptotic properties of Gaussian random fields

In this paper we study continuous mean. zero Gaussian random fields X(p) with an N-dimensional parameter and having a correlation function p(p, q) for which 1 p(p, q) is asymptotic to a regularly varying (at zero) function of the distance dis (p, q) with exponent 0 < a < 2. For such random fields, we obtain the asymptotic tail distribution of the maximum of X(p) and an asymptotic almost sure property for X(p) as IPI N. Both results generalize ones previously given by the authors for N = 1.