Asymptotic properties of Gaussian random fields

Abstract:

In this paper we study continuous mean zero Gaussian random fields $X(p)$ with an N-dimensional parameter and having a correlation function $\rho (p,q)$ for which $1 - \rho (p,q)$ is asymptotic to a regularly varying (at zero) function of the distance ${\text {dis}}\;(p,q)$ with exponent $0 < \alpha \leq 2$. 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 $|p| \to \infty$. Both results generalize ones previously given by the authors for $N = 1$.