Extreme Value Statistics Compatible with Random Field Theory

Abstract

Classical extreme value distributions represent maxima or minima of a large number of independent, identically distributed random variables. Their application to modeling extremes of homogeneous random media is usually motivated by the idealization of a solid body as an assemblage of elements with statistically independent “element properties”. In this paper we take a direct approach to deriving extreme value distributions, based on the theory of homogeneous random fields. It accounts explicitly for statistical dependence and provides information about the size and occurrence frequency of isolated regions of excursion above prescribed high (or below low) levels. After summarizing the derivation of excursion statistics for n-dimensional homogeneous random fields, some specific results are reviewed for Gaussian fields [1]. The approach further leads to new interpretations of the Weibull and Gumbel Type I extreme value distributions and their parameters.