Random marked sets

Abstract

We aim to link random fields and marked point processes, and, therefore, introduce a new class of stochastic processes which are defined on a random set in R^d. Unlike for random fields, the mark covariance function of a random marked set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.