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 Rd. 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.
"Random marked sets." Adv. in Appl. Probab. 44 (3) 603 - 616, September 2012. https://doi.org/10.1239/aap/1346955256