Empirical Mark Covariance and Product Density Function of Stationary Marked Point Processes—A Survey on Asymptotic Results

Abstract

Marked point processes are stochastic models to describe random patterns of marked points {[X i ,M i ], i ≥ 1} in some bounded subset of the d-dimensional Euclidean space (usually d = 1, 2 or 3 in applications), where each point X i carries additional random information expressed as mark M i taking values in some metric space. To study the correlations between distinct points and between marks located at distinct points we use kernel-type estimators of the second-order product density and the mark covariance function of a spatially homogeneous marked point process. Both functions and their empirical counterparts are suitable characteristics to identify point process models by construction of statistical goodness-of-fit tests.