On the empty space function of some germ-grain models

Abstract

We derive and discuss formulas for the density and the hazard rate of the empty space function of a germ-grain model Ξ in R d generated by a stationary point process Φ and i.i.d. convex primary grains Ξ n, n∈ N , that are independent of Φ. Our formulas are based on the Palm probability of the germ process and the mean generalized curvature measure of the grain. Particular attention is paid to cluster models, where the grains form a Poisson cluster process. Our discussion of specific Gauss–Poisson models with spherical grains provides some motivation for the use of the failure rate of F to detect clustering effects. In the general case we propose a family of functions comparing the behaviour in the neighbourhood of a typical germ with the neighbourhood of an arbitrary point in space. These characteristics can be used to measure effects of clustering and spatial interactions between the locations of the individual grains.