Matérn's hard core models of types I and IIwith arbitrary compact grains

By Markus Kiderlen and Mario Hörig
CSGB Research Reports
No. 05, July 2013

Matérn's classical hard core models can be interpreted as models obtained from a stationary marked Poisson process by dependent thinning. The marks are balls of fixed radius, and a point is retained when its associated ball does not hit any other balls (type I) or when its random birth time is strictly smaller than the birth times of all balls hitting it (type II). Extending ideas of [M. Månsson and M. Rudemo. Random patterns of nonoverlapping convex grains. Adv. in Appl. Probab., 34:718--738, 2002.], who considered grains that are isotropic rotations or random scalings of a fixed convex set, we discuss these two models in $d$-dimensional space when the marks are arbitrary random compact grains. We determine the intensity and the mark distribution after thinning, and find the second order factorial moment density of the ground process for model II under weak additional assumptions. By Brunn-Minkowski's inequality, the volume density associated to model II turns out to be bounded by $2^{-d}$. This bound is sharp. It is attained asymptotically (when the proposal intensity tends to infinity) only when all grains coincide with one deterministic origin-symmetric convex set. We also discuss how known connections of this model with the process of intact grains of the dead leaves model and the Stienen model leads to analogous results for the latter.

Keywords: Matern hard core models of types I and II, Palm distribution, dependently thinned Poisson point process, germ grain model, soft core particle process, dead leaves model, Stienen model, volume density

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