On existence and mixing properties of germ-grain models

Abstract

We give a rigorous definition of germ-grain models (ggm's) which were introduced in [6] as at most countable unions of random closed sets (called grains) in translated by the atoms (called germs) of a point process in , and establish conditions under which the random set Z in a.s. closed. In case of i.i.d. grains we prove a continuity theorem for ggm's in terms of weak convergence. Further, we characterize ergodicity and (weak) mixing of stationary ggm's with a.s. compact grains by the corresponding properties of the underlying stationary point process. As a consequence we apply an ergodic theorem of Nguyen and Zessin [9] to spatial averages of certain geometric functionals of ggm's with a.s. compact convex grains.

AMS 1991 subject classifications::

• Primary 60 G55
• 60 G60
• secondary 60 G57

Keywords:

• Random closed sets
• point processes
• hitting function
• weak convergence
• ergodicity
• mixing