On existence and mixing properties of germ-grain models

摘要

We give a rigorous definition of germ-grain models (ggm's) which were introduced in 6 as at most countable unionsof random closed sets(called grains) intranslated by the atoms(called germs) of a point process in, and establish conditions under which the random setZin a.s. closed. In case ofi.i.d. grains we prove a continuity theorem forggm'sin terms of weak convergence. Further, we characterize ergodicity and (weak) mixing of stationaryggm'swith 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 ofggm'swith a.s. compact convex grains.