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On existence and mixing properties of germ-grain models. (English) Zbl 0811.60034
Summary: We give a rigorous definition of germ-grain models (ggm’s) which were introduced by K.-H. Hanisch [Serdica 7, 160-166 (1981; Zbl 0493.60020)] as at most countable unions \(Z= \bigcup_{i\in\mathbb{N}} (X_ i+ Z_ i)\) of random closed sets \(\{Z_ i\), \(i\in\mathbb{N}\}\) (called grains) in \(\mathbb{R}^ d\) translated by the atoms \(\{X_ i\), \(i\in\mathbb{N}\}\) (called germs) of a point process in \(\mathbb{R}^ d\), and establish conditions under which the random set \(Z\) is 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 X. X. Nguyen and H. Zessin [Z. Wahrscheinlichkeitstheorie Verw. Geb. 48, 133-158 (1979; Zbl 0397.60080)] to spatial averages of certain geometric functionals of ggm’s with a.s. compact convex grains.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05 Geometric probability and stochastic geometry
60G60 Random fields
60G57 Random measures
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