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Generalized contact distributions of inhomogeneous Boolean models

Part of: Normed linear spaces and Banach spaces; Banach lattices Global differential geometry General convexity Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  19 February 2016

Daniel Hug ,
Günter Last  and
Wolfgang Weil
Daniel Hug*
Affiliation:
Albert-Ludwigs-Universität, Freiburg
Günter Last*
Affiliation:
Universität Karlsruhe
Wolfgang Weil*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstr. 1, D-79104 Freiburg i. Br., Germany.
∗∗ Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe (TH), Englerstr. 2, D-76128 Karlsruhe, Germany. Email address: [email protected]
∗∗∗ Mathematisches Institut II, Universität Karlsruhe (TH), Englerstr. 2, D-76128 Karlsruhe, Germany.
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Abstract

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

Keywords

Stochastic geometrycontact distribution functiongerm-grain modelBoolean modelsupport (curvature, surface area) measuremarked point processPalm probabilitiesrandom measure

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G57: Random measures 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 53C65: Integral geometry 46B20: Geometry and structure of normed linear spaces
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 21 - 47
Copyright
Copyright © Applied Probability Trust 2002 

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