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Comparisons and asymptotics for empty space hazard functions of germ-grain models

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Günter Last  and
Ryszard Szekli
Günter Last*
Affiliation:
Karlsruher Institut für Technologie
Ryszard Szekli*
Affiliation:
University of Wrocław
*
∗ Postal address: Institut für Stochastik, Karlsruher Institut für Technologie, 76128 Karlsruhe, Germany. Email address: [email protected]
∗∗ Postal address: Mathematical Institute, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
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Abstract

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We study stochastic properties of the empty space for stationary germ-grain models in Rd; in particular, we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower dimensional. We consider Poisson cluster and mixed Poisson germ-grain models, and show in several situations that more variability results in stochastically greater empty space in terms of the empty space hazard function. Furthermore, we study the asymptotic behaviour of the empty space hazard functions at 0 and at ∞.

Keywords

Germ-grain modelempty space hazardrelative distancepoint processPoisson cluster processmixed Poisson processhazard rate ordering

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 943 - 962
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

This work was supported by MNiSW Research Grant N N201 394137.

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