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Random Marked Sets

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

F. Ballani ,
Z. Kabluchko  and
M. Schlather
F. Ballani*
Affiliation:
TU Bergakademie Freiberg
Z. Kabluchko*
Affiliation:
Universität Ulm
M. Schlather*
Affiliation:
Universität Göttingen
*
∗ Postal address: Institut für Stochastik, TU Bergakademie Freiberg, D-09596 Freiberg, Germany. Email address: ballani@math.tu-freiberg.de
∗∗ Postal address: Institut für Stochastik, Universität Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany. Email address: zakhar.kabluchko@uni-ulm.de
∗∗∗ Current address: Institut für Mathematik, Universität Mannheim, A5, 6, D-68131 Mannheim, Germany. Email address: schlather@math.uni-mannheim.de
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Abstract

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We aim to link random fields and marked point processes, and, therefore, introduce a new class of stochastic processes which are defined on a random set in . Unlike for random fields, the mark covariance function of a random marked set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.

Keywords

Random fieldrandom setmarked point processmark correlation functionmark covariance function

MSC classification

Primary: 60G60: Random fields 60G55: Point processes
Secondary: 60G15: Gaussian processes 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 603 - 616
Copyright
© Applied Probability Trust 

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