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Stationary partitions and Palm probabilities

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Günter Last
Günter Last*
Affiliation:
Universität Karlsruhe (TH)
*
∗ Postal address: Institut für Stochastik, Universität Karlsruhe (TH), 76128 Karlsruhe, Germany. Email address: last@math.uni-karlsruhe.de
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Abstract

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A stationary partition based on a stationary point process N in ℝd is an ℝd-valued random field π={π(x): x∈ℝd} such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd: π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

Keywords

Point processPalm probabilitystationary partitionstationary tessellationstochastic geometryshift coupling

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 60G60: Random fields
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 602 - 620
Copyright
Copyright © Applied Probability Trust 2006 

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