Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T18:11:28.127Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary partitions and Palm probabilities

Part of: Stochastic processes Geometric probability and stochastic geometry

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Heveling, M. and Last, G. (2005). Characterization of Palm measures via bijective point-shifts. Ann. Prob. 33, 1698–1715.Google Scholar
Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Prob. 33, 31–52.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. John Wiley, Chichester.Google Scholar
Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
Mecke, J. (1975). Invarianzeigenschaften allgemeiner Palmscher Masse. Math. Nachr. 65, 335–344.CrossRefGoogle Scholar
Neveu, J. (1977). Processus ponctuels. In École d'Été de Probabilités de Saint-Flour, VI (Lecture Notes Math. 598). Springer, Berlin, pp. 249–445.Google Scholar
Nieuwenhuis, G. (1989). Equivalence of functional limit theorems for stationary point processes and their Palm distributions. Prob. Theory Relat. Fields 81, 593–608.Google Scholar
Nieuwenhuis, G. (1994). Bridging the gap between a stationary point process and its Palm distribution. Statist. Neerlandica 48, 37–62.Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Prob. 24, 2057–2064.CrossRefGoogle Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar