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The Palm-duality for random subsets of d-dimensional grids

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Hermann Thorisson
Hermann Thorisson*
Affiliation:
University of Iceland
*
∗ Postal address: Department of Mathematics, University of Iceland, 107 Reykjavik, Iceland. Email address: hermann@hi.is
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Abstract

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The Palm version of a stationary random subset of a d-dimensional grid is contructed using the two-step change-of-origin and change-of-measure method. An elementary proof is given of the fact that the Palm version is characterized by point-stationarity (distributional invariance under bijective shifts of the origin from a point of the set to another point of the set).

Keywords

Palm theorypoint processesstationary random field

MSC classification

Primary: 60G55: Point processes
Secondary: 60G60: Random fields
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 318 - 325
Copyright
Copyright © Applied Probability Trust 2007 

References

Daley, D. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, Berlin.Google Scholar
Ferrari, P. A., Landim, C. and Thorisson, H. (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Prob. Statist. 40, 141–152.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. (2003). Trees and matchings from point processes. Electron. Commun. Prob. 8, 17–27.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. John Wiley, Chichester.Google Scholar
Mecke, J. (1975). Invarianzeigenschaften allgemeiner Palmscher Mase. Math. Nachr. 65, 335–344.Google Scholar
Neveu, J. (1977). Processus ponctuels. In École d'Été de Probabilités de Saint-Flour VI–1976 (Lecture Notes Math. 598), Springer, Berlin, pp. 249–445.Google Scholar
Thorisson, H. (1999). Point-stationarity in d dimensions and Palm theory. Bernoulli 5, 797–831.CrossRefGoogle Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar
Timar, A. (2004). Tree and grid factors of general point processes, Electron. Commun. Prob. 9, 53–59.Google Scholar