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Nearest-neighbour Markov point processes on graphs with Euclidean edges

Part of:Stochastic processesGeometric probability and stochastic geometry

Published online by Cambridge University Press:  29 November 2018

M. N. M. van Lieshout
M. N. M. van Lieshout*
Affiliation:
CWI and University of Twente
*
* Postal address: CWI, PO Box 94079, NL-1090 GB Amsterdam, The Netherlands. Email address: marie-colette.van.lieshout@cwi.nl
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Abstract

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We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

Keywords

Delaunay neighbourgraph with Euclidean edgeslinear networkMarkov point processnearest-neighbour interactionrenewal process

MSC classification

Primary:60G55: Point processes
Secondary:60D05: Geometric probability and stochastic geometry

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018, pp. 1275 - 1293
Copyright
Copyright © Applied Probability Trust 2018 

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