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On Comparison of Clustering Properties of Point Processes

Part of: Geometric probability and stochastic geometry Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  22 February 2016

Bartłomiej Błaszczyszyn  and
D. Yogeshwaran
Bartłomiej Błaszczyszyn*
Affiliation:
INRIA/ENS
D. Yogeshwaran*
Affiliation:
Technion, Israel Institute of Technology
*
∗ Postal address: INRIA/ENS, 23 Av. d'Italie, 75214 Paris Cedex 13, France. Email address: bartek.blaszczyszyn@ens.fr
∗∗ Postal address: Department of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, 32000, Israel. Email address: yogesh@ee.technion.ac.il
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Abstract

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In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure. We provide some motivating results on percolation and coverage processes, and preview further ones on other stochastic geometric models, such as minimal spanning forests, Lilypond growth models, and random simplicial complexes, showing that the new tool is relevant for a systemic approach to the study of macroscopic properties of non-Poisson point processes. This new comparison is also implied by the directionally convex ordering of point processes, which has already been shown to be relevant to the comparison of the spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point processes as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity nor the total independence property are realistic assumptions.

Keywords

Point processclusteringdirectionally convex orderingassociationperturbed latticedeterminantal point processpermanental point processsub-Poisson point processsuper-Poisson point process

MSC classification

Primary: 60G55: Point processes 60E15: Inequalities; stochastic orderings
Secondary: 60D05: Geometric probability and stochastic geometry 60G60: Random fields
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 1 - 20
Copyright
© Applied Probability Trust 

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