# On comparison of clustering properties of point processes

1 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
Abstract : 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 and preview further ones, showing that the new tool is relevant in the study of macroscopic, percolative properties of point processes. This new comparison is also implied by the directionally convex ($dcx$ ordering of point processes, which has already been shown to be relevant to comparison of 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 process 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 not realistic assumptions.
Type de document :
Article dans une revue
Advances in Applied Probability, Applied Probability Trust, 2014, 46 (1), pp.1--21. 〈10.1239/aap/1396360100〉
Domaine :

https://hal.inria.fr/hal-00651480
Contributeur : Bartlomiej Blaszczyszyn <>
Soumis le : mardi 13 décembre 2011 - 16:39:24
Dernière modification le : vendredi 25 mai 2018 - 12:02:06

### Citation

Bartlomiej Blaszczyszyn, Dhandapani Yogeshwaran. On comparison of clustering properties of point processes. Advances in Applied Probability, Applied Probability Trust, 2014, 46 (1), pp.1--21. 〈10.1239/aap/1396360100〉. 〈hal-00651480〉

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