Advances in Applied Probability

On comparison of clustering properties of point processes

Bartłomiej Błaszczyszyn and D. Yogeshwaran

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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 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.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 1 (2014), 1-20.

Dates
First available in Project Euclid: 1 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1396360100

Digital Object Identifier
doi:10.1239/aap/1396360100

Mathematical Reviews number (MathSciNet)
MR3189045

Zentralblatt MATH identifier
1295.60059

Subjects
Primary: 60G55: Point processes 60E15: Inequalities; stochastic orderings
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G60: Random fields

Keywords
Point process clustering directionally convex ordering association perturbed lattice determinantal point process permanental point process sub-Poisson point process super-Poisson point process

Citation

Błaszczyszyn, Bartłomiej; Yogeshwaran, D. On comparison of clustering properties of point processes. Adv. in Appl. Probab. 46 (2014), no. 1, 1--20. doi:10.1239/aap/1396360100. https://projecteuclid.org/euclid.aap/1396360100


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