## Advances in Applied Probability

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

### On comparison of clustering properties of point processes

Bartłomiej Błaszczyszyn and D. Yogeshwaran

#### 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