On Comparison of Clustering Properties of Point Processes

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.

On negative association of some finite point processes on general state spaces

“…From the definition of the log-concave ordering < lc in [46] it follows that η K (B) < lc Y , where Y is described above, which in turn implies that η K (B) < cx Y , see Theorem 1 in [46]. The above corollary for the case of jointly observable sets and X = R d was observed by Blaszczyszyn and Yogeshwaran in [4], Proposition 5.3, using a different argument. for any t, a > 0, and the upper bounds can be replaced by the values taken from the dominating in < dcx process.…”

“…Directly from the definition of sNA we conclude that P(η(B) = 0), η(B ′ ) = 0) ≤ P(η(B) = 0)P(η(B ′ ) = 0). Now, from Proposition 3.1 in [4] we get that P(η(B) = 0) ≤ exp( −Eη(B) 1 (B)). Then…”

Robustness of interdependent random geometric networks

“…The critical node density of a clustered point process is not the same as the critical node density of the homogeneous Poisson point process for percolation. More detailed study on the percolation of a clustered point process can be found in [21]. Fig.…”

On the Effect of Shadowing Correlation on Wireless Network Performance

“…To quantify the impact of clustering properties among point processes, the directionally convex order on point processes [24] and the properties of positive and negative association [25], [26] have been proposed. This was for instance used to compare certain point processes with the Poisson point process [27].…”