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On comparison of clustering properties of point processes. (English) Zbl 1295.60059

The usual statistical approach to the study of clustering in point processes consists of the evaluation of Ripley’s \(K\)-function, pair-correlation function, or contact distribution function (also called the empty space function). However, such a comparison of local characteristics seems a weak tool for the study of the impact of clustering on some macroscopic properties of point processes such as those required in continuum percolation models. It was observed in [B. Błaszczyszyn and D. Yogeshwaran, Adv. Appl. Probab. 41, No. 3, 623–646 (2009; Zbl 1181.60022)] that the directionally convex (DCX) order on point process implies the ordering of \(K\)-functions as well as pair-correlation functions, in the sense that point processes in the DCX order have larger \(K\)-functions and pair-correlation functions, while having the same mean number of points in any given set. Unfortunately, the examples from [Zbl 1181.60022] are mostly only some doubly stochastic Poisson point processes, which are DCX larger than Poisson (called super-Poisson). In order to provide more examples of DCX ordered point processes, in particular those that are smaller than Poisson (called sub-Poisson), the authors study in this paper a notion of perturbation of a point process consisting of independent replication and translation of points from some given original point process. A key observation is that such a perturbation is DCX monotone with respect to the convex order on the number of point replications. In particular, perturbing a deterministic lattice in the above sense, give examples of both sub- and super- Poisson point processes, with the Poisson itself obtained when the number of point replications has a Poisson distribution. In this paper, the authors use these examples to illustrate the aforementioned heuristic on the impact of clustering on the percolation of Boolean models. However, many examples of point processes considered as clustering less or more than the Poisson point process of the same intensity escape from the DCX comparison. The rest of the paper is organized as follows. The necessary notions, notations, and basic facts are introduced and recalled in Section 2. In Section 3, the authors define classes of strongly and weakly sub- and super-Poisson point processes and as a main result, they prove that weak sub-Poissonianity or super-Poissonianity is implied by negative or positive association, respectively. The authors study the perturbed-lattice point process in Section 4 and determinantal and permanental point processes in Section 5. Section 6 discusses some further theoretical implications (especially percolation) of the present ideas as well as their connections to other stochastic geometric models and modeling applications.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60E15 Inequalities; stochastic orderings
60D05 Geometric probability and stochastic geometry
60G60 Random fields

Citations:

Zbl 1181.60022

References:

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