Clustering Comparison of Point Processes, with Applications to Random Geometric Models

Błaszczyszyn, Bartłomiej; Yogeshwaran, D.

doi:10.1007/978-3-319-10064-7_2

Cited by 39 publications

(50 citation statements)

References 61 publications

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“…More recently, the community has started to look at point processes that go far beyond the simplistic Poisson model. In particular, this includes sub-Poisson [4,5], Ginibre [12] and Gibbsian point processes [15,34].…”

Section: Introductionmentioning

confidence: 99%

Continuum percolation for Cox point processes

Hirsch

Jahnel

Cali

2019

Stochastic Processes and their Applications

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We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub-and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in largeradius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.celebrated concept of stabilization [17,[26][27][28]31] suffices to guarantee the existence of a subcritical phase. In contrast, for the existence of a super-critical phase, stabilization alone is not enough since percolation is impossible unless the support of the random measure has sufficiently good connectivity properties itself. Hence, our proof for the existence of a super-critical phase relies on a variant of the notion of asymptotic essential connectedness from [2].Second, when considering the Poisson point process, the high-density or large-radius limit of the percolation probability tends to 1 exponentially fast and is governed by the isolation probability. In the random environment, the picture is more subtle since the regime of a large radius is no longer equivalent to that of a high density. Since we rely on a refined largedeviation analysis, we assume that the random environment is not only stabilizing, but in fact b-dependent.Since the high-density and the large-radius limit are no longer equivalent, this opens up the door to an analysis of coupled limits. As we shall see, the regime of a large radius and low density is of highly averaging nature and therefore results in a universal limiting behavior. On the other hand, in the converse limit the geometric structure of the random environment remains visible in the limit. In particular, a different scaling balance between the radius and density is needed when dealing with absolutely continuous and singular random measures, respectively. Finally, we illustrate our results with specific examples and simulations. Model definition and main resultsLoosely speaking, Cox point processes are Poisson point processes in a random environment. More precisely, the random environment is given by a random element Λ in the space M of Borel measures on R d equipped with the usual evaluation σ-algebra. Throughout the manuscript we assume that Λ is stationary, but at this point we do not impose any additional conditions. In particular, Λ could be an absolutely continuous or singular random intensity measure. Nevertheless, in some of the presented results, completely different behavior will appear. Example 2.1 (Absolutely continuous environment). Let Λ(dx) = x dx with = { x } x∈R d a stationary non-negative random field. For example, this includes random measures modulated by a random closed set Ξ, [7, Section 5.2.2]. Here, x = λ 1 1{x ∈ Ξ} + λ 2 1{x ∈ Ξ} with λ 1 , λ 2 ≥ 0. Another example are random measures induced by shot-noise fields, [7...

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Section: Introductionmentioning

confidence: 99%

Continuum percolation for Cox point processes

Hirsch

Jahnel

Cali

2019

Stochastic Processes and their Applications

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“…From the definition of < sm we obtain directly that E(η(B 1 ) · · · η(B n ))) ≤ E(η(B 1 )) · · · E(η(B n )). Now from Proposition 1 in [5] we obtain that this implies E(exp( R d h(x)η(dx) ≤ exp( R d (e h(x) − 1)Eη(dx)), for all h ≥ 0. Regarding void probabilities, since (η(B 1 ) · · · η(B n )) is sNA for all bounded Borel sets B 1 , .…”

Section: Na and Dependence Orderings For Point Processesmentioning

confidence: 79%

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

Last

Szekli

2019

J. Appl. Probab.

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“…Our goal in this section is to prove a crucial covariance inequality and to deduce an α-mixing property for associated point processes. We recall that associated point processes are defined the following way (see Definitions 2.11-2.12 in [6] for example).…”

Section: Negative and Positive Associationmentioning

confidence: 99%

Mixing properties and central limit theorem for associated point processes

Poinas¹,

Delyon²,

Lavancier³

2019

Bernoulli

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Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove α-mixing properties for associated spatial point processes by controlling their α-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the α-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.We then establish in Section 3 a general central limit theorem (CLT) for random fields defined as a function of an associated point process (Theorem 3.1). A standard method for proving this kind of theorem is to rely on sufficiently fast decaying α-mixing coefficients along with some moment assumptions. We use an alternative procedure that exploits both the mixing properties and the association property. This results in weaker assumptions on the underlying point process, that can have slower decaying mixing coefficients. This improvement allows in particular to include all standard DPPs, some of them being otherwise excluded with the first approach (like for instance DPPs associated to the Bessel-type kernels [4]).Section 4 discusses in detail the case of DPPs, where we derive a tight explicit bound for their α-mixing coefficients and prove a central limit theorem for certain functionals of a DPP (Theorem 4.4). Specifically, these functionals write as a sum of a bounded function of the DPP, over subsets of close-enough points of the DPP. A particular case concerns sums over p-tuple of close enough points of the DPP, which are frequently encountered in asymptotic inference. Limit theorems in this setting have been established in [38] when p " 1, and in [3] for stationary DPPs and p ě 1. We thus extend these studies to sums over any subsets and without the stationary assumption. As a statistical application, we consider the parametric estimation of second-order intensity reweighted stationary DPPs. These DPPs have an inhomogeneous first order intensity, but translation-invariant higher order (reweighted) intensities. We prove that the two-step estimator introduced in [39], designed for this kind of inhomogeneous point process models, is consistent and asymptotically normal when applied to DPPs. Associated point processes and α-mixing NotationIn this paper, we consider locally finite simple point processes on R d , for a fixed d P N. Some theoretical background on point processes can be found in [12,34]. We denote by Ω the set of locally finite point configurations in R d . For X P Ω and A Ă R d , we write N pAq :" cardpX ∩ Aq for the random variable representing the number of points of X that fall in A. W...

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Clustering Comparison of Point Processes, with Applications to Random Geometric Models

Cited by 39 publications

References 61 publications

Continuum percolation for Cox point processes

Continuum percolation for Cox point processes

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

Mixing properties and central limit theorem for associated point processes

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