The scaling limit of Poisson-driven order statistics with applications in geometric probability

Schulte, Matthias; Thäle, Christoph

doi:10.1016/j.spa.2012.08.011

Cited by 47 publications

(89 citation statements)

References 38 publications

(106 reference statements)

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“…Our last example deals with non-intersecting k-flats. The result generalizes Example 1 in [30] and one of the results in [31] to arbitrary distance powers a > 0.…”

Section: Examplessupporting

confidence: 80%

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Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Schulte

Thäle

2016

Stochastic Analysis for Poisson Point Processes

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Let η t be a Poisson point process with intensity measure tµ, t > 0, over a Borel space X, where µ is a fixed measure. Another point process ξ t on the real line is constructed by applying a symmetric function f to every k-tuple of distinct points of η t . It is shown that ξ t behaves after appropriate rescaling like a Poisson point process, as t → ∞, under suitable conditions on η t and f . This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints and non-intersecting k-flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process.

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“…Our last example deals with non-intersecting k-flats. The result generalizes Example 1 in [30] and one of the results in [31] to arbitrary distance powers a > 0.…”

Section: Examplessupporting

confidence: 80%

“…To motivate our general theory, let us describe a particular situation to which our results can be applied (see Remark 4 and also Example 4 in [30] for more details). Let K be a convex body in R d , d ≥ 2, (that is a compact convex set with interior points) whose volume is denoted by ℓ d (K).…”

Section: Introductionmentioning

confidence: 99%

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Schulte

Thäle

2016

Stochastic Analysis for Poisson Point Processes

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“…Scaling limits of order statistics of Poisson PPs has garnered some interest in recent times; see [30]. Our techniques can easily yield that first moments of the order statistics are ordered for α-weakly ordered PPs but the question of further asymptotics remains open.…”

Section: First Passage Percolationmentioning

confidence: 99%

On Comparison of Clustering Properties of Point Processes

Błaszczyszyn

Yogeshwaran

2014

Adv. Appl. Probab.

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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.

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“…The standard literature on EVT includes [31,18,16,32]. In recent years many attempts have been made at understanding the geometric and topological features of multivariate extremes, among them [4,5,33,15] as well as [1,29] cited above.…”

Section: Introductionmentioning

confidence: 99%

Convergence of persistence diagrams for topological crackle

Owada¹,

Bobrowski²

2020

Bernoulli

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In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of homological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to homological cycles generated by "noisy" samples where the support is unbounded. We aim to establish weak convergence results for persistence diagramsa point process representation for persistent homology, where each homological cycle is represented by its (birth, death) coordinates. In this work we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. In this framework we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a short lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.2010 Mathematics Subject Classification. Primary 60G70. Secondary 55U10, 60F05, 60G55.

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The scaling limit of Poisson-driven order statistics with applications in geometric probability

Cited by 47 publications

References 38 publications

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

On Comparison of Clustering Properties of Point Processes

Convergence of persistence diagrams for topological crackle

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