Extreme values for characteristic radii of a Poisson-Voronoi Tessellation

Calka, Pierre; Chenavier, Nicolas

doi:10.1007/s10687-014-0184-y

Cited by 35 publications

(57 citation statements)

References 51 publications

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“…are i.i.d. uniformly distributed on the unit cube [0, 1] d , then (2a) and (2c) of Theorem 1 in Calka and Chenavier [3] denotes the distribution function of the Gumbel extreme value distribution. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

The limit distribution of the maximum probability nearest-neighbour ball

2019

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Let X 1 , . . . , X n be independent random points drawn from an absolutely continuous probability measure with density f in R d . Under mild conditions on f , we derive a Poisson limit theorem for the number of large probability nearest neighbor balls. Denoting by P n the maximum probability measure of nearest neighbor balls, this limit theorem implies a Gumbel extreme value distribution for nP n − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nP n − ln n, which does not depend on f .

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Section: Introductionmentioning

confidence: 99%

The limit distribution of the maximum probability nearest-neighbour ball

2019

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“…Lemmas concerning an upper bound for b 2 The arguments showing that b 2 converges to 0 are mainly inspired by [4], and they rely on a "local condition", i.e. on an upper bound for the probability that the incircles of two cells with centers in a short distance simultaneously exceed the threshold v ρ .…”

Section: Technical Lemmasmentioning

confidence: 99%

“…Our paper complements [2,3] where Poisson-Voronoi tessellations are treated. However, although Theorem 1 in [3] is a general result on extremes for random tessellations, the conditions of this theorem are too restrictive to be applied to STIT tessellations.…”

Section: Introductionmentioning

confidence: 97%

The largest order statistics for the inradius in an isotropic STIT tessellation

Chenavier

Nagel

2019

Extremes

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A planar stationary and isotropic STIT tessellation at time t > 0 is observed in the window Wρ = t −1 √ π ρ · [− 1 2 , 1 2 ] 2 , for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest disk contained in the cell. Using the Chen-Stein method, we compute the limit distributions of the largest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to infinity.This notation N B (v) is consistent with (1.3).Moreover, for any finite set A, we denote by |A| the number of elements of A. For any real number x, we denote by x the integer part of x. Given a probability space (Ω, E, P), and two random variables X and Y defined on Ω, the notation X d = Y indicates that X and Y have the same distribution. We write E [ Y |X ] for the conditional expectation of Y with respect to X.

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“…Here, we have written "r(C) < v ρ (τ )" instead of "r(C) > v ρ (τ )" in the probability because we consider the smallest inradii. Moreover, according to [7], we know that…”

Section: Reciprocal Of the Inradiusmentioning

confidence: 99%

Cluster size distributions of extreme values for the Poisson–Voronoi tessellation

Chenavier¹,

Robert²

2018

Ann. Appl. Probab.

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We consider the Voronoi tessellation based on a homogeneous Poisson point process in R d . For a geometric characteristic of the cells (e.g. the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson-Delaunay tessellation is also discussed.where #S denotes the cardinality of any finite set S. Thanks to the stationarity of m, the intensity does not depend on the choice of A. Without loss of generality, we assume that γ m = 1.The typical cell C of a stationary tessellation m is a random polytope with distribution given by(1.1)where f : K d → R is any bounded measurable function on the set of convex bodies K d (endowed with the Hausdorff topology). Let χ be a locally finite subset of R d . The Voronoi cell with nucleus x ∈ χ is the set of all sites y ∈ R d whose distance from x is smaller or equal than the distances to all other points of χ, i.e. C χ (x) := {y ∈ R d : |y − x| ≤ |y − x |, x ∈ χ}. When χ = η is a homogeneous Poisson point process, the family m := {C η (x) : x ∈ η} is the so-called Poisson-Voronoi tessellation. The intensity of such a tessellation equals the intensity of η. A consequence of the theorem of Slivnyak (see e.g. Theorem 3.3.5 in [28]) shows that C

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Extreme values for characteristic radii of a Poisson-Voronoi Tessellation

Cited by 35 publications

References 51 publications

The limit distribution of the maximum probability nearest-neighbour ball

The limit distribution of the maximum probability nearest-neighbour ball

The largest order statistics for the inradius in an isotropic STIT tessellation

Cluster size distributions of extreme values for the Poisson–Voronoi tessellation

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