Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Schulte, Matthias; Thäle, Christoph

doi:10.1007/978-3-319-05233-5_8

Cited by 10 publications

(14 citation statements)

References 40 publications

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“…For the cells minimising the inradius, we show that asymptotically, m Wρ [r] has the same behaviour as the r-th smallest value associated with a carefully chosen U -statistic. This will allow us to apply the theorem in Schulte and Thäle [22]. The main difficulties we encounter will be in checking the conditions for their theorem, and to deal with boundary effects.…”

Section: Contributionsmentioning

confidence: 99%

“…As stated above, Schulte and Thäle established a general theorem to deal with U -statistcs (Theorem 1.1 in Schulte and Thäle [23]). In this work we make use of a new version of their theorem (to appear in Schulte and Thäle [22]), which we modify slightly to suit our requirements. Let g : A 3 → R be a measurable symmetric function and take m g,Wρ [r] to be the r-th smallest value of g(H 1:3 ) over all 3-tuples of lines H 1: (10).)…”

Section: Lemmamentioning

confidence: 99%

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Extremes for the inradius in the Poisson line tessellation

Chenavier

Hemsley

2016

Adv. Appl. Probab.

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A Poisson line tessellation is observed in the window Wρ := B(0, π −1/2 ρ 1/2 ), for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to infinity. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.The Poisson line tessellation LetX be a stationary and isotropic Poisson line process of intensitŷ γ = π in R 2 endowed with its scalar product ·, · and its Euclidean norm | · |. By A, we shall denote the set of affine lines which do not pass through the origin 0 ∈ R 2 . Each line can be written asfor some t ∈ R, u ∈ S, where S is the unit sphere in R 2 . When t > 0, this representation is unique. The intensity measure ofX is then given byfor all Borel subsets E ⊆ A, where A is endowed with the Fell topology (see for example Schneider and Weil [21], p563) and where σ(·) denotes the uniform measure on S with the normalisation σ(S) = 2π. The set of closures of the connected components of R 2 \X defines a stationary and isotropic random tessellation with intensity γ (2) = π (see for example (10.46) in Schneider and Weil [21]) which is the so-called Poisson line tessellation, m pht . By a slight abuse of notation, we also writeX to denote the union of lines. An example of the Poisson line tessellation in R 2 is depicted in Figure 1. Let B(z, r) denote the (closed) disc of radius r ∈ R + , centred at z ∈ R 2 and let K be the family of convex bodies (i.e. convex compact sets in R 2 with non-empty interior), endowed with the Hausdorff topology. With each convex body K ∈ K, we may now define the inradius,When there exists a unique z ∈ R 2 such that B(z , r(K)) ⊂ K, we define z(C) := z to be the incentre of K. If no such z exists, we take z(K) := 0 ∈ R 2 . Note that each cell C ∈ m pht has a unique z

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Section: Contributionsmentioning

confidence: 99%

Section: Lemmamentioning

confidence: 99%

Extremes for the inradius in the Poisson line tessellation

Chenavier

Hemsley

2016

Adv. Appl. Probab.

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“…The limiting distributions of the minimal distance between the points of a Poisson process and of large inradii and small circumscribed radii of Poisson-Voronoi tessellations have been studied before in e.g. [11,13,38,39]. Some of these works provide quantitative bounds for the difference of the distribution functions at a fixed u ∈ R, which depend on u.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Poisson approximation with applications to stochastic geometry

Pianoforte

Schulte

2021

Electron. J. Probab.

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This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference between the cumulative distribution functions. Special attention is dedicated to estimating the difference when the cumulative distribution functions are evaluated at 0. This permits to approximate the minimum (or maximum) of a collection of random variables by a suitable random variable in the Kolmogorov distance. The main theoretical results are obtained by combining the Chen-Stein method with size-bias coupling and a generalization of size-bias coupling for integer-valued random variables developed herein. A wide variety of applications are then discussed with a focus on stochastic geometry. In particular, transforms of the minimal circumscribed radius and the maximal inradius of Poisson-Voronoi tessellations as well as the minimal inter-point distance of the points of a Poisson process are considered and bounds for their Kolmogorov distances to extreme value distributions are derived.

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“…where M < ∞ and each f q is bounded and such that its support is contained in a rectangle of the type C × • • • × C, where C ∈ Z verifies μ(C) < ∞. Such a class of random variables contains most U -statistics that are relevant for geometric applications [see the surveys Lachièze-Rey and , Schulte and Thäle (2016) and the references therein], as well as nonlinear functionals of Volterra Lévy processes Peccati and Zheng (2010), , and the finite homogeneous sums in independent Poisson random variables considered in Peccati and Zheng (2014). A similar remark applies to the assumptions appearing in the statement of our main abstract bounds in Proposition 4.1 and Proposition 4.3.…”

Section: Main Results For Normal Approximationsmentioning

confidence: 99%

The fourth moment theorem on the Poisson space

Döbler

Peccati

2018

Ann. Probab.

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We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result-that has been elusive for several years-shows that the so-called 'fourth moment phenomenon', first discovered by Nualart and Peccati [Ann. Probab. 33 (2005) 177-193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein's method, Malliavin calculus and Mecketype formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [Ann. Probab. 40 (2012) 2439-2459 and Azmoodeh, Campese and Poly [J. Funct. Anal. 266 (2014) 2341-2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.

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Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Cited by 10 publications

References 40 publications

Extremes for the inradius in the Poisson line tessellation

Extremes for the inradius in the Poisson line tessellation

Poisson approximation with applications to stochastic geometry

The fourth moment theorem on the Poisson space

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