Weak convergence of the intersection point process of Poisson hyperplanes

Anastas Baci, Gilles Bonnet, Christoph Thäle

OnderzoeksoutputAcademicpeer review

1 Citaat (Scopus)

Samenvatting

This paper deals with the intersection point process of a stationary and isotropic Poisson hyperplane process in Rd of intensity t > 0, where only hyperplanes that intersect a centred ball of radiusR >0 are considered. Taking R = t - d d+1 it is shown that this point process converges in distribution, as t →∞, to a Poisson point process on Rd \ {0} whose intensity measure has powerlaw density proportional to ∥x∥-(d+1) with respect to the Lebesgue measure. A bound on the speed of convergence in terms of the Kantorovich-Rubinstein distance is provided as well. In the background is a general functional Poisson approximation theorem on abstract Poisson spaces. Implications on the weak convergence of the convex hull of the intersection point process and the convergence of its f -vector are also discussed, disproving and correcting thereby a conjecture of Devroye and Toussaint (J. Algorithms 14 (1993) 381-394) in computational geometry.

Originele taal-2English
Pagina's (van-tot)1208-1227
Aantal pagina's20
TijdschriftAnnales de l institut henri poincare-Probabilites et statistiques
Volume58
Nummer van het tijdschrift2
DOI's
StatusPublished - mei-2022
Extern gepubliceerdJa

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