Translator Disclaimer
May 2022 Weak convergence of the intersection point process of Poisson hyperplanes
Anastas Baci, Gilles Bonnet, Christoph Thäle
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 58(2): 1208-1227 (May 2022). DOI: 10.1214/21-AIHP1201

Abstract

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 radius R>0 are considered. Taking R=tdd+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 power-law 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.

Cet article traite du processus ponctuel d’intersection d’un processus d’hyperplans de Poisson stationnaire et isotrope dans Rd d’intensité t>0, où seuls les hyperplans qui intersectent une boule centrée de rayon R>0 sont considérés. En prenant R=tdd+1, on montre que ce processus ponctuel converge en distribution, lorsque t, vers un processus ponctuel de Poisson dans Rd{0} dont la mesure d’intensité a une densité proportionnelle à x(d+1) par rapport à la mesure de Lebesgue. Une borne sur la vitesse de convergence en termes de distance de Kantorovich–Rubinstein est également fournie. En arrière-plan se trouve un théorème général d’approximation de Poisson fonctionnel sur les espaces de Poisson abstraits. Les implications sur la convergence faible de l’enveloppe convexe du processus de point d’intersection et la convergence de son vecteur f sont également discutées, réfutant et corrigeant ainsi une conjecture de Devroye et Toussaint (J. Algorithms 14 (1993) 381–394) en géométrie computationelle.

Funding Statement

AB and GB were supported by the German Research Foundation (DFG) via GRK 2131 “High-Dimensional Phenomena in Probability – Fluctuations and Discontinuity”.

Citation

Download Citation

Anastas Baci. Gilles Bonnet. Christoph Thäle. "Weak convergence of the intersection point process of Poisson hyperplanes." Ann. Inst. H. Poincaré Probab. Statist. 58 (2) 1208 - 1227, May 2022. https://doi.org/10.1214/21-AIHP1201

Information

Received: 13 August 2020; Revised: 13 May 2021; Accepted: 14 June 2021; Published: May 2022
First available in Project Euclid: 15 May 2022

Digital Object Identifier: 10.1214/21-AIHP1201

Subjects:
Primary: 60D05 , 60F05
Secondary: 52A22 , 53C65

Keywords: Convex hull , integral geometry , Intersection point process , Poisson hyperplane process , Poisson point process approximation , rate of convergence , weak convergence

Rights: Copyright © 2022 Association des Publications de l’Institut Henri Poincaré

JOURNAL ARTICLE
20 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.58 • No. 2 • May 2022
Back to Top