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-2 | English |
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Pagina's (van-tot) | 1208-1227 |
Aantal pagina's | 20 |
Tijdschrift | Annales de l institut henri poincare-Probabilites et statistiques |
Volume | 58 |
Nummer van het tijdschrift | 2 |
DOI's | |
Status | Published - mei-2022 |
Extern gepubliceerd | Ja |