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Large deviation probabilities for the number of vertices of random polytopes in the ball

Part of: Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  01 July 2016

Pierre Calka  and
Tomasz Schreiber
Pierre Calka*
Affiliation:
Université René Descartes Paris 5
Tomasz Schreiber*
Affiliation:
Nicolaus Copernicus University, Toruń
*
∗ Postal address: Université René Descartes Paris 5, MAP5, UFR Math-Info, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France. Email address: pierre.calka@math-info.univ-paris5.fr
∗∗ Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. Email address: tomeks@mat.uni.torun.pl
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Abstract

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In this paper we establish large deviation results on the number of extreme points of a homogeneous Poisson point process in the unit ball of Rd. In particular, we deduce an almost-sure law of large numbers in any dimension. As an auxiliary result we prove strong localization of the extreme points in an annulus near the boundary of the ball.

Keywords

Convex hullcovering of the spherelarge deviationmeasure concentrationrandom polytope

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F10: Large deviations
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 47 - 58
Copyright
Copyright © Applied Probability Trust 2006 

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