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Gaussian polytopes: variances and limit theorems

Part of: Geometric probability and stochastic geometry Multivariate analysis General convexity Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Daniel Hug  and
Matthias Reitzner
Daniel Hug*
Affiliation:
Albert-Ludwigs-Universität Freiburg
Matthias Reitzner*
Affiliation:
Technische Universität Wien
*
∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstr. 1, D-79104 Freiburg i. Br., Germany. Email address: daniel.hug@math.uni-freiburg.de
∗∗ Postal address: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria. Email address: mreitzne@mail.zserv.tuwien.ac.at
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Abstract

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The convex hull of n independent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the variance of the ith intrinsic volume of a Gaussian polytope in ℝd, d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number of i-faces and for the ith intrinsic volume of a Gaussian polytope as n→∞.

Keywords

Random pointconvex hullf-vectorintrinsic volumegeometric probabilitynormal distributionGaussian samplestochastic geometryvariancelaw of large numberslimit theorem

MSC classification

Primary: 52A22: Random convex sets and integral geometry 60D05: Geometric probability and stochastic geometry
Secondary: 60C05: Combinatorial probability 62H10: Distribution of statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 297 - 320
Copyright
Copyright © Applied Probability Trust 2005 

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