Daniel Fresen
Published 2011-01-25, updated 2013-10-21Version 2
We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case. We provide quantitative bounds in terms of the number of points and the dimension of the ambient space.
Comments: Published in at http://dx.doi.org/10.1214/12-AOP804 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Journal: Annals of Probability 2013, Vol. 41, No. 5, 3051-3080
Strong laws of large numbers for capacities
Fragmentation processes and the convex hull of the Brownian motion in a disk
A particle system with explosions: law of large numbers for the density of particles and the blow-up time
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