A multivariate Gnedenko law of large numbers
Abstract
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.
Keywords
Cite
@article{arxiv.1101.4887,
title = {A multivariate Gnedenko law of large numbers},
author = {Daniel Fresen},
journal= {arXiv preprint arXiv:1101.4887},
year = {2013}
}
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)