Mathematics – Probability
Scientific paper
2011-01-25
Mathematics
Probability
18 pages
Scientific paper
We show that the convex hull of a large i.i.d. sample from a non-vanishing log-concave distribution approximates a pre-determined body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For p-log-concave distributions with p>1 (such as the normal distribution where p=2) we also have approximation in the Hausdorff distance. These are multivariate versions of the Gnedenko law of large numbers which gaurantees concentration of the maximum and minimum in the one dimensional case. We give three different deterministic bodies that serve as approximants to the random body. The first is the floating body that serves as a multivariate quantile, the second body is given as a contour of the density function, and the third body is given in terms of the Radon transform. We end the paper by constructing a probability measure with an interesting universality property.
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