The Annals of Probability

A multivariate Gnedenko law of large numbers

Daniel Fresen

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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.

Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3051-3080.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991831

Digital Object Identifier
doi:10.1214/12-AOP804

Mathematical Reviews number (MathSciNet)
MR3127874

Zentralblatt MATH identifier
1293.60012

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F99: None of the above, but in this section
Secondary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 52B11: $n$-dimensional polytopes

Keywords
Random polytope log-concave law of large numbers

Citation

Fresen, Daniel. A multivariate Gnedenko law of large numbers. Ann. Probab. 41 (2013), no. 5, 3051--3080. doi:10.1214/12-AOP804. https://projecteuclid.org/euclid.aop/1378991831


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