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September 2013 A multivariate Gnedenko law of large numbers
Daniel Fresen
Ann. Probab. 41(5): 3051-3080 (September 2013). DOI: 10.1214/12-AOP804


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.


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Daniel Fresen. "A multivariate Gnedenko law of large numbers." Ann. Probab. 41 (5) 3051 - 3080, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1293.60012
MathSciNet: MR3127874
Digital Object Identifier: 10.1214/12-AOP804

Primary: 60D05 , 60F99
Secondary: 52A20 , 52A22 , 52B11

Keywords: Law of Large Numbers , log-concave , random polytope

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 5 • September 2013
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