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

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.

Citation

Download Citation

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

Information

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

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

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

Rights: Copyright © 2013 Institute of Mathematical Statistics

JOURNAL ARTICLE
30 PAGES


SHARE
Vol.41 • No. 5 • September 2013
Back to Top