A MULTIVARIATE GNEDENKO LAW OF LARGE NUMBERS

• 成果类型: Article
• 署名作者: Fresen, Daniel
• 署名单位: Yale University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP804
• 发表日期: 2013
• 页码: 3051-3080
• 关键词: convex; geometry; points; volume

摘要

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.