A multivariate Gnedenko law of large numbers

TL;DR

This paper extends the Gnedenko law of large numbers to multivariate settings, showing that the convex hull of large i.i.d. samples from log-concave distributions approximates a convex body in various distances, with quantitative bounds.

Contribution

It introduces multivariate versions of the Gnedenko law of large numbers for log-concave distributions, providing approximation results and bounds in multiple distances.

Findings

Convex hulls approximate a convex body in logarithmic Hausdorff distance.

Approximation also holds in Banach-Mazur distance.

Quantitative bounds depend on sample size and dimension.

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.