Convex hulls of uniform samples from a convex polygon

TL;DR

This paper proves a central limit theorem for the number of vertices and remaining area of the convex hull of uniform samples from a convex polygon, extending previous results and clarifying variance relations.

Contribution

It derives a new representation linking the number of vertices and remaining area, and proves the central limit theorem for these quantities, correcting earlier scaling constants.

Findings

Central limit theorem for vertices and area of convex hulls

Representation of remaining area as exponential variables

Correction of previous scaling constants

Abstract

In Groeneboom (1988) a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of convex polygon was derived. In the unpublished preprint Nagaev and Khamdamov (1991) (in Russian) a central limit result for the joint distribution of the number of vertices and the remaining area is given, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of the number of vertices and the remaining area in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Nagaev (1995) and Cabo and Groeneboom (1994).