On joint probability distribution of the number of vertices and area of the convex hulls generated by a Poisson point process

Abstract

Consider a convex hull generated by a homogeneous Poisson point process in a cone in the plane. In the present paper the central limit theorem is proved for the joint probability distribution of the number of vertices and the area of a convex hull in a cone bounded by the disk of radius T (the center of the disk is at the cone vertex), for T→∞. From the results of the present paper the previously known results of Groeneboom (1988) and Cabo and Groeneboom (1994) are followed, in which the central limit theorem was proved for the number of vertices and the area of the convex hull in a square by approximating the binomial point process by a homogeneous Poisson point process.