Supporting-points processes and some of their applications

Abstract

We introduce a stochastic point process of S-supporting points and prove that upon rescaling it converges to a Gaussian field. The notion of S-supporting points specializes (for adequately chosen S) to Pareto (or, more generally, cone) extremal points or to vertices of convex hulls or to centers of generalized Voronoi tessellations in the models of large scale structure of the Universe based on Burgers equation. The central limit theorems proven here imply i.a. the asymptotic normality for the number of convex hull vertices in large Poisson sample from a simple polyhedra or for the number of Pareto (vector extremal) points in Poisson samples with independent coordinates.