The scaling limit of Poisson-driven order statistics with applications in geometric probability

Abstract

Let eta(t) be a Poisson point process of intensity t >= 1 on some state space Y and let f be a non-negative symmetric function on Y-k for some k >= 1. Applying f to all k-tuples of distinct points of eta(t) generates a point process xi(t) on the positive real half-axis. The scaling limit of xi(t) as t tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m-th smallest point of xi(t) is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener-Ito chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen-Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry. (C) 2012 Elsevier B.V. All rights reserved.