Weak Laws in Geometric Probability

TL;DR

This paper proves a general weak law of large numbers for functionals of binomial point processes in multiple dimensions, with applications to various geometric graphs and models, highlighting the dependence on point density.

Contribution

It introduces a coupling-based approach to establish weak laws for geometric functionals, accommodating non-uniform densities and diverse graph structures.

Findings

Weak laws for total edge length and degree counts in geometric graphs

Application to Boolean model statistics and marked point processes

Explicit dependence on point process density

Abstract

Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly non-uniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph, and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specifed degree, and number of components. We also obtain weak laws for functionals of marked point processes, including statistics of Boolean models.