Asymptotic theory for statistics of the Poisson--Voronoi approximation

TL;DR

This paper develops asymptotic formulas for various geometric statistics of the Poisson--Voronoi approximation as the point process intensity grows, covering volume, surface area, and face counts.

Contribution

It introduces a general limit theorem framework for stabilizing functionals, enabling asymptotic analysis of complex geometric features in Voronoi approximations.

Findings

Expectation and variance asymptotics for volume and surface area

Limit theorems for face counts and Hausdorff measures

Analysis of Voronoi zone complexity and iterated approximations

Abstract

This paper establishes expectation and variance asymptotics for statistics of the Poisson--Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approximation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formula applicable in the setting of sums of stabilizing functionals.