Almost Sure Central Limit Theorems in Stochastic Geometry

TL;DR

This paper establishes almost sure central limit theorems on the Poisson space, specifically tailored for stabilizing functionals in stochastic geometry, providing new probabilistic convergence results for geometric graph functionals.

Contribution

It introduces almost sure CLTs for stabilizing functionals in stochastic geometry, extending classical results to the Poisson space with applications to geometric graphs.

Findings

Almost sure CLT for total edge length of k-nearest neighbors graph

Almost sure CLT for clique counts in random geometric graphs

Almost sure CLT for volume of Poisson-Voronoi set approximation

Abstract

We prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals emerging in stochastic geometry. As a consequence, we provide almost sure central limit theorems for $(i)$ the total edge length of the $k$-nearest neighbors random graph, $(ii)$ the clique count in random geometric graphs, $(iii)$ the volume of the set approximation via the Poisson-Voronoi tessellation.