Bounds to the normal for proximity region graphs

TL;DR

This paper establishes bounds to the normal distribution for functionals of proximity region graphs in Euclidean space, including edge counts and total length, using probabilistic and geometric techniques.

Contribution

It introduces broad conditions under which normal approximation bounds are derived for functionals of proximity graphs, extending previous results to more general settings.

Findings

Bounds on Kolmogorov and Wasserstein distances to normal distribution.

Variance lower bounds for graph functionals without strong stabilization.

Applicability to Poisson and binomial point processes.

Abstract

In a proximity region graph ${\cal G}$ in $\mathbb{R}^d$, two distinct points $x,y$ of a point process $\mu$ are connected when the 'forbidden region' $S(x,y)$ these points determine has empty intersection with $\mu$. The Gabriel graph, where $S(x,y)$ is the open disc with diameter the line segment connecting $x$ and $y$, is one canonical example. When $\mu$ is a Poisson or binomial process, under broad conditions on the regions $S(x,y)$, bounds on the Kolmogorov and Wasserstein distances to the normal are produced for functionals of ${\cal G}$, including the total number of edges and the total length. Variance lower bounds, not requiring strong stabilization, are also proven to hold for a class of such functionals.