Bounds to the normal for proximity region graphs

Abstract

In a proximity region graph $\mathcal{G}$ 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 $\mathcal{G}$, 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.

33 pages; changes in response to referees' comments

References in corpus (3)

• A new method of normal approximation
• Normal approximation for coverage models over binomial point processes
• New Kolmogorov bounds for functionals of binomial point processes