Normal approximation of Gibbsian sums in geometric probability

TL;DR

This paper investigates the asymptotic normality of sums of functionals over Gibbsian spatial point processes, providing variance estimates and rates of convergence to normality for various geometric and probabilistic models.

Contribution

It establishes conditions under which Gibbsian sums exhibit volume order fluctuations and derives normal approximation rates using Stein's method.

Findings

Variance asymptotics for Gibbsian sums

Central limit theorems for geometric graphs with Gibbsian input

Normal approximation rates for diverse spatial models

Abstract

This paper concerns the asymptotic behavior of a random variable $W_\lambda$ resulting from the summation of the functionals of a Gibbsian spatial point process over windows $Q_\lambda \uparrow R^d$. We establish conditions ensuring that $W_\lambda$ has volume order fluctuations, that is they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of normal approximation for $W_\lambda$, as $\lambda\to\infty$. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.