Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

TL;DR

This paper develops variance asymptotics and a central limit theorem for geometric functionals of Poisson cylinder processes, generalizing models like Boolean and hyperplane processes, with detailed variance analysis and quantitative CLT results.

Contribution

It introduces new variance asymptotics and CLT results for Poisson cylinder processes, including a detailed analysis of the asymptotic variance constant and degeneracy phenomena.

Findings

Berry-Esseen bounds for the volume of union sets

Asymptotic normality of geometric functionals including intrinsic volumes

Analysis of variance degeneracy phenomena in cylinder processes

Abstract

This paper deals with the union set of a stationary Poisson process of cylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an $m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or $m$-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of geometric functionals, including the intrinsic volumes. In this context the asymptotic variance constant is analysed in detail, which in contrast to the Boolean model leads to a new degeneracy phenomenon. A quantitative central limit theory is developed in a multivariate set-up as well.