Second order analysis of geometric functionals of Boolean models

TL;DR

This paper derives asymptotic covariance formulas and CLTs for geometric functionals of stationary Boolean models, emphasizing anisotropic cases like aligned rectangles, with explicit formulas and simulation comparisons.

Contribution

It introduces second order asymptotic covariance formulas and CLTs for a broad class of geometric functionals, including Minkowski tensors, with explicit results for anisotropic models.

Findings

Explicit covariance formulas for anisotropic Boolean models

Validation of formulas through simulations

Insights into grain distribution second moments

Abstract

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.