Second-order properties and central limit theorems for geometric functionals of Boolean models

Abstract

Let $Z$ be a Boolean model based on a stationary Poisson process $\eta$ of compact, convex particles in Euclidean space $\mathbb{R}^{d}$. Let $W$ denote a compact, convex observation window. For a large class of functionals $\psi$, formulas for mean values of $\psi(Z\cap W)$ are available in the literature. The first aim of the present work is to study the asymptotic covariances of general geometric (additive, translation invariant and locally bounded) functionals of $Z\cap W$ for increasing observation window $W$, including convergence rates. Our approach is based on the Fock space representation associated with $\eta$. For the important special case of intrinsic volumes, the asymptotic covariance matrix is shown to be positive definite and can be explicitly expressed in terms of suitable moments of (local) curvature measures in the isotropic case. The second aim of the paper is to prove multivariate central limit theorems including Berry–Esseen bounds. These are based on a general normal approximation result obtained by the Malliavin–Stein method.