Annals of Applied Probability

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

Daniel Hug, Günter Last, and Matthias Schulte

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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.

Article information

Ann. Appl. Probab., Volume 26, Number 1 (2016), 73-135.

Received: August 2013
Revised: April 2014
First available in Project Euclid: 5 January 2016

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes 60H07: Stochastic calculus of variations and the Malliavin calculus 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

Boolean model central limit theorem covariance matrix intrinsic volume additive functional curvature measure Fock space representation Malliavin calculus Wiener–Itô chaos expansion Berry–Esseen bound integral geometry


Hug, Daniel; Last, Günter; Schulte, Matthias. Second-order properties and central limit theorems for geometric functionals of Boolean models. Ann. Appl. Probab. 26 (2016), no. 1, 73--135. doi:10.1214/14-AAP1086.

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