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

Hug, Daniel; Last, Günter; Schulte, Matthias (2016). Second-order properties and central limit theorems for geometric functionals of Boolean models. Annals of applied probability, 26(1), pp. 73-135. Institute of Mathematical Statistics 10.1214/14-AAP1086

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Let Z be a Boolean model based on a stationary Poisson process
η of compact, convex particles in Euclidean space Rᵈ. Let W denote
a compact, convex observation window. For a large class of function-
als, formulas for mean values of ψ(Z ∩ W) are available in the
literature. The first aim of the present work is to study the asymp-
totic covariances of general geometric (additive, translation invariant
and locally bounded) functionals of Z ∩ W for increasing observation
window W, including convergence rates. Our approach is based on
the Fock space representation associated with η. 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.

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