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

By Daniel Hug, Günter Last, Matthias Schulte. 2013

53 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/427ec530e57ad3c989b78994529d25368210db13

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Random geometric complexes in the thermodynamic regime (2014, result): “es ergodic theorems for χ(CB(Φ,r)) when the underlying point process Φ is itself ergodic. More recently, a slew of results have been established for χ(CB(P,r)) (i.e. the Poisson case) in the preprint [17]. The arguments in this paper replace more classic integral geometric arguments, and are based o”
Lectures on the Poisson Process (2017, background): “9 is new but closely related to a result from [52].”