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 η of compact, convex particles in Euclidean space R d . Let W denote a compact, convex observation window. For a large class of functionals ψ, formulas for mean values of ψ(Z ∩ 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 ∩ 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.

Multivariate second order Poincaré inequalities for Poisson functionals

“…In the following, we derive quantitative multivariate central limit theorems for Boolean models, extending previous findings in [12] and [17,Chapter 22]. Our proofs rely on the general bounds from Subsection 1.2 as well as arguments from [12] and [17,Chapter 22].…”

“…(1.5) may be evaluated since the first two difference operators have a clear interpretation via the operation of adding additional points. This is the advantage of these findings over Malliavin-Stein bounds for normal approximation of Poisson functionals which either require the knowledge of the chaos expansion of F (see, for example, [7,12,23,30]) or which involve bounds expressed in terms of gradient operators and conditional expectations as in [25]. Inequality (1.5) yields rates of normal approximation for some classic problems in stochastic geometry and some non-linear functionals of Poisson-shot-noise processes [16], as well as for functionals of convex hulls of random samples in a smooth convex body, statistics of nearest neighbors graphs, the number of maximal points in a random sample, and estimators of surface area and volume arising in set approximation [15].…”

Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

“…Last and Ochsenreither [77] have proven that this coincides with the global maximum of the asymptotic variance of the Euler characteristic. Moreover, Hug et al [73] found for overlapping discs that both the local minimum of the variance of the perimeter and the local minimum of the covariance of Figure 13. Asymptotic variances σ WµWµ of the Minkowski functionals as functions of the intensity γ of the Boolean models with squares (left) or rectangles with aspect ration 1 2 (right), the rectangles are either fully aligned (top) or follow an isotropic orientation distribution (bottom).…”

Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

“…For Boolean models, not only the mean values of the Minkowski functionals and tensors but also their second moments and joint probability distributions are known analytically …”

“…For Boolean models, not only the mean values of the Minkowski functionals and tensors but also their second moments and joint probability distributions are known analytically. 54,[65][66][67] The Minkowski functionals have already been used to adjust a Boolean model to an experimental structure, resulting in an excellent match of the mechanical and transport properties. 26,68 They have also been used to predict properties of nanoscale flow through porous materials.…”