Anisotropic Poisson Processes of Cylinders

Abstract

Main characteristics of stationary anisotropic Poisson processes of cylinders (dilated k-dimensional flats) in d-dimensional Euclidean space are studied. Explicit formulae for the capacity functional, the covariance function, the contact distribution function, the volume fraction, and the intensity of the surface area measure are given which can be used directly in applications.

Mixing properties of stationary Poisson cylinder models

“…o d ∈ S. Straightforward calculations carried out in [8] and [10], see also [20] for a different approach, yield that …”

“…1 frequently used Boolean model) was recognized just recently, see [20] and [8], [10] for central limit theorems of the volume and surface content in expanding windows.…”

“…In stochastic geometry, a k-cylinder in R d is defined as Minkowski sum B ⊕ L of a direction space L ∈ G(d, k) (= the Grassmannian of k-dimensional subspaces of R d ) and a compact base B in the orthogonal complement L ⊥ , see e.g. [19] or [20]. In the following we go along the line suggested in [8], [10] (which slightly differs from that in [14] and [20] …”

“…[19] or [20]. In the following we go along the line suggested in [8], [10] (which slightly differs from that in [14] and [20] …”

Multivariate Poisson distributions associated with Boolean models

“…We only mention that the proof of (3.16) relies on the fact (by applying the mean value theorem and Slutsky's lemma) that the distributional limits of 17) where the variance σ λ…”