Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains

Abstract

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d−1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d−k)-dimensional orthogonal complement. If the second moment of the (d−k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k) th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

Mixing properties of stationary Poisson cylinder models

“…, d − 1, the stationary P-k-CM Ξ λ,Q d,k has long range correlations. It is easily checked (and already mentioned in [10]) that the events…”

“…Next, we recall the notion of ergodicity and various mixing properties of RACSs, see [6], [10] and [19] for details. For this we need a family of shift operators {S x : x ∈ R d } defined by …”

“…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.…”

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