Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

Abstract

This paper deals with the union set of a stationary Poisson process of cylinders in R n having an (n−m)-dimensional base and an m-dimensional direction space, where m ∈ {0, 1, . . . , n−1} and n ≥ 2. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or m-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of geometric functionals, including the intrinsic volumes. In this context the asymptotic variance constant is analysed in detail, which in contrast to the Boolean model leads to a new degeneracy phenomenon. A quantitative central limit theory is developed in a multivariate set-up as well.

Variance asymptotics for the area of planar cylinder processes generated by Brillinger-mixing point processes

“…red q , where 1 q < ∞. Formally, we may put γ (1) red T V := 1 and c (1) red q = c (1) red * q := 1. Note that the existence and integrability of c…”

“…where v(Φ 0 ), X := x∈X v(Φ 0 ), x with v(ϕ), x = x (1) cos ϕ + x (2) sin ϕ for x = (x (1) , x (2) ) ∈ R 2 .…”

“…red (dp 2 ) dp 1 with integration over points x = (x (1) , x (2) ) ∈ K and y = (y (1) , y (2) ) ∈ K. As in the proof of Lemma 4, we express the product of the probabilities…”

“…Meanwhile CPs also receive much attention due to numerous applications (for d = 2, 3) in modern technologies as models for materials consisting of long fibers or to model dynamic telecommunication networks. Until now, so far as we know, asymptotic properties of CPs in expanding domains were exclusively studied under Poisson assumptions; see [13,14,20] and [1]. The Poisson property of the generating stationary point process (PP) provides, among others, the stationarity of the associated CP.…”

Asymptotics of a time bounded cylinder model

“…Assumptions of convexity on the cylinder bases and isotropy of 𝑍 were made. In the very recent preprint [3], Betken et al show central limit theorems for geometric funtionals for Poisson cylinder processes. The special case of 𝑞 = 1 and spherical cylinder bases is studied in [4], where Broman and Tykesson focus on connectivity properties.…”