Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane

Abstract

The point process of vertices of an iteration infinitely divisible or, more specifically, of an iteration stable random tessellation in the Euclidean plane is considered. We explicitly determine its covariance measure and its pair-correlation function, as well as the crosscovariance measure and the cross-correlation function of the vertex point process and the random length measure in the general nonstationary regime. We also give special formulae in the stationary and isotropic setting. Exact formulae are given for vertex count variances in compact and convex sampling windows, and asymptotic relations are derived. Our results are then compared with those for a Poisson line tessellation having the same length density parameter. Moreover, a functional central limit theorem for the joint process of suitably rescaled total edge counts and edge lengths is established with the process (ξ, tξ ), t > 0, arising in the limit, where ξ is a centered Gaussian variable with explicitly known variance.

Spatial Stit Tessellations: Distributional Results for I-Segments

“…Using Lemma 5, we obtain¯ s−r ϕ=u = 3/2 ([u])(s − r). Combining (16) with (17) and again using Lemma 5, we find that…”

“…Besides the classical Poisson hyperplane and Poisson-Voronoi tessellations, random tessellations constructed by subsequent cell division have attracted particular interest in recent times in stochastic geometry and spatial statistics (see [1], [2], [14], and especially [3], and the references cited therein). Among these models, the so-called STIT tessellations (which are stable under iteration-see below) introduced in [9] and [12] are of particular interest, because of the number of analytically available results [7], [10], [13], [16]- [18], [20]- [23]; see Figure 1 for illustrations. The model shows the potential to become a new reference model for crack or fissure structures.…”

Spatial Stit Tessellations: Distributional Results for I-Segments

“…Combining (16) with (17) and again using Lemma 5, we find that Using Lemma 5, we obtain¯ s−r ϕ=u = 3/2 ([u])(s − r).…”

“…Whereas in the past mainly mean values and their relations were considered, current research focuses on second-order parameters, limit theorems, and distributional results; see [4], [5], and [6], among others. Among these models, the so-called STIT tessellations (which are stable under iteration-see below) introduced in [9] and [12] are of particular interest, because of the number of analytically available results [7], [10], [13], [16]- [18], [20]- [23]; see Figure 1 for illustrations. Among these models, the so-called STIT tessellations (which are stable under iteration-see below) introduced in [9] and [12] are of particular interest, because of the number of analytically available results [7], [10], [13], [16]- [18], [20]- [23]; see Figure 1 for illustrations.…”

“…After detailed analyses of planar STIT tessellations in [10], [16], and [20], in this paper we consider the three-dimensional case. We study so-called I-segments in homogeneous threedimensional STIT tessellations, which form the one-dimensional building blocks of a STIT tessellation.…”