Advances in Applied Probability
- Adv. in Appl. Probab.
- Volume 42, Number 4 (2010), 913-935.
Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane
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 cross-covariance 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.
Adv. in Appl. Probab. Volume 42, Number 4 (2010), 913-935.
First available in Project Euclid: 23 December 2010
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Schreiber, Tomasz; Thäle, Christoph. Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane. Adv. in Appl. Probab. 42 (2010), no. 4, 913--935. doi:10.1239/aap/1293113144. https://projecteuclid.org/euclid.aap/1293113144