Advances in Applied Probability

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

Tomasz Schreiber and Christoph Thäle

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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.

Article information

Adv. in Appl. Probab. Volume 42, Number 4 (2010), 913-935.

First available in Project Euclid: 23 December 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes 60J75: Jump processes 60F05: Central limit and other weak theorems

Central limit theorem covariance measure cross correlation Markov process iteration/nesting martingale pair-correlation function random tessellation stochastic stability stochastic geometry


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.

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