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

Autor(en): Schreiber, T.
Thäle, C.
Stichwörter: Central limit theorem; Covariance measure; Cross correlation; Cross correlations; Iteration/nesting; Iterative methods; Markov process; Markov processes; Martingale; Pair correlation functions; Pair-correlation function; Random tessellation; Stochastic geometry; Stochastic stability; Stochastic stability, Geometry; Stochastic systems, Correlation methods
Erscheinungsdatum: 2010
Journal: Advances in Applied Probability
Volumen: 42
Ausgabe: 4
Startseite: 913
Seitenende: 935
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 (?, t? ), t > 0, arising in the limit, where ? is a centered Gaussian variable with explicitly known variance. © Applied Probability Trust 2010.
ISSN: 00018678
DOI: 10.1239/aap/1293113144
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