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

Part of: Limit theorems Markov processes Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Tomasz Schreiber  and
Christoph Thäle
Tomasz Schreiber*
Affiliation:
Nicolaus Copernicus University, Toruń
Christoph Thäle*
Affiliation:
University of Fribourg
*
∗ Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ulica Chopina 12/18, 87-100 Toruń, Poland. Email address: tomeks@mat.umk.pl
∗∗ Current address: Department of Mathematics, University of Osnabrück, D-49076 Osnabrück, Germany. Email address: christoph.thaele@uni-osnabrueck.de
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Abstract

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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.

Keywords

Central limit theoremcovariance measurecross correlationMarkov processiteration/nestingmartingalepair-correlation functionrandom tessellationstochastic stabilitystochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 60J75: Jump processes 60F05: Central limit and other weak theorems

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 913 - 935
Copyright
Copyright © Applied Probability Trust 2010 

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