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Limit theorems for stationary tessellations with random inner cell structures

Part of: Limit theorems Geometric probability and stochastic geometry Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Lothar Heinrich ,
Hendrik Schmidt  and
Volker Schmidt
Lothar Heinrich*
Affiliation:
University of Augsburg
Hendrik Schmidt*
Affiliation:
University of Ulm
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Institute of Mathematics, University of Augsburg, D-86135 Augsburg, Germany.
∗∗ Postal address: Department of Stochastics, University of Ulm, D-89069 Ulm, Germany.
∗∗ Postal address: Department of Stochastics, University of Ulm, D-89069 Ulm, Germany.
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Abstract

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We consider stationary and ergodic tessellations X = Ξnn≥1 in Rd, where X is observed in a bounded and convex sampling window Wp ⊂ Rd. It is assumed that the cells Ξn of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J0 acting on Rd. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in Wp, as Wp ↑ Rd. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

Keywords

Stationary random tessellationinner random cell structurecumulative random functionalergodic theoremcentral limit theoremPoisson nestingmodel choicetelecommunication network

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F05: Central limit and other weak theorems 90B15: Network models, stochastic
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 25 - 47
Copyright
Copyright © Applied Probability Trust 2005 

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