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Central limit theorems for functionals of stationary germ-grain models

Part of: Limit theorems Inference from stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Ursa Pantle ,
Volker Schmidt  and
Evgueni Spodarev
Ursa Pantle*
Affiliation:
Universität Ulm
Volker Schmidt*
Affiliation:
Universität Ulm
Evgueni Spodarev*
Affiliation:
Universität Ulm
*
∗ Postal address: Abteilung Stochastik, Universität Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
∗ Postal address: Abteilung Stochastik, Universität Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
∗ Postal address: Abteilung Stochastik, Universität Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
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Abstract

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Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.

Keywords

Random closed setBoolean modelstationary random fieldm-dependent random fieldvaluationasymptotic normalityβ-mixingspecific intrinsic volumeEuler number

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F05: Central limit and other weak theorems 62M40: Random fields; image analysis
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 76 - 94
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Presented at the ICMS Workshop on Spatial Stochastic Modelling with Applications to Communications Networks (Edinburgh, June 2004).

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