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Central limit theorem for a class of random measures associated with germ-grain models

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

Published online by Cambridge University Press:  01 July 2016

Lothar Heinrich  and
Ilya S. Molchanov
Lothar Heinrich*
Affiliation:
University of Augsburg
Ilya S. Molchanov*
Affiliation:
University of Glasgow
*
∗ Postal address: Institut für Mathematik, Universität Augsburg, D-86135 Augsburg, Germany.
∗∗ Postal address: Department of Statistics, University of Glasgow, Glasgow G12 8QW, Scotland, U.K. Email address: [email protected]
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Abstract

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

Keywords

β-mixingBoolean modelgerm-grain modelintrinsic volumesm-dependent random fieldrandom measurerandom setweak dependence

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 60F05: Central limit and other weak theorems 60G57: Random measures
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 283 - 314
Copyright
Copyright © Applied Probability Trust 1999 

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