Refined convergence for the Boolean model
Pierre Calka, Julien Michel, and Katy Paroux
Source: Adv. in Appl. Probab. Volume 41, Number 4
(2009), 940-957.
Abstract
In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1261669579
Digital Object Identifier: doi:10.1239/aap/1261669579
Zentralblatt MATH identifier: 05706684
Mathematical Reviews number (MathSciNet): MR2663229
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