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Asymptotic geometry of high-density smooth-grained Boolean models in bounded domains

Part of: Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Tomasz Schreiber
Tomasz Schreiber*
Affiliation:
Nicolaus Copernicus University, Toruń
*
∗ Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland. Email address: [email protected]
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Abstract

The purpose of the paper is to study the asymptotic geometry of a smooth-grained Boolean model (X[t])t≥0 restricted to a bounded domain as the intensity parameter t goes to ∞. Our approach is based on investigating the asymptotic properties as t → ∞ of the random sets X[t;β], β≥0, defined as the Gibbsian modifications of X[t] with the Hamiltonian given by βtμ(·), where μ is a certain normalized measure on the setting space. We show that our model exhibits a phase transition at a certain critical value of the inverse temperature β and we prove that at higher temperatures the behaviour of X[t;β] is qualitatively very similar to that of X[t] but it becomes essentially different in the low-temperature region. From these facts we derive information about the asymptotic properties of the original process X[t]. The results obtained include large- and moderate-deviation principles. We conclude the paper with an example application of our methods to analyse the asymptotic moderate-deviation properties of convex hulls of large uniform samples on a multidimensional ball. To translate the above problem to the Boolean model setting considered we use an appropriate representation of convex sets in terms of their support functions.

Keywords

Random closed setsBoolean modelsGibbs measuresphase transitionlarge deviationsconvex hullmean width

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F10: Large deviations
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 913 - 936
Copyright
Copyright © Applied Probability Trust 2003 

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