Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T00:05:10.807Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bräker, H. and Hsing, T. (1998). On the area and perimeter of a random convex hull in a bounded convex set. Prob. Theory Relat. Fields 111, 517–550.CrossRefGoogle Scholar
Cabo, A. J. and Groeneboom, P. (1994). Limit theorems for functionals of convex hulls. Prob. Theory Relat. Fields 100, 31–55.CrossRefGoogle Scholar
Chayes, J. T., Chayes, L. and Kotecký, R. (1995). The analysis of the Widom–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston, MA.Google Scholar
Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, Chichester.CrossRefGoogle Scholar
Efron, B. (1965). The convex hull of a random set of points. Biometrika 52, 331–343.CrossRefGoogle Scholar
Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. De Gruyter, Berlin.CrossRefGoogle Scholar
Groeneboom, P. (1988). Limit theorems for convex hulls. Prob. Theory Relat. Fields 79, 327–368.CrossRefGoogle Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Hsing, T. (1994). On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Prob. 4, 478–493.CrossRefGoogle Scholar
Hueter, I. (1999). Limit theorems for the convex hull of random points in higher dimensions. Trans. Amer. Math. Soc. 11, 4337–4363.CrossRefGoogle Scholar
Kendall, W. S., Mecke, J. and Stoyan, D. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Khamdamov, I. M. and Nagaev, A. V. (1991). Limiting distributions for functionals of the convex hull generated by uniformly distributed variables. Dokl. Akad. Nauk UzSSR 7, 8–9 (in Russian).Google Scholar
Küfer, K. H. (1994). On the approximation of a ball by random polytopes. Adv. Appl. Prob. 26, 876–892.CrossRefGoogle Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Molchanov, I. S. (1993). Limit Theorems for Unions of Random Closed Sets (Lecture Notes Math. 1561). Springer, Berlin.Google Scholar
Molchanov, I. S. (1995). On the convergence of random processes generated by polyhedral approximation of convex compacts. Theory Prob. Appl. 40, 383–390.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 75–84.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1964). Über die konvexe Hülle von n zufällig gewählten Punkten. II. Z. Wahrscheinlichkeitsth. 3, 138–147.CrossRefGoogle Scholar
Schneider, R. (1988). Random approximation of convex sets. J. Microscopy 151, 211–227.CrossRefGoogle Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopaedia Math. Appl. 44). Cambridge University Press.Google Scholar
Schreiber, T. (2000). Large deviation principle for set-valued union processes. Prob. Math. Statist. 20, 273–285.Google Scholar
Schreiber, T. (2002a). Limit theorems for certain functionals of unions of random closed sets. Theory Prob. Appl. 47, 130–142.Google Scholar
Schreiber, T. (2002b). Variance asymptotics and central limit theorems for volumes of unions of random closed sets. Adv. Appl. Prob. 34, 520–539.CrossRefGoogle Scholar
Schreiber, T. (2002c). A note on deviation probabilities for volumes of unions of random closed sets. Preprint 2/2002, Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń. Available at http://www.mat.uni.torun.pl/preprints/.Google Scholar
Van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York.CrossRefGoogle Scholar