Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-01T06:58:11.120Z Has data issue: false hasContentIssue false
  • English
  • Français

Central limit theorems for functionals of stationary germ-grain models

Part of: Inference from stochastic processes Limit theorems 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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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).

References

Böhm, S., Heinrich, L. and Schmidt, V. (2004). Asymptotic properties of estimators for the volume fraction of Jointly stationary random sets. Statist. Neerlandica 58, 388–406.CrossRefGoogle Scholar
Doukhan, P. (1994). Mixing: Properties and Examples (Lecture Notes Statist. 85). Springer, New York.CrossRefGoogle Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Heinrich, L. (1988). Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster processes. Math. Nachr. 136, 131–148.Google Scholar
Heinrich, L. (1993). Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika 31, 349–360.Google Scholar
Heinrich, L. (1994). Normal approximations for some mean-value estimates of absolutely regular tessellations. Math. Meth. Statist. 3, 1–24.Google Scholar
Heinrich, L. and Molchanov, I. (1999). Central limit theorem for a class of random measures associated with germ–grain models. Adv. Appl. Prob. 31, 283–314.Google Scholar
Ivanov, A. V. and Leonenko, N. N. (1989). Statistical Analysis of Random Fields. Kluwer, Dordrecht.Google Scholar
Mase, S. (1982). Asymptotic properties of stereological estimators of volume fraction for stationary random sets. J. Appl. Prob. 19, 111–126.Google Scholar
Molchanov, I. S. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, Chichester.Google Scholar
Schmidt, V. and Spodarev, E. (2005). Joint estimators for the specific intrinsic volumes of stationary random sets. Stoch. Process. Appl. 115, 959–981.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Spodarev, E. and Schmidt, V. (2005). On the local connectivity number of stationary random closed sets. In Proc. 7th Internat. Symp. Math. Morphology, eds Ronse, C. et al., Kluwer, Dordrecht, pp. 343–356.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester.Google Scholar