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Limit theorems for stationary tessellations with random inner cell structures

Part of: Geometric probability and stochastic geometry Limit theorems Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Lothar Heinrich ,
Hendrik Schmidt  and
Volker Schmidt
Lothar Heinrich*
Affiliation:
University of Augsburg
Hendrik Schmidt*
Affiliation:
University of Ulm
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Institute of Mathematics, University of Augsburg, D-86135 Augsburg, Germany.
∗∗ Postal address: Department of Stochastics, University of Ulm, D-89069 Ulm, Germany.
∗∗ Postal address: Department of Stochastics, University of Ulm, D-89069 Ulm, Germany.
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Abstract

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We consider stationary and ergodic tessellations X = Ξnn≥1 in Rd, where X is observed in a bounded and convex sampling window Wp ⊂ Rd. It is assumed that the cells Ξn of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J0 acting on Rd. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in Wp, as Wp ↑ Rd. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

Keywords

Stationary random tessellationinner random cell structurecumulative random functionalergodic theoremcentral limit theoremPoisson nestingmodel choicetelecommunication network

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F05: Central limit and other weak theorems 90B15: Network models, stochastic
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 25 - 47
Copyright
Copyright © Applied Probability Trust 2005 

References

Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Prob. 3, 1033–1046.Google Scholar
Baccelli, F., Gloaguen, C. and Zuyev, S. (2000). Superposition of planar Voronoi tessellations. Commun. Statist. Stoch. Models 16, 69–98.Google Scholar
Baddeley, A. J. (1980). A limit theorem for statistics of spatial data. Adv. Appl. Prob. 12, 447–461.Google Scholar
Böhm, S., Heinrich, L. and Schmidt, V. (2004). Asymptotic properties of estimators for the volume fractions of Jointly stationary random sets. Statist. Neerlandica 58, 388–406.CrossRefGoogle Scholar
Cowan, R. (1980). Properties of ergodic random mosaic processes. Math. Nachr. 97, 89–102.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to The Theory of Point Processes. Springer, New York.Google Scholar
Dellacherie, C. and Meyer, P.-A. (1994). Probabilities and Potential. North-Holland, Amsterdam.Google Scholar
Gloaguen, C., Coupé, P., Maier, R. and Schmidt, V. (2002). Stochastic modelling of urban access networks. In Proc. 10th Internat. Telecommun. Network Strategy Planning Symp. (Munich, June 2002), VDE, Berlin, pp. 99–104.Google Scholar
Heinrich, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Meth. Statist. 3, 1–24.Google Scholar
Heinrich, L. and Molchanov, I. S. (1999). Central limit theorem for a class of random measures associated with germ-grain models. Adv. Appl. Prob. 31, 283–314.CrossRefGoogle Scholar
Leichtweiss, K. (1980). Konvexe Mengen. Springer, Berlin.Google Scholar
Maier, R. and Schmidt, V. (2003). Stationary iterated tessellations. Adv. Appl. Prob. 35, 337–353.Google Scholar
Maier, R., Mayer, J. and Schmidt, V. (2004). Distributional properties of the typical cell of stationary iterated tessellations. Math. Meth. Operat. Res. 59, 287–302.Google Scholar
Mase, S. (1982). Asymptotic properties of stereological estimators of the volume fraction for stationary random sets. J. Appl. Prob. 19, 111–126.Google Scholar
Mayer, J., Schmidt, V. and Schweiggert, F. (2004). A unified simulation framework for spatial stochastic models. Simul. Modelling Practice Theory 12, 307–326.Google Scholar
Møller, J. (1989). Random tessellations in R d . Adv. Appl. Prob. 21, 37–73.Google Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations, 2nd edn. John Wiley, Chichester.Google Scholar
Paroux, K. (1998). Quelques théorèmes centraux limites pour les processus poissoniens de droites dans le plan. Adv. Appl. Prob. 30, 640–656.Google Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables (Oxford Studies Prob. 4). Oxford University Press.Google Scholar
Schmidt, V. and Spodarev, E. (2005). Joint estimators for the specific intrinsic volumes of stationary random sets. To appear in Stoch. Process. Appl. Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar