## Bernoulli

• Bernoulli
• Volume 13, Number 3 (2007), 868-891.

### Limit theorems for functionals on the facets of stationary random tessellations

#### Abstract

We observe stationary random tessellations X={Ξn}n≥1 in ℝd through a convex sampling window W that expands unboundedly and we determine the total (k−1)-volume of those (k−1)-dimensional manifold processes which are induced on the k-facets of X (1≤kd−1) by their intersections with the (d−1)-facets of independent and identically distributed motion-invariant tessellations Xn generated within each cell Ξn of X. The cases of X being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in W are approximately normally distributed when W is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson–Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).

#### Article information

Source
Bernoulli, Volume 13, Number 3 (2007), 868-891.

Dates
First available in Project Euclid: 7 August 2007

https://projecteuclid.org/euclid.bj/1186503491

Digital Object Identifier
doi:10.3150/07-BEJ6131

Mathematical Reviews number (MathSciNet)
MR2348755

Zentralblatt MATH identifier
1156.60010

#### Citation

Heinrich, Lothar; Schmidt, Hendrik; Schmidt, Volker. Limit theorems for functionals on the facets of stationary random tessellations. Bernoulli 13 (2007), no. 3, 868--891. doi:10.3150/07-BEJ6131. https://projecteuclid.org/euclid.bj/1186503491

#### References

• Brakke, K.A. (1987). Statistics of random plane Voronoi tessellations. Preprint, Dept. Math. Sciences, Susquehanna, Univ.
• Calka, P. (2002). The distributions of the smallest disk containing Poisson--Voronoi typical cells and the crofton cell in the plane., Adv. Appl. Probab. (SGSA) 34 702--717.
• Chadœuf, J. and Monestiez, P. (1992). Parameter estimation in tessellation models derived from the Voronoi model., Acta Stereol. 11 53--58.
• Chiu, S.N. and Quine, M.P. (1997). Central limit theory for the number of seeds in a growth model in $\mathbbR^d$ with inhomogeneous arrivals., Ann. Appl. Probab. 7 802--814.
• Chiu, S.N. and Quine, M.P. (2001). Central Limit Theorem for Germination--Growth models in $\mathbbR^d$ with non-Poisson locations., Adv. Appl. Probab. (SGSA) 33 751--755.
• Daley, D.J. and Vere-Jones, D. (1988)., An Introduction to the Theory of Point Processes. New York: Springer.
• Favis, G. and Weiss, V. (1998). Mean values of weighted cells of stationary Poisson hyperplane tessellations of $\mathbbR^d$., Math. Nachr. 193 37--48.
• Gloaguen, C., Fleischer, F., Schmidt, H. and Schmidt, V. (2006). Fitting of stochastic telecommunication network models via distance measures and Monte Carlo tests., Telecommunication Systems 31 353--378.
• Gloaguen, C., Fleischer, F., Schmidt, H. and Schmidt, V. (2007). Analysis of shortest paths and subscriber line lengths in telecommunication access networks., Networks and Spatial Economics 8.
• Heinrich, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations., Math. Methods Statist. 3 1--24.
• Heinrich, L., Körner, R., Mehlhorn, N. and Muche, L. (1998). Numerical and analytical computation of some second-order characteristics of spatial Poisson--Voronoi tessellations., Statistics 31 235--259.
• Heinrich, L. and Molchanov, I.S. (1999). Central limit theorem for a class of random measures associated with germ--grain models., Adv. Appl. Probab. 31 283--314.
• Heinrich, L. and Muche, L. (2007). Second-order properties of the point process of nodes in a stationary Voronoi tessellation., Math. Nachr. To appear.
• Heinrich, L., Schmidt, H. and Schmidt, V. (2005). Limit theorems for stationary tessellations with random inner cell structures., Adv. Appl. Probab. 37 25--47.
• Heinrich, L., Schmidt, H. and Schmidt, V. (2006). Central limit theorems for Poisson hyperplane tessellations., Ann. Appl. Probab. 16 919--950.
• Heinrich, L. and Schertz, S. (2007). On asymptotic normality, covariance matrices, and the associated zonoid of stationary Poisson hyperplane processes. Preprint, Institute of Mathematics, Univ., Augsburg.
• Karr, A.F. (1992)., Probability. New York: Springer.
• Mecke, J. (1981). Stereological formulas for manifold processes., Probab. Math. Statist. 2 31--35.
• Maier, R. and Schmidt, V. (2003). Stationary iterated tessellations., Adv. Appl. Probab. (SGSA) 35 337--353.
• Matheron, G. (1975)., Random Sets and Integral Geometry. New York: Wiley.
• Møller, J. (1989). Random tessellations in $\mathbbR^d$., Adv. Appl. Probab. 21 37--73.
• Møller, J. (1994)., Lectures on Random Voronoi Tessellations. Lecture Notes in Statist. 87. New York: Springer.
• Nahapetian, B. (1991)., Limit Theorems and Some Applications to Statistical Physics. Teubner--Texte zur Mathematik 123. Stuttgart--Leipzig: Teubner.
• Okabe, A., Boots, B., Sugihara K. and Chiu, S.N. (2000)., Spatial Tessellations, 2nd ed. Chichester: Wiley.
• Schmidt, H. (2006)., Asymptotic analysis of stationary random tessellations with applications to network modelling. Dissertation Thesis, Ulm Univ., http://vts.uni-ulm.de/doc.asp?id=5702.
• Schneider, R. (1993)., Convex Bodies: The Brunn--Minkowski Theory. Cambridge: Cambridge University Press.
• Schneider, R. and Weil, W. (2000)., Stochastische Geometrie. Stuttgart: Teubner.
• Schwella, A. (2001)., Special inequalities for Poisson and Cox hyperplane processes. Dissertation Thesis, Univ. Jena.
• Stoyan, D. and Stoyan, H. (1985). On one of Matérn's hard core point process models., Math. Nachr. 122 205--214.
• Stoyan, D., Kendall, W.S. and Mecke, J. (1995)., Stochastic Geometry and Its Applications, 2nd ed. Chichester: Wiley.
• Weil, W. (1979). Centrally symmetric convex bodies and distributions. II., Israel J. Math. 32 173--182.
• Zähle, M. (1988). Random cell complexes and generalised sets., Ann. Probab. 16 1742--1766.