Bernoulli

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

Limit theorems for functionals on the facets of stationary random tessellations

Lothar Heinrich, Hendrik Schmidt, and Volker Schmidt

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1186503491

Digital Object Identifier
doi:10.3150/07-BEJ6131

Mathematical Reviews number (MathSciNet)
MR2348755

Zentralblatt MATH identifier
1156.60010

Keywords
asymptotic variance β-mixing central limit theorem k-facet process nesting of tessellation Poisson hyperplane process Poisson–Voronoi tessellation weakly dependent tessellation

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


Export citation

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.