Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T11:29:01.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane

Part of: Stochastic processes Limit theorems Markov processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Tomasz Schreiber  and
Christoph Thäle
Tomasz Schreiber*
Affiliation:
Nicolaus Copernicus University, Toruń
Christoph Thäle*
Affiliation:
University of Fribourg
*
∗ Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ulica Chopina 12/18, 87-100 Toruń, Poland. Email address: tomeks@mat.umk.pl
∗∗ Current address: Department of Mathematics, University of Osnabrück, D-49076 Osnabrück, Germany. Email address: christoph.thaele@uni-osnabrueck.de
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.

The point process of vertices of an iteration infinitely divisible or, more specifically, of an iteration stable random tessellation in the Euclidean plane is considered. We explicitly determine its covariance measure and its pair-correlation function, as well as the cross-covariance measure and the cross-correlation function of the vertex point process and the random length measure in the general nonstationary regime. We also give special formulae in the stationary and isotropic setting. Exact formulae are given for vertex count variances in compact and convex sampling windows, and asymptotic relations are derived. Our results are then compared with those for a Poisson line tessellation having the same length density parameter. Moreover, a functional central limit theorem for the joint process of suitably rescaled total edge counts and edge lengths is established with the process (ξ, tξ), t > 0, arising in the limit, where ξ is a centered Gaussian variable with explicitly known variance.

Keywords

Central limit theoremcovariance measurecross correlationMarkov processiteration/nestingmartingalepair-correlation functionrandom tessellationstochastic stabilitystochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 60J75: Jump processes 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 913 - 935
Copyright
Copyright © Applied Probability Trust 2010 

References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I. Springer, New York.Google Scholar
Heinrich, L. (2009). Central limit theorems for motion-invariant Poisson hyperplanes in expanding convex windows. Rend. Circ. Mat. Palermo 81, 187–212.Google Scholar
Heinrich, L. and Muche, L. (2008). Second-order properties of the point process of nodes in a stationary Voronoi tessellation. Math. Nachr. 281, 350–375.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Mecke, J., Nagel, W. and Weiss, V. (2008). A global construction of homogeneous random planar tessellations that are stable under iteration. Stochastics 80, 51–67.Google Scholar
Nagel, W. and Weiss, V. (2005). Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Prob. 37, 859–883.Google Scholar
Nagel, W. and Weiss, V. (2006). STIT tessellations in the plane. Rend. Circ. Mat. Palermo 77, 441–458.Google Scholar
Schmidt, H. (2008). Asymptotic Analysis of Stationary Random Tessellations. VDM Verlag, Saarbrücken.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Schreiber, T. and Thäle, C. (2010). Typical geometry, second-order properties and central limit theory for iteration stable tessellations. Preprint. Available at http://arxiv.org/abs/1001.0990v3.Google Scholar
Stoyan, D. and Ohser, J. (1982). Correlations between planar random structures, with an ecological application. Biometrical J. 24, 631–647.Google Scholar
Stoyan, D. and Ohser, J. (1985). Cross-correlation measures for weighted random measures and their estimation. Theory Prob. Appl. 29, 345–355.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Weiss, V., Ohser, J. and Nagel, W. (2010). Second moment measure and K-function for planar STIT tessellations. Image Anal. Stereology 29, 121–131.CrossRefGoogle Scholar