, Volume 22, Issue 4, pp 571–598 | Cite as

The largest order statistics for the inradius in an isotropic STIT tessellation

  • Nicolas ChenavierEmail author
  • Werner Nagel


A planar stationary and isotropic STIT tessellation at time t > 0 is observed in the window \(W_{\rho }={t^{-1}}\sqrt {\pi \ \rho }\cdot [-\frac {1}{2},\frac {1}{2}]^{2}\), for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest disk contained in the cell. Using the Chen-Stein method, we compute the limit distributions of the largest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to infinity.


Stochastic geometry Random tessellations Extreme values Poisson approximation 

AMS 2000 Subject Classifications

60D05 60G70 60F05 62G32 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was partially supported by the French ANR grant ASPAG (ANR-17-CE40-0017). The authors thank the referees for their helpful comments.


  1. Arratia, R., Goldstein, L., Gordon, L.: Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab. 17(1), 9–25 (1989)MathSciNetCrossRefGoogle Scholar
  2. Calka, P., Chenavier, N.: Extreme values for characteristic radii of a Poisson-Voronoi tessellation. Extremes 17(3), 359–385 (2014)MathSciNetCrossRefGoogle Scholar
  3. Chenavier, N.: A general study of extremes of stationary tessellations with examples. Stoch. Process. Appl. 124(9), 2917–2953 (2014)MathSciNetCrossRefGoogle Scholar
  4. Chenavier, N., Hemsley, R.: Extremes for the inradius in the P,oisson line tessellation. Adv. Appl. Probab. 48(2), 544–573 (2016)MathSciNetCrossRefGoogle Scholar
  5. de Haan, L., Ferreira, A.: Extreme value theory. An introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)Google Scholar
  6. Lachièze-Rey, R.: Mixing properties for STIT tessellations. Adv. Appl. Probab. 43(1), 40–48 (2011)MathSciNetCrossRefGoogle Scholar
  7. Lazarus, V., Pauchard, L.: From craquelures to spiral crack patterns: influence of layer thickness on the crack patterns induced by desiccation. Soft Matter 7, 2552–2559 (2011)CrossRefGoogle Scholar
  8. Martínez, S., Nagel, W.: Ergodic description of STIT tessellations. Stochastics 84(1), 113–134 (2012)MathSciNetCrossRefGoogle Scholar
  9. Martínez, S., Nagel, W.: STIT, tessellations have trivial tail σ-algebra. Adv. Appl. Probab. 46(3), 643–660 (2014)MathSciNetCrossRefGoogle Scholar
  10. Martínez, S., Nagel, W.: The β-mixing rate of STIT tessellations. Stochastics 88(3), 396–414 (2016)MathSciNetCrossRefGoogle Scholar
  11. Mecke, J., Nagel, W., Weiss, V.: A global construction of homogeneous random planar tessellations that are stable under iteration. Stochastics 80(1), 51–67 (2008)MathSciNetCrossRefGoogle Scholar
  12. Mecke, J., Nagel, W., Weiss, V.: Some distributions for I,-segments of planar random homogeneous STIT tessellations. Math Nachr. 284(11-12), 1483–1495 (2011)MathSciNetCrossRefGoogle Scholar
  13. Mosser, L., Matthai, S.: Tessellations stable under iteration: Evaluation of application as an improved stochastic discrete fracture modeling algorithm. International Discrete Fracture Network Engineering Conference (2014)Google Scholar
  14. Nagel, W., Weiss, V.: Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Probab. 37(4), 859–883 (2005)MathSciNetCrossRefGoogle Scholar
  15. Nagel, W., Nguyen, N.L., Thäle, C., Weiss, V.: A Mecke-type formula and Markov properties for STIT tessellation processes. ALEA Lat. Am. J. Probab. Math. Stat. 14(2), 691–718 (2017)MathSciNetzbMATHGoogle Scholar
  16. Nagel, W., Weiss, V.: Mean values for homogeneous STIT, tessellations in 3d Image. Anal. Stereol. 27(1), 29–37 (2008)CrossRefGoogle Scholar
  17. Santaló, L. A.: Integral geometry and geometric probability. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, With a foreword by Mark Kac (2004)CrossRefGoogle Scholar
  18. Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications (New York). Springer, Berlin (2008)CrossRefGoogle Scholar
  19. Schulte, M., Thäle, C.: The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stoch. Process. Appl. 122(12), 4096–4120 (2012)MathSciNetCrossRefGoogle Scholar
  20. Stoyan, D., Kendall, W., Mecke, J.: Stochastic Geometry and its applications. Wiley, New York (2008)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université du Littoral Côte d’OpaleLMPA Joseph LiouvilleCalais CedexFrance
  2. 2.Fakultät für Mathematik und InformatikFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations