Abstract
A homogeneous Poisson-Voronoi tessellation of intensity γ is observed in a convex body W. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in W. We prove that when \(\gamma \rightarrow \infty \), these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between W and its so-called Poisson-Voronoi approximation.
Similar content being viewed by others
References
Arratia, R., Goldstein, L., Gordon, L.: Poisson approximation and the Chen-Stein method. Stat. Sci. 5(4), 403—434 (1990)
Baccelli, F.B.: Blaszczyszyn. Stochastic geometry and wireless networks volume 2: APPLICATIONS. Foundations and TrendsⓇ in Networking 4(1–2), 1–312 (2009)
Baumstark, V., Last, G.: Some distributional results for Poisson-Voronoi tessellations. Adv. Appl. Probab. 39(1), 16–40 (2007)
Calka, P.: The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Probab. 34(4), 702–717 (2002)
Calka, P.: An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell. Adv. Appl. Probab. 35(4), 863–870 (2003)
Calka, P.: Tessellations. In: New Perspectives in Stochastic Geometry, pp. 145–169. Oxford University Press, Oxford (2010)
Capasso, V., Villa, E.: On the geometric densities of random closed sets. Stoch. Anal. Appl. 26(4), 784–808 (2008)
de Haan, L., Ferreira, A.: Extreme value theory. In: Springer Series in Operations Research and Financial Engineering. Springer, New York (2006). An introduction
Foss, S.G., Zuyev, S.A.: On a Voronoi aggregative process related to a bivariate Poisson process. Adv. Appl. Probab. 28(4), 965–981 (1996)
Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, vol. 1730. Springer-Verlag, Berlin (2000)
Heinrich, L., Muche, L.: Second-order properties of the point process of nodes in a stationary Voronoi tessellation. Math. Nachr. 281(3), 350–375 (2008)
Heinrich, L., Schmidt, H., Schmidt, V.: Limit theorems for stationary tessellations with random inner cell structures. Adv. Appl. Probab. 37(1), 25–47 (2005)
Henze, N.: The limit distribution for maxima of “weighted” rth-nearest-neighbour distances. J. Appl. Probab. 19(2), 344–354 (1982)
Heveling, M., Reitzner, M.: Poisson-Voronoi approximation. Ann. Appl. Probab. 19(2), 719–736 (2009)
Hlubinka, D.: Stereology of extremes; shape factor of spheroids. Extremes 6(1), 5–24 (2003)
Hug, D., Reitzner, M., Schneider, R.: Large Poisson-Voronoi cells and Crofton cells. Adv. Appl. Probab. 36(3), 667–690 (2004)
Jammalamadaka, S.R., Janson, S.: Limit theorems for a triangular scheme of U-statistics with applications to inter-point distances. Ann. Probab. 14(4), 1347–1358 (1986)
Janson, S.: Random coverings in several dimensions. Acta Math. 156(1-2), 83–118 (1986)
Ju, L., Gunzburger, M., Zhao, W.: Adaptive finite element methods for elliptic PDEs based on conforming centroidal Voronoi-Delaunay triangulations. SIAM J. Sci. Comput. 28(6), 2023–2053 (2006)
Khmaladze, E., Toronjadze, N.: On the almost sure coverage property of Voronoi tessellation: the ℝ1 case. Adv. Appl. Probab. 33(4), 756–764 (2001)
Lantuéjoul, C., Bacro, J.N., Bel, L.: Storm processes and stochastic geometry. Extremes 14(4), 413–428 (2011)
Leadbetter, M.R.: On extreme values in stationary sequences. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28, 289–303 (1973/74)
Leadbetter, M.R.: Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65(2), 291–306 (1983)
LeCaer, G., Ho, J.S.: The Voronoi tessellation generated from eigenvalues of complex random matrices. J. Phys. A :Math. Gen. 23, 3279–3295 (1990)
Loynes, R.M.: Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Stat. 36, 993–999 (1965)
Mayer, M., Molchanov, I.: Limit theorems for the diameter of a random sample in the unit ball. Extremes 10(3), 129–150 (2007)
Møller, J.: Random tessellations in R d. Adv. Appl. Probab. 21(1), 37–73 (1989)
Møller, J.: Lectures on Random Voronoı̆ Tessellations. Lecture Notes in Statistics, volume 87. Springer-Verlag, New York (1994)
Møller, J., Waagepetersen, R.P.: Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability, vol. 100. Chapman & Hall/CRC, Boca Raton, FL (2004)
Muche, L.: The Poisson-Voronoi tessellation: relationships for edges. Adv. Appl. Probab. 37(2), 279–296 (2005)
Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial tessellations: concepts and applications of Voronoi diagrams, 2nd edn. Wiley Series in Probability and Statistics. Wiley, Chichester (2000)
Pawlas, Z.: Local stereology of extremes. Image Anal. Stereol. 31(2), 99–108 (2012)
Penrose, M.: Random geometric graphs. Oxford Studies in Probability, vol. 5. Oxford University Press, Oxford (2003)
Poupon, A.: Voronoi and Voronoi-related tessellations in studies of protein structure and interaction. Curr. Opin. Struct. Biol. 14(2), 233–241 (2004)
Ramella, M., Boschin, W., Fadda, D., Nonino, M.: Finding galaxy clusters using Voronoi tessellations. Astron. Astrophys. 368, 776–786 (2001)
Reitzner, M., Spodarev, E., Zaporozhets, D.: Set reconstruction by Voronoi cells. Adv. Appl. Probab. 44, 938–953 (2012)
Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Applied Probability, vol. 4. A Series of the Applied Probability Trust. Springer-Verlag, New York (1987)
Roque, W.L.: Introduction to Voronoi diagrams with applications to robotics and landscape ecology. Proc. II Escuela de Matematica Aplicada 01, 1–27 (1997)
Schneider, R., Weil, W.: Stochastic and integral geometry. Probability and its Applications (New York). Springer-Verlag, Berlin (2008)
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)
Smith, R.L.: Extreme value theory for dependent sequences via the Stein-Chen method of Poisson approximation. Stoch. Process. Appl. 30(2), 317–327 (1988)
Zessin, H.: Point processes in general position. J. Contemp. Math. Anal., Armen. Acad. Sci. 43(1), 59–65 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Calka, P., Chenavier, N. Extreme values for characteristic radii of a Poisson-Voronoi Tessellation. Extremes 17, 359–385 (2014). https://doi.org/10.1007/s10687-014-0184-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-014-0184-y