Skip to main content Accessibility help
×
×
Home

On the measure of Voronoi cells

  • Luc Devroye (a1), László Györfi (a2), Gábor Lugosi (a3) and Harro Walk (a4)
Abstract

We study the measure of a typical cell in a Voronoi tessellation defined by n independent random points X 1, . . ., X n drawn from an absolutely continuous probability measure μ with density f in ℝ d . We prove that the asymptotic distribution of the measure – with respect to dμ = f(x)dx – of the cell containing X 1 given X 1 = x is independent of x and the density f. We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as d becomes large. In particular, we show that the variance converges to 0 exponentially fast in d. We also obtain a bound independent of the density for the rate of convergence of the diameter of a typical Voronoi cell.

Copyright
Corresponding author
* Postal address: School of Computer Science, McGill University, 3480 University Street, Montreal, H3A 0E9, Canada.
** Postal address: Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Magyar Tudósok krt. 2., Budapest, H-1117, Hungary.
*** Postal address: Department of Economics and Business, Pompeu Fabra University, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain.
**** Postal address: Institut für Stochastik und Anwendungen, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany.
References
Hide All
[1] Baccelli, F., Klein, M., Lebourges, M. and Zuyev, S. A. (1997). Stochastic geometry and architecture of communication networks. Telecommun. Systems 7, 209227.
[2] Biau, G. and Devroye, L. (2015). Lectures on the Nearest Neighbor Method. Springer, Cham.
[3] Brakke, K. A. (2005). Statistics of random plane Voronoi tessellations. Unpublished manuscript. Available at http://facstaff.susqu.edu/brakke/aux/downloads/papers/vorplane.pdf.
[4] Brakke, K. A. (2005). Statistics of three dimensional random Voronoi tessellations. Unpublished manuscript. Available at http://facstaff.susqu.edu/brakke/aux/downloads/papers/3d.pdf.
[5] Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and its Applications, 3rd edn. John-Wiley, Chichester.
[6] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The L 1 View. John Wiley, New York.
[7] Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958972.
[8] Hayen, A. and Quine, M. P. (2002). Areas of components of a Voronoi polygon in a homogeneous Poisson process in the plane. Adv. Appl. Prob. 34, 281291.
[9] Heinrich, L. and Muche, L. (2008). Second-order properties of the point process of nodes in a stationary Voronoi tessellation. Math. Nachr. 281, 350375.
[10] 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, 235259.
[11] Maier, R., Mayer, J. and Schmidt, V. (2004). Distributional properties of the typical cell of stationary iterated tessellations. Math. Meth. Operat. Res. 59, 287302.
[12] Møller, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.
[13] Møller, J. and Stoyan, D. (2007). Stochastic geometry and random tessellations. In Tessellations in the Sciences: Virtues, Techniques and Applications of Geometric Tilings eds. R. van de Weijgaert et al., Springer, 32pp.
[14] Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.
[15] Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral. Marcel Dekker, New York.
[16] Zuyev, S. A. (1992). Estimates for distributions of the Voronoi polygon's geometric characteristics. Random Structures Algorithms 3, 149162.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed