On the measure of Voronoi cells

Luc Devroye, L. Györfi, Gábor Lugosi, Harro Walk

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 Rd. 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.

Original languageEnglish
Pages (from-to)394-408
Number of pages15
JournalJournal of Applied Probability
Volume54
Issue number2
DOIs
Publication statusPublished - Jun 1 2017

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Voronoi Cell
Asymptotic distribution
Voronoi Tessellation
Cell
Absolutely Continuous
Probability Measure
Rate of Convergence
Moment
Converge

Keywords

  • Random pointset
  • stochastic geometry
  • Voronoi tessellation

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

Cite this

On the measure of Voronoi cells. / Devroye, Luc; Györfi, L.; Lugosi, Gábor; Walk, Harro.

In: Journal of Applied Probability, Vol. 54, No. 2, 01.06.2017, p. 394-408.

Research output: Contribution to journalArticle

Devroye, L, Györfi, L, Lugosi, G & Walk, H 2017, 'On the measure of Voronoi cells', Journal of Applied Probability, vol. 54, no. 2, pp. 394-408. https://doi.org/10.1017/jpr.2017.7
Devroye, Luc ; Györfi, L. ; Lugosi, Gábor ; Walk, Harro. / On the measure of Voronoi cells. In: Journal of Applied Probability. 2017 ; Vol. 54, No. 2. pp. 394-408.
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