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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.

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* 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

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On the measure of Voronoi cells

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

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