### 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.

Original language | English |
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Pages (from-to) | 394-408 |

Number of pages | 15 |

Journal | Journal of Applied Probability |

Volume | 54 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2017 |

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### Keywords

- Random pointset
- stochastic geometry
- Voronoi tessellation

### ASJC Scopus subject areas

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

### Cite this

*Journal of Applied Probability*,

*54*(2), 394-408. https://doi.org/10.1017/jpr.2017.7

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

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 54, no. 2, pp. 394-408. https://doi.org/10.1017/jpr.2017.7

}

TY - JOUR

T1 - On the measure of Voronoi cells

AU - Devroye, Luc

AU - Györfi, L.

AU - Lugosi, Gábor

AU - Walk, Harro

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

AB - 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.

KW - Random pointset

KW - stochastic geometry

KW - Voronoi tessellation

UR - http://www.scopus.com/inward/record.url?scp=85021058947&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021058947&partnerID=8YFLogxK

U2 - 10.1017/jpr.2017.7

DO - 10.1017/jpr.2017.7

M3 - Article

AN - SCOPUS:85021058947

VL - 54

SP - 394

EP - 408

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 2

ER -