The Annals of Applied Probability

Poisson–Voronoi approximation

Matthias Heveling and Matthias Reitzner

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Abstract

Let X be a Poisson point process and K⊂ℝd a measurable set. Construct the Voronoi cells of all points xX with respect to X, and denote by vX(K) the union of all Voronoi cells with nucleus in K. For K a compact convex set the expectation of the volume difference V(vX(K))−V(K) and the symmetric difference V(vX(KK) is computed. Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a Poisson point process. Concentration inequalities for both quantities are proved using Azuma’s inequality.

Article information

Source
Ann. Appl. Probab. Volume 19, Number 2 (2009), 719-736.

Dates
First available in Project Euclid: 7 May 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1241702248

Digital Object Identifier
doi:10.1214/08-AAP561

Mathematical Reviews number (MathSciNet)
MR2521886

Zentralblatt MATH identifier
1172.60003

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60C05: Combinatorial probability

Keywords
Poisson point process Poisson–Voronoi cell jackknife estimate of variance approximation of convex sets valuation

Citation

Heveling, Matthias; Reitzner, Matthias. Poisson–Voronoi approximation. Ann. Appl. Probab. 19 (2009), no. 2, 719--736. doi:10.1214/08-AAP561. https://projecteuclid.org/euclid.aoap/1241702248


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