Poisson–Voronoi approximation

Poisson–Voronoi approximation

Matthias Heveling and Matthias Reitzner

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

Abstract

Let X be a Poisson point process and K⊂ℝ^d a measurable set. Construct the Voronoi cells of all points x∈X with respect to X, and denote by v_X(K) the union of all Voronoi cells with nucleus in K. For K a compact convex set the expectation of the volume difference V(v_X(K))−V(K) and the symmetric difference V(v_X(K)ΔK) 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.

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Primary Subjects: 60D05

Secondary Subjects: 60G55, 52A22, 60C05

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

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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1241702248
Digital Object Identifier: doi:10.1214/08-AAP561
Mathematical Reviews number (MathSciNet): MR2521886
Zentralblatt MATH identifier: 1172.60003

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