Poisson--Voronoi approximation

Abstract

Let $X$ be a Poisson point process and $K\subset {\mathbb{R}}^d$ a measurable set. Construct the Voronoi cells of all points $x\in X$ 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 $V(v_X(K)△K)$ 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.

Published in at http://dx.doi.org/10.1214/08-AAP561 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Laws of large numbers in stochastic geometry with statistical applications