%0 Journal Article
%T Poisson--Voronoi approximation
%A Matthias Heveling
%A Matthias Reitzner
%J Mathematics
%D 2009
%I arXiv
%R 10.1214/08-AAP561
%X 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 $V(v_X(K)\triangle K)$ 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.
%U http://arxiv.org/abs/0906.4238v1