POisson-voronoi approximation

Abstract

Let X be a Poisson point process and K ℝd a measurable set. Construct the Voronoi cells of all points x X 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(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. © Institute of Mathematical Statistics, 2009.