Poisson–Voronoi approximation

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.