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.
"Poisson–Voronoi approximation." Ann. Appl. Probab. 19 (2) 719 - 736, April 2009. https://doi.org/10.1214/08-AAP561