Nonparametric estimation of a function from noiseless observations at random points

We study the measure of a typical cell in a Voronoi tessellation defined by n independent random points drawn from a density f in Rd. In particular, we prove that the asymptotic distribution of the measure–with respect to dμ = f(x)dx–of the cell centered at a point x ∈ Rd is independent of x and the density f . We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as d becomes large. In particular, we show that the variance converges to zero exponentially fast in d. We also obtain a bound independent of the density for the rate of convergence of the diameter of a typical Voronoi cell.