Asymptotics for statistical distances based on Voronoi tessellations
The Voronoi tessellation of a set of points \(\{x_1,\ldots,x_n\}\) in \(\mathbb R^d\) is a partition which associates with \(x_i\) the `cell' of all elements of \(\mathbb R^d\) which are closer to \(x_i\) than to any other \(x_j\), \(j\neq i\). The authors obtain an information-type inequality and a strong law for a wide class of statistical distances between empirical estimates and random measures, based on such Voronoi tessellations. Their results extend some basic results in the asymptotic theory of sample spacings, when the Voronoi cells are interpreted as \(d\)-dimensional spacings. The study exploits a monotonicity property of Voronoi tessellations analogous to that found in 1-dimensional spacings.
scientific article; zbMATH DE number 1784216