Numerical and Analytical Computation of Some Second-Order Characteristics of Spatial Poisson-Voronoi Tessellations

摘要

We describe and discuss the explicit calculation of the pair correlation function of the point process of nodes associated with a three-dimensional stationary Poisson – Voronoi tessellation. Moreover, the precise asymptotics for the variance of the number of nodes in an expanding region and the variance of the number of vertices of the typical Poisson – Voronoi polyhedron are obtained. This gives rise to an asymptotically exact confidence interval for the number of nodes and cells when the sampling region is large enough. A geometric interpretation of our formulae shows that, among others, an essential problem is to calculate the mean volume of a tetrahedron whose vertices are uniformly distributed on a circular domain of the unit sphere.