Second‐order properties of the point process of nodes in a stationary Voronoi tessellation

Abstract

Abstract In this paper we derive representation formulae for the second factorial moment measure of the point process of nodes and the second moment of the number of vertices of the typical cell associated with a stationary normal Voronoi tessellation in ℝ d . In case the Voronoi tessellation is generated by a stationary Poisson process with intensity λ > 0 the corresponding pair correlation function g V,λ ( r ) can be expressed by a weighted sum of d +2 (numerically tractable) multiple parameter integrals. The asymptotic variance of the number of nodes in an increasing cubic domain as well as the second moment of the number of vertices of the typical Poisson Voronoi cell are calculated exactly by means of these parameter integrals. The existence of a ( d − 1)st‐order pole of g V,λ ( r ) at r = 0 is proved and the exact value of lim r →0 r d –1 g V,λ ( r ) is determined. In the particular cases d = 2 and d = 3 the graph of g V ,1 ( r ) including its local extreme points, the points of level 1 of gV, 1( r ) and other characteristics are computed by numerical integration. Furthermore, an asymptotically exact confidence interval for the intensity of nodes is obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)