On a Voronoi aggregative process related to a bivariate Poisson process

Abstract

We consider two independent homogeneous Poisson processes Π 0 and Π 1 in the plane with intensities λ 0 and λ 1 , respectively. We study additive functionals of the set of Π 0 -particles within a typical Voronoi Π 1 -cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π 0 -particles to the nucleus within a typical Voronoi Π 1 -cell.