On a Voronoi aggregative process related to a bivariate Poisson process

Abstract

We consider two independent homogeneous Poisson processes Π₀and Π₁in the plane with intensities λ₀and λ₁, respectively. We study additive functionals of the set of Π₀-particles within a typical Voronoi Π₁-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 Π₀-particles to the nucleus within a typical Voronoi Π₁-cell.