On a Voronoi aggregative process related to a bivariate Poisson process

We consider two independent homogeneous Poisson processes Pi(0) and Pi(1) in the plane with intensities lambda(0) and lambda(1), respectively. We study additive functionals of the set of Pi(0)-particles within a typical Voronoi Pi(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 Pi(0)-particles to the nucleus within a typical Voronoi Pi(1)-cell.