Multivariate Poisson Distributions Associated with Boolean Models

We consider a d-dimensional Boolean model Z = (Z_1+X_1) cup (Z_2+X_2) cup ... generated by a Poisson point process X_i, i = 1,2,... with some intensity measure and a sequence Z_i, i = 1,2,... of independent copies of some random compact set Z_0. Given compact sets K_1,...,K_l, we show that the discrete random vector (N(K_1),...,N(K_l)), where N(K_j) equals the number of shifted sets Z_i+X_i hitting K_j, obeys a l-variate Poisson distribution with 2^l-1 parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set Z in some unboundedly expanding window W_n (as n --> infty) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k-cylinders for k =1,...,d-1 and more general set-valued functionals of independently marked Poisson processes