Multivariate Poisson distributions associated with Boolean models

Abstract

We consider a d−dimensional Boolean model Ξ = (Ξ 1 + X 1 ) ∪ (Ξ 2 + X 2 ) ∪ · · · generated by a Poisson point process {X i , i ≥ 1} with intensity measure Λ and a sequence {Ξ i , i ≥ 1} of independent copies of some random compact set Ξ 0 . Given compact sets K 1 , ..., K , we show that the discrete random vector (N (K 1 ), . . . , N (K )), where N (K j ) equals the number of shifted sets Ξ i +X i hitting K j , obeys a −variate Poisson distribution with 2 −1 parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set Ξ in some unboundedly expanding window W n (as n → ∞) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k−cylinders for 1 ≤ k < d and more general set-valued functionals of independently marked Poisson processes.