Multivariate Poisson distributions associated with Boolean models

Abstract

We consider a d-dimensional Boolean model $\varXi = (\varXi _1+X_1)\cup (\varXi _2+X_2)\cup \cdots$ generated by a Poisson point process $\{X_i, i\ge 1\}$ with intensity measure $\varLambda$ and a sequence $\{\varXi _i, i\ge 1\}$ of independent copies of some random compact set $\varXi _0\,$. Given compact sets $K_1,\ldots ,K_{\ell }$, we show that the discrete random vector $(N(K_1),\ldots ,N(K_\ell ))$, where $N(K_j)$ equals the number of shifted sets $\varXi _i+X_i$ hitting $K_j$, obeys an $\ell$-variate Poisson distribution with $2^{\ell }-1$ parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set $\varXi$ in some unboundedly expanding window $W_n$ (as $n \rightarrow \infty$) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k-cylinders for $1\le k < d$ and more general set-valued functionals of independently marked Poisson processes.