Title: 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.