Asymptotic independence of counts in isotropic planar point processes of phase-type

The isotropic planar point processes of phase-type are natural generalizations of the Poisson process on the plane. On the one hand, those processes are isotropic and stationary for the mean count, as in the case of the Poisson process. On the other hand, they exhibit dependence of counts in disjoint sets. In a recent paper, we have proved that the number of points in a square window has a Poisson distribution asymptotically as the window is located far away from the origin of the process. We extend our work to the case of a window of arbitrary shape.