Kybernetika

Consider a stationary Boolean model $X$ with convex grains in $\mathbb{R}^d$ and let any exposed lower tangent point of $X$ be shifted towards the hyperplane $N_0=\{x\in \mathbb{R}^d: x_1 = 0\}$ by the length of the part of the segment between the point and its projection onto the $N_0$ covered by $X$. The resulting point process in the halfspace (the Laslett's transform of $X$) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie \cite{Cressie}) although the proof based on discretization is partly heuristic and not complete. Starting from the same idea we present a rigorous proof in the $d$-dimensional case. As a technical tool equivalent characterization of vague convergence for locally finite integer valued measures is formulated. Another proof based on the martingale approach was presented by A.\,D. Barbour and V. Schmidt~\cite{barb+schm}.