Geodesic
Laslett’s transform for the Boolean model in $\Bbb R^d$
Kybernetika, Tome 42 (2006) no. 5, pp. 569-584 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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=\lbrace x\in \mathbb{R}^d: x_1 = 0\rbrace $ 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 [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 [barb+schm].
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=\lbrace x\in \mathbb{R}^d: x_1 = 0\rbrace $ 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 [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 [barb+schm].
Classification : 60D05, 60G55
Keywords: Boolean model; Laslett’s transform 
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Černý, Rostislav. Laslett’s transform for the Boolean model in $\Bbb R^d$. Kybernetika, Tome 42 (2006) no. 5, pp. 569-584. http://geodesic.mathdoc.fr/item/KYB_2006_42_5_a3/

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