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Boolean model; Laslett’s transform
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].
[1] Barbour A. D., Schmidt V.: On Laslett’s transform for the Boolean model. Adv. in Appl. Probab. 33 (2001), 1–5 DOI 10.1239/aap/999187893 | MR 1825312 | Zbl 0978.60017
[2] Billingsley P.: Convergence of Probability Measures. Second edition. Wiley, New York 1999 MR 1700749 | Zbl 0944.60003
[3] Cressie N. A. C.: Statistics for Spatial Data. Second edition. Wiley, New York 1993 MR 1239641 | Zbl 0799.62002
[4] Molchanov I. S.: Statistics of the Boolean model: From the estimation of means to the estimation of distribution. Adv. Appl. Probab. 27 (1995), 63–86 DOI 10.2307/1428096 | MR 1315578
[5] Molchanov I. S.: Statistics of the Boolean Model for Practioners and Mathematicions. Wiley, Chichester 1997
[6] Rataj J.: Point Processes (in Czech). Karolinum, Prague 2000
[7] Stoyan D., Kendall W. S., Mecke J.: Stochastic Geometry and Its Applications. Akademie–Verlag, Berlin 1987 MR 0879119 | Zbl 1155.60001
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