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Laslett’s transform for the Boolean model in \(\mathbb {R}^d\). (English) Zbl 1249.60012

Summary: 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. A. C. Cressie, Statistics for spatial data. New York etc.: John Wiley and Sons (1991; Zbl 0799.62002)]) although the proof based on discretization is partly heuristic and not complete. Starting with the same idea, we present a rigorous proof in the \(d\)-dimensional case. As a technical tool, an 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 [Adv. Appl. Probab. 33, No. 1, 1–5 (2001; Zbl 0978.60017)].

MSC:

60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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References:

[1] Barbour A. D., Schmidt V.: On Laslett’s transform for the Boolean model. Adv. in Appl. Probab. 33 (2001), 1-5 · Zbl 0978.60017 · doi:10.1239/aap/999187893
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[3] Cressie N. A. C.: Statistics for Spatial Data. Second edition. Wiley, New York 1993 · Zbl 0799.62002
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