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On Laslett’s transform for the Boolean model. (English) Zbl 0978.60017

Consider the planar Boolean model, where the germs form a homogeneous Poisson point process with intensity \(\lambda\) and the grains are convex compact random sets. With every grain we associate its lower tangent point. These tangent points do not form a Poisson point process and exhibit rather strong dependencies, see I. Molchanov and D. Stoyan [Adv. Appl. Probab. 26, No. 2, 301-323 (1994; Zbl 0806.62078)]. According to the Laslett’s transform, the exposed tangent points in the right half-plane are moved horizontally to the left by the distance equal to the total length of the covered part of the perpendicular drawn from the corresponding tangent point to the \(y\)-axis. It is known [see N. A. C. Cressie, “Statistics for spatial data” (1991; Zbl 0799.62002), Section 9.5.3] that the transformed exposed points form a Poisson point process in the right half-plane with intensity \(\lambda\). The authors give a new and very elegant proof of this result, which is based on a martingale argument. They also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity \(\lambda\) can be embedded in it.

MSC:

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