Laslett’s transform for the Boolean model in ${ℝ}^d$

Černý, Rostislav. "Laslett’s transform for the Boolean model in $\mathbb {R}^d$." Kybernetika 42.5 (2006): 569-584. <http://eudml.org/doc/33825>.

@article{Černý2006,
abstract = {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].},
author = {Černý, Rostislav},
journal = {Kybernetika},
keywords = {Boolean model; Laslett’s transform; Boolean model; Laslett's transform},
language = {eng},
number = {5},
pages = {569-584},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Laslett’s transform for the Boolean model in $\mathbb \{R\}^d$},
url = {http://eudml.org/doc/33825},
volume = {42},
year = {2006},
}

TY - JOUR
AU - Černý, Rostislav
TI - Laslett’s transform for the Boolean model in $\mathbb {R}^d$
JO - Kybernetika
PY - 2006
PB - Institute of Information Theory and Automation AS CR
VL - 42
IS - 5
SP - 569
EP - 584
AB - 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].
LA - eng
KW - Boolean model; Laslett’s transform; Boolean model; Laslett's transform
UR - http://eudml.org/doc/33825
ER -