On the distribution of random lines in the general case
type: Item
instance of: scholarly article
title: On the distribution of random lines in the general case
zbMATH Open document ID: 0657.60019
author: Ming Zheng
published in: Chinese Annals of Mathematics. Series B
publication date: 1988-01-01T00:00:00Z
review text: L is a line in a plane through the origin, with an angle \(\alpha\) to the x-axis, \(0<\alpha <\pi\). M is a point process on the positive x-axis. Through the nth point of M draw a line with a random angle \(\theta_ n\) to the x-axis, \(\phi^+\) is the set of intersections of those lines with \(L^+.\) Let \(m=EM\). If, for every \(c>0\), \(Em(c\theta^{-1})<\infty\), then \(\phi^+\) is locally finite on L, and let \(\tilde M\) be the point process constructed by \(\phi^+\), then \(E\tilde M\) exists. If, for all intervals \(L\subset L^+\), \[ \int^{\infty}_{0}r(I,x) M(dx)=\infty \quad a.s., \] then \(\phi^+\) is dense on \(L^+\). If L is drawn parallel to the x-axis, the same results can be obtained, and this time \(\tilde M\) is a cluster point process with cluster center M.
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 4074058
zbMATH Keywords: intersections of random lines
zbMATH Keywords: random angle
zbMATH Keywords: cluster point process
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description: scientific article; zbMATH DE number 4074058