On the distribution of random lines in the general case (Q1110893)

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.