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On the distribution of random lines

Published online by Cambridge University Press:  14 July 2016

Peter Hall
Peter Hall*
Affiliation:
The Australian National University
*
∗ Postal address: Department of Statistics, SGS, The Australian National University, P.O. Box 4, Canberra, A.C.T. 2600, Australia.
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Abstract

Draw two lines, L 1 and L 2, in a plane. Along L 1 place the points of a Poisson process, and through each point draw a line, the angles of intersection with L 1 being distributed independently and uniformly on (0, π) . The intercepts of these random lines with L 2 form a new process, which is Poisson if and only if L 1 and L 2 are parallel. Curiously, if L 1 and L 2 are inclined then, with probability 1, the new process forms a dense subset of L 2. It is not really even a point process. In the present paper we shall investigate this anomaly and some of its generalizations.

Keywords

LIMIT THEOREMPOISSON POINT PROCESSRANDOM ANGLERANDOM LINE

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 3 , September 1981 , pp. 606 - 616
Copyright
Copyright © Applied Probability Trust 

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References

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