Random, nonuniform distribution of line segments on a circle

Peter Hall

Stochastic Processes and their Applications, 1984, vol. 18, issue 2, 239-261

Abstract: The random distribution of line segments on a circle is used in several applications to model a physical surface being impinged upon at random by foreign bodies. In this context, the hypothesis of uniformity suggests that the bodies come from all directions with equal probabilities, and have no preference for one region of the surface over any other. The present paper examines the distribution of line segments when the hypothesis of uniformity breaks down. We adopt a limit theoretic approach, and identify the principal aspects of nonuniformity which influence asymptotic behaviour, as the number of segments increases and their length decreases. Our results are stated formally as limit theorems, and described informally by means of brief summaries.

Date: 1984
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