Random, nonuniform distribution of line segments on a circle
Suppose n points are distributed independently on a circle of unit circumference according to a distribution with density f, and each point forms the left endpoint of a segment of length \(a_ n\). Let \(L(n,a_ n)\) be the total number of gaps built by the union of these n segments, let \(M(n,a_ n)\) be the length of the largest gap, and let \(V(n,a_ n)\) be the total length of the gaps. The author studies the distributional behaviour of L, M, and V and gives limit theorems, as \(n\to \infty\), distinguishing three different cases according to the asymptotic behaviour of \(a_ n\). The limit theory follows the development given by \textit{J. Hüsler} [J. Appl. Probab. 19, 578-587 (1982; Zbl 0495.60018)] in the uniform case (f\(\equiv 1)\). Since for some of the theorems f is required to have a unique minimum point, the results do not cover the uniform case, in general.
scientific article; zbMATH DE number 3890428
scientific article