Random, nonuniform distribution of line segments on a circle
type: Item
instance of: scholarly article
title: Random, nonuniform distribution of line segments on a circle
zbMATH Open document ID: 0558.60016
DOI: 10.1016/0304-4149(84)90298-9
published in: Stochastic Processes and their Applications
publication date: 1984-01-01T00:00:00Z
review text: 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.
reviewed by: Wolfgang Weil
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 60G99
zbMATH DE Number: 3890428
zbMATH Keywords: central limit theorem
zbMATH Keywords: line segments
zbMATH Keywords: order statistics
zbMATH Keywords: spacings
zbMATH Keywords: asymptotic behaviour
MaRDI profile type: Publication
full work available at URL: 0304-4149(84)90298-9
OpenAlex ID: W2069652934
cites work: Moments of coverage and coverage spaces
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cites work: On the lengths of the pieces of a stick broken at random
cites work: On convergence of the coverage by random arcs on a circle and the largest spacing
cites work: Random coverage of the circle and asymptotic distributions
cites work: Random circles on a sphere
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author: Peter Hall
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description: scientific article; zbMATH DE number 3890428