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August, 1981 On Convergence of the Coverage by Random Arcs on a Circle and the Largest Spacing
Lars Holst
Ann. Probab. 9(4): 648-655 (August, 1981). DOI: 10.1214/aop/1176994370


Consider $n$ points taken at random on the circumference of a unit circle. Let the successive arc-lengths between these points be $S_1, S_2, \cdots, S_n$. Convergence of the moment generating function of $\max_{1 \leq k \leq n} S_k - \ln n$ is proved. Let each point be associated with an arc, each of length $a_n$, and let the length of the circumference which is not covered by any arc, the vacancy, be $V_n$. Convergence of the vacancy after suitable scaling is obtained. The methods used are general and can, e.g., be used to obtain asymptotic results for other spacings and coverage problems.


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Lars Holst. "On Convergence of the Coverage by Random Arcs on a Circle and the Largest Spacing." Ann. Probab. 9 (4) 648 - 655, August, 1981.


Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0468.60018
MathSciNet: MR624691
Digital Object Identifier: 10.1214/aop/1176994370

Primary: 60D05
Secondary: 60E05 , 60F99 , 62E99

Keywords: coverage distribution , Extreme values , Geometrical probability , limit theorems , random arcs , spacings , uniform distribution

Rights: Copyright © 1981 Institute of Mathematical Statistics


Vol.9 • No. 4 • August, 1981
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