The Annals of Probability

On Convergence of the Coverage by Random Arcs on a Circle and the Largest Spacing

Lars Holst

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab., Volume 9, Number 4 (1981), 648-655.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994370

Digital Object Identifier
doi:10.1214/aop/1176994370

Mathematical Reviews number (MathSciNet)
MR624691

Zentralblatt MATH identifier
0468.60018

JSTOR
links.jstor.org

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60E05: Distributions: general theory 60F99: None of the above, but in this section 62E99: None of the above, but in this section

Keywords
Spacings uniform distribution random arcs coverage distribution limit theorems geometrical probability extreme values

Citation

Holst, Lars. On Convergence of the Coverage by Random Arcs on a Circle and the Largest Spacing. Ann. Probab. 9 (1981), no. 4, 648--655. doi:10.1214/aop/1176994370. https://projecteuclid.org/euclid.aop/1176994370


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