## The Annals of Probability

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

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

#### 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