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.