Limit Distribution of Spacing on the Unit Circle

Abstract

Order statistics of continuous uniform random variable on unit interval are related to several distributions. It is known that spacings, one of combina- tions of order statistics, follow the Dirichlet distribution. It can be applied to circular uniform random variables also with mild modication. In this paper, we review the derivation of the joint distribution of spacings on the unit interval and circle. Using this, we prove that the distribution of discrete random variables from a circular arrangement that consist of two distinct elements converges to Dirichlet distribution.