On random covering of a circle

Abstract

Let XjXj, j=1,2,…,nj=1,2,…,n be the independent and identically distributed random vectors which take the values on the unit circumference. Let SnSn be the area of the convex polygon having XjXj as vertices. The paper by Nagaev and Goldfield (1989) has proved the asymptotic normality of random variableSn. Our main aim is to show that the random variableSn can be represented as a sum of functions of uniform spacings. This allows us to apply known results related to uniform spacings for the analysis of SnSn.