On the random coverage of the circle

Abstract

Place n arcs of equal length a uniformly at random on the circumference of a circle. We discuss the joint limit distributions of the number of gaps, the uncovered proportion of the circle and the lengths of the largest gap and of the smallest gap, depending on how a → 0 as n →∞. We show that the results may be proved in a unified and simple way by using a result of Le Cam.