On the polygon generated by nn random points on a circle

Let AnAn denote the surface area of the random polygon generated by nn independent points uniformly distributed on the unit circle in R2R2. We investigate the asymptotic properties of AnAn. In particular, we show that E[An]=π−4π3/n2+o(1/n5/2)E[An]=π−4π3/n2+o(1/n5/2), Var[An]=160π6/n5+o(1/n5)Var[An]=160π6/n5+o(1/n5) and that the distribution of (An−E[An])/Var[An] is asymptotically normal. Similar results are obtained for the perimeter. As a byproduct of this investigation, we give a simple proof of a general convergence theorem for sums of powers of the spacings in a sample from the uniform distribution on an interval.