Limiting distribution of the maximal distance between random points on a circle: A moments approach

Motivated by the problem of computing the distribution of the largest distance $d_{\max}$ between $n$ random points on a circle we derive an explicit formula for the moments of the maximal component of a random vector following a Dirichlet distribution with concentration parameters $(1,\ldots,1)$. We use this result to give a new proof of the fact that the law of $n\,d_{\max}-\log n$ converges to a Gumbel distribution as $n$ tends to infinity.