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 dmax 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,…,1). We use this result to give a new proof of the fact that the law of ndmax−logn converges to a Gumbel distribution as n tends to infinity.