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

TL;DR

This paper derives explicit moments for the maximum distance between random points on a circle, providing a new proof that its scaled distribution converges to a Gumbel distribution as the number of points grows large.

Contribution

It introduces a moments-based approach to analyze the distribution of the maximum distance, offering a new proof of the Gumbel convergence result.

Findings

Explicit formula for moments of the maximum distance

Proof of Gumbel distribution convergence for scaled maximum distance

Method applicable to similar geometric probability problems

Abstract

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.