A limit theorem for the maximal interpoint distance of a random sample in the unit ball

Abstract

We prove a limit theorem for the the maximal interpoint distance (also called the diameter) for a sample of n i.i.d. points in the unit ball of dimension 2 or more. The exact form of the limit distribution and the required normalisation are derived using assumptions on the tail of the interpoint distance for two i.i.d. points. The results are specialised for the cases when the points have spherical symmetric distributions, in particular, are uniformly distributed in the whole ball and on its boundary.