The limit law of the largest interpoint distance in a $d$-dimensional ellipsoid

Abstract

We consider the largest interpoint distance $M_n={\max }_{1\le i<j\le n}\Vert X_i-X_j\Vert$ among independent random points $X_1,\dots ,X_n$, uniformly distributed on a $d$-dimensional ellipsoid. We assume that the largest semi-axis has length 1 and multiplicity $k\ge 2$, whereas the remaining semi-axes are strictly smaller. In this situation, the diameter is attained on a manifold of dimension $k-1$, and the extremal points are no longer isolated. We establish a weak limit law for the diameter deficit $2-M_n$. Writing $q=d-k$ and $\alpha =q+(k+3)/2$, we show that $n^{2/\alpha }(2-M_n)$ converges in distribution to a Weibull random variable. The proof is based on a local analysis near the diameter manifold, a sharp asymptotic formula for the two-point tail probability, and a Chen--Stein Poisson approximation for rare nearly diametral pairs.

12 pages