A limit theorem for the largest interpoint distance of $p$ independent and identically distributed points in $\mathbb{R}^n$ to the Gumbel distribution is proved, where the number of points $p=p_n$ tends to infinity as the dimension of the points $n\to\infty$. The theorem holds under moment assumptions and corresponding conditions on the growth rate of $p$. We obtain a plethora of ancillary results such as the joint convergence of maximum and minimum interpoint distances. Using the inherent sum structure of interpoint distances, our result is generalized to maxima of dependent random walks with non-decaying correlations and we also derive point process convergence. An application of the maximum interpoint distance to testing the equality of means for high-dimensional random vectors is presented. Moreover, we study the largest off-diagonal entry of a sample covariance matrix. The proofs are based on the Chen-Stein Poisson approximation method and Gaussian approximation to large deviation probabilities.

翻译：本文证明了在$\mathbb{R}^n$中独立同分布的$p$个点的最大点间距离服从Gumbel分布的极限定理，其中点数$p=p_n$随维度$n\to\infty$而趋于无穷。该定理在矩条件及关于$p$增长率的相应条件下成立。我们获得了一系列辅助结果，如最大与最小点间距离的联合收敛性。利用点间距离固有的求和结构，我们将结果推广至具有非衰减相关性的相依随机游走最大值，并推导出点过程收敛性。文中给出了最大点间距离在高维随机向量均值相等性检验中的应用，同时研究了样本协方差矩阵的最大非对角元。证明基于Chen-Stein泊松逼近方法和大型偏差概率的高斯逼近。