In this paper, we considier the limiting distribution of the maximum interpoint Euclidean distance $M_n=\max _{1 \leq i<j \leq n}\left\|\boldsymbol{X}_i-\boldsymbol{X}_j\right\|$, where $\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots, \boldsymbol{X}_n$ be a random sample coming from a $p$-dimensional population with dependent sub-gaussian components. When the dimension tends to infinity with the sample size, we proves that $M_n^2$ under a suitable normalization asymptotically obeys a Gumbel type distribution. The proofs mainly depend on the Stein-Chen Poisson approximation method and high dimensional Gaussian approximation.

翻译：本文考虑最大点间欧氏距离 $M_n=\max _{1 \leq i<j \leq n}\left\|\boldsymbol{X}_i-\boldsymbol{X}_j\right\|$ 的极限分布，其中 $\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots, \boldsymbol{X}_n$ 是来自具有相依次高斯分量的 $p$ 维总体的随机样本。当维数随样本量趋于无穷大时，我们证明 $M_n^2$ 在适当归一化下渐近服从 Gumbel 型分布。证明主要依赖于 Stein-Chen 泊松逼近方法和高维高斯逼近。