Limit Law for the Maximum Interpoint Distance of High Dimensional Dependent Variables

Abstract: In this paper, we considier the limiting distribution of the maximum interpoint Euclidean distance $M_n=\max _{1 \leq i&lt;j \leq n}\left|\boldsymbol{X}_i-\boldsymbol{X}_j\right|$, 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 $M_n<sup>2$ 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.