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In this paper, we considier the limiting distribution of the maximuminterpoint Euclidean distance $M_n=\max _{1 \leq i<j \leqn}\left\|\boldsymbol{X}_i-\boldsymbol{X}_j\right\|$, 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 thedimension tends to infinity with the sample size, we proves that $M_n^2$ undera suitable normalization asymptotically obeys a Gumbel type distribution. Theproofs mainly depend on the Stein-Chen Poisson approximation method and highdimensional Gaussian approximation.

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