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

In this paper, we consider the limiting distribution of the maximum interpoint Euclidean distance M_n=\max _1 \leqslant i<j \leqslant n\left\|\boldsymbolX_i-\boldsymbolX_j\right\|, where \boldsymbolX_1, \boldsymbolX_2, \cdots, \boldsymbolX_n be a random sample drawn from a p dimensional population with dependent sub-Gaussian components. When the dimension tends to infinity with the sample size, we prove that M_n^2 under a suitable normalization asymptotically obeys a Gumbeltype distribution. The proofs mainly rely on the Chen-Stein Poisson approximation method and highdimensional Gaussian approximation.