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, \ldots, \boldsymbolX_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.