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 \leq i<j \leq 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 Gumbel-type distribution. The proofs mainly rely on the Chen-Stein Poisson approximation method and high-dimensional Gaussian approximation.