Abstract
Let X1,X2,…,Xn be a random sample coming from a p-dimensional population with independent sub-exponential components. Denote the maximum interpoint Euclidean distance by Mn=max1≤i<j≤n‖Xi−Xj‖. When the dimension p=p(n)→∞ with the sample size n→∞, it proves that Mn2 under a suitable normalization asymptotically obeys a Gumbel type distribution. The proofs mainly depend on the Stein–Chen Poisson approximation method and the moderate deviation of the sum of independent random variables.
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