Maximum interpoint distance of high-dimensional random vectors

TL;DR

This paper establishes a limit theorem for the maximum interpoint distance among high-dimensional i.i.d. vectors, showing convergence to the Gumbel distribution and exploring related statistical applications.

Contribution

It introduces a new limit theorem for maximum interpoint distances in high dimensions, including dependent cases and applications to covariance matrices and hypothesis testing.

Findings

Maximum interpoint distance converges to Gumbel distribution as dimension grows.

Joint convergence results for maximum and minimum interpoint distances.

Application to testing mean equality in high-dimensional data.

Abstract

A limit theorem for the largest interpoint distance of $p$ independent and identically distributed points in $\mathbb{R}^n$ to the Gumbel distribution is proved, where the number of points $p=p_n$ tends to infinity as the dimension of the points $n\to\infty$. The theorem holds under moment assumptions and corresponding conditions on the growth rate of $p$. We obtain a plethora of ancillary results such as the joint convergence of maximum and minimum interpoint distances. Using the inherent sum structure of interpoint distances, our result is generalized to maxima of dependent random walks with non-decaying correlations and we also derive point process convergence. An application of the maximum interpoint distance to testing the equality of means for high-dimensional random vectors is presented. Moreover, we study the largest off-diagonal entry of a sample covariance matrix. The proofs are based on the Chen-Stein Poisson approximation method and Gaussian approximation to large deviation probabilities.