  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.
