  We study point process convergence for sequences of iid random walks. The
objective is to derive asymptotic theory for the extremes of these random
walks. We show convergence of the maximum random walk to the Gumbel
distribution under the existence of a $(2+\delta)$th moment. We make heavily
use of precise large deviation results for sums of iid random variables. As a
consequence, we derive the joint convergence of the off-diagonal entries in
sample covariance and correlation matrices of a high-dimensional sample whose
dimension increases with the sample size. This generalizes known results on the
asymptotic Gumbel property of the largest entry.
