  Let $\mathbf{X}(n) \in \mathbb{R}^d$ be a sequence of random vectors, where
$n\in\mathbb{N}$ and $d = d(n)$. Under certain weakly dependence conditions, we
prove that the distribution of the maximal component of $\mathbf{X}$ and the
distribution of the maximum of their independent copies are asymptotically
equivalent. Our result on extremal independence relies on new lower and upper
bounds for the probability that none of a given finite set of events occurs. As
applications, we obtain the distribution of various extremal characteristics of
random discrete structures such as maximum codegree in binomial random
hypergraphs and the maximum number of cliques sharing a given vertex in
binomial random graphs. We also generalise Berman-type conditions for a
sequence of Gaussian random vectors to possess the extremal independence
property.
