  Extreme Value Theory plays an important role to provide approximation results
for the extremes of a sequence of independent random variables when their
distribution is unknown. An important one is given by the {generalised Pareto
distribution} $H_\gamma(x)$ as an approximation of the distribution
$F_t(s(t)x)$ of the excesses over a threshold $t$, where $s(t)$ is a suitable
norming function. In this paper we study the rate of convergence of
$F_t(s(t)\cdot)$ to $H_\gamma$ in variational and Hellinger distances and
translate it into that regarding the Kullback-Leibler divergence between the
respective densities.
