  Assessing the probability of occurrence of extreme events is a crucial issue
in various fields like finance, insurance, telecommunication or environmental
sciences. In a multivariate framework, the tail dependence is characterized by
the so-called stable tail dependence function (STDF). Learning this structure
is the keystone of multivariate extremes. Although extensive studies have
proved consistency and asymptotic normality for the empirical version of the
STDF, non-asymptotic bounds are still missing. The main purpose of this paper
is to fill this gap. Taking advantage of adapted VC-type concentration
inequalities, upper bounds are derived with expected rate of convergence in
O(k^-1/2). The concentration tools involved in this analysis rely on a more
general study of maximal deviations in low probability regions, and thus
directly apply to the classification of extreme data.
