  It is shown that distributions arising in Renyi-Tsallis maximum entropy
setting are related to the Generalized Pareto Distributions (GPD) that are
widely used for modeling the tails of distributions. The relevance of such
modelization, as well as the ubiquity of GPD in practical situations follows
from Balkema-De Haan-Pickands theorem on the distribution of excesses (over a
high threshold). We provide an entropic view of this result, by showing that
the distribution of a suitably normalized excess variable converges to the
solution of a maximum Tsallis entropy, which is the GPD. This highlights the
relevance of the so-called Tsallis distributions in many applications as well
as some relevance to the use of the corresponding entropy.
