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 result resembles the entropic approach to the central limit theorem; however, the convergence in entropy proved here is weaker than the convergence in supremum norm given by Pickandspsila theorem.