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Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution ; it is well-known that Fu (x) , where F u is the d.f of the excesses over u , converges, when u tends to s+ (F) , the end-point of F , to , where is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for , a function Λ which verifies and is such that converges to 0 faster than .

Penultimate approximation for the distribution of the excesses