Let F be a distribution function (d.f.) in the maximum domain of attraction of an extreme value distribution H-gamma; then, F-u, d.f. of the excesses over u, converges, when u tends to s(+)(F), the end-point of F. to G(gamma,sigma (u)) (x), the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure, for gamma > -1, the existence of a function Lambda satisfying lim(u-->s+(F))Lambda (u) = gamma such that [GRAPHICS] converges to 0 faster than [GRAPHICS] (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.