THE ASYMPTOTIC PROPERTY OF ADDITIVE EXCESSES AND THE EXTREME VALUES THEORY - THE CASE OF GUMBEL EXTREMAL DISTRIBUTION

A classical necessary and sufficient condition for a distribution P to belong to the domain of attraction of the Gumbel extremal distributio n is: there exists a mapping g from ] - infinity, s(P)[ into R+ such t hat: For All x is an element of R, lim(t - s(P)) (F) over bar (t + g(t )x) / (F) over bar (t) = - ln H-0 (x) = e(-x), where (F) over bar = 1 - F and s(P) = sup{x is an element of R; (F) over bar (x) > 0}. This p roperty has no probabilistic interpretation for negative x; this is wh y we study its restriction to R+ and show that it provides a sufficien t condition for P to belong to the domain of attraction of the Gumbel extremal distribution. (C) Academie des Sciences/Elsevier, Paris.