A distribution \(P\) on \(\mathbb{R}\) is said to lie in the domain of attraction of the Gumbel extremal distribution if there exist sequences \((a_n)\) in \(\mathbb{R}^*_+\) and \((b_n)\) in \(\mathbb{R}\) such that \[ (\forall x\in\mathbb{R})\;\lim_nF^n (a_nx+b_n)=\exp(-e^{-x}).\tag{D} \] Here the author proposes a sufficient condition to ensure that (D) holds. This condition, the additive excesses property, is more restrictive than the classical condition given by J. Galambos [“The asymptotic theory of extreme order statistics” (1978; Zbl 0381.62039)], but is more suitable in allowing a probabilistic interpretation.