We isolate various su cient conditions for a Banach space X to have the so-called Blum-Hanson property. In particular, we show that X has the Blum-Hanson property if either the modulus of asymptotic smoothness of X has an extremal behaviour at in nity, or if X is uniformly G^ateaux smooth and embeds isometrically into a Banach space with a 1-unconditional nite-dimensional decomposition
