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We show that the predual of a JBW$^*$-triple has the weak Banach-Saksproperty, that is, reflexive subspaces of a JBW$^*$-triple predual aresuper-reflexive. We also prove that JBW$^*$-triple preduals satisfy theKoml\'os property (which can be considered an abstract version of the weak lawof large numbers). The results rely on two previous papers from which we inferthe fact that, like in the classical case of $L^1$, a subspace of aJBW$^*$-triple predual contains $\ell_1$ as soon as it contains uniform copiesof $\ell_1^n$.

Weak Banach-Saks property and Komlos theorem for preduals of JBW$^*$-triples