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