  We study nonlinear wave equations on $\mathbb R^{2+1}$ with quadratic
derivative nonlinearities, which include in particular nonlinearities
exhibiting a null form structure, with random initial data in $H_x^1\times
L^2_x$. In contrast to the counterexamples of Zhou \cite{Zhou} and
Foschi-Klainerman \cite{FK}, we obtain a uniform time interval $I$ on which the
Picard iterates of all orders are almost surely bounded in $C_t(I ; \dot
H_x^1)$.
