  Turbulence closure for the weakly nonlinear stochastic waves requires,
besides weak nonlinearity, randomness in both the phases and the amplitudes of
the Fourier modes. This randomness, once present initially, must remain over
the nonlinear evolution time. Finding out to what extent is this true is the
main goal of the present Letter. For this analysis we derive an evolution
equation for the full probability density function (PDF) of the wave field. We
will show that, for any statistics of the amplitudes, phases tend to stay
random if they were random initially. If in addition the initial amplitudes are
independent variables they will remain independent in a coarse-grained sense,
i.e. when considered in small subsets which are much less than the total set of
modes.
