  Random Phase Approximation (RPA) provides a very convenient tool to study the
ensembles of weakly interacting waves, commonly called Wave Turbulence. In its
traditional formulation, RPA assumes that phases of interacting waves are
random quantities but it usually ignores randomness of their amplitudes.
Recently, RPA was generalised in a way that takes into account the amplitude
randomness and it was applied to study of the higher momenta and probability
densities of wave amplitudes. However, to have a meaningful description of wave
turbulence the RPA properties assumed for the initial fields must be proven to
survive over the nonlinear evolution time, and such a proof is the main goal of
the present paper. We derive an evolution equation for the full probability
density function which contains the complete information about the joint
statistics of all wave amplitudes and phases. We show that, for any initial
statistics of the amplitudes, the phase factors remain statistically
independent uniformly distributed variables. If in addition the initial
amplitudes are also independent variables (but with arbitrary distributions)
they will remain independent when considered in small sets which are much less
than the total number of modes. However, if the size of a set is of order of
the total number of modes then the joint probability density for this set is
not factorisable into the product of one-mode probabilities. In the other
words, the modes in such a set are involved in a ``collective'' (correlated)
motion. We also study new type of correlators describing the phase statistics.
