Following the approach of [22,23], we now define a ``Random
Phase and Amplitude'' (RPA) field.^{2}We say that the field is of
RPA type if it possesses the following statistical properties:

- All amplitudes and their phase factors are
independent random variables, i.e. their joint PDF is equal to the
product of the one-mode PDF's corresponding to each individual
amplitude and phase,
- The phase factors are uniformly distributed on the unit
circle in the complex plane, i.e. for any mode

In [22,23] RPA was *assumed* to hold over the nonlinear
time. In [24] this assumption was examined *a posteriori*,
i.e. based on the evolution equation for the multi-point PDF obtained with RPA
initial fields. Below we will describe this work.
We will see that RPA fails to hold in its pure form as
formulated above but it survives in the leading order so that the WT
closure built using the RPA is valid. We will also see that
independence of the the phase factors is quite straightforward,
whereas the amplitude independence is subtle. Namely, amplitudes
are independent only up to a correction. Based on this
knowledge, and leaving justification for later on in this paper, we
thus reformulate RPA in a weaker form which holds over the nonlinear
time and which involves -mode PDF's with rather than the
full -mode PDF.