Often in WT, studies are restricted to wavefields with fastly decaying spatial
correlations [2]. For such fields, the statistics of the Fourier modes
is close to being Gaussian. Indeed, it the correlation length is much smaller
than the size of the box, then this box can be divided into many smaller
boxes, each larger than the correlation length. The Fourier transform over the
big box will be equal to the sum of the Fourier transforms over the smaller
boxes which are statistically independent quantities. Therefore, by the
Central Limit Theorem, the big-box Fourier transform has a Gaussian
distribution. Corrections to Gaussianity are small as the ratio of the
correlation volume to the box volume. On the other hand, in the RPA defined
above the amplitude PDF is not specified and can significantly deviate from
the Rayleigh distribution (corresponding to Gaussian wavefields). Such fields
correspond to long correlations of order or greater than the box size. In
fact, long correlated fields are quite typical for WT because, due to weak
nonlinearity, wavepackets can propagate over long distance preserving their
identity. Moreover, restricting ourselves to short-correlated fields would
render our study of the PDF evolution meaningless because the later would be
fixed at the Gaussian state. Note that long correlations modify the usual
Wick's rule for the correlator splitting by adding a *singular
cumulant*, e.g. for the forth-order correlator,