  We study the k-space fluctuations of the waveaction about its mean spectrum
in the turbulence of dispersive waves. We use a minimal model based on the
Random Phase Approximation (RPA) and derive evolution equations for the
arbitrary-order one-point moments of the wave intensity in the wavenumber
space. The first equation in this series is the familiar Kinetic Equation for
the mean waveaction spectrum, whereas the second and higher equations describe
the fluctuations about this mean spectrum. The fluctuations exhibit a
nontrivial dynamics if some long coordinate-space correlations are present in
the system, as it is the case in typical numerical and laboratory experiments.
Without such long-range correlations, the fluctuations are trivially fixed at
their Gaussian values and cannot evolve even if the wavefield itself is
non-Gaussian in the coordinate space. Unlike the previous approaches based on
smooth initial k-space cumulants, the RPA model works even for extreme cases
where the k-space fluctuations are absent or very large and intermittent. We
show that any initial non-Gaussianity at small amplitudes propagates without
change toward the high amplitudes at each fixed wavenumber. At each fixed
amplitude, however, the PDF becomes Gaussian at large time.
