  The recent development of a statistical model for incompressible
Navier-Stokes (NS) fluids based on inverse kinetic theory (IKT, 2004-2008)
poses the problem of searching for particular realizations of the theory which
may be relevant for \textit{the statistical description of turbulence} and in
particular for the so-called \textit{homogeneous, isotropic}and
\textit{stationary} turbulence (HIST). Here the problem is set in terms of the
$1-$point velocity probability density function (PDF) which determines a
complete IKT-statistical model for NS fluids. This raises the interesting
question of identifying the statistical assumptions under which a Gaussian PDF
can be achieved in such a context. In this paper it is proven that for the IKT
statistical model, HIST requires necessarily that $f_{1}$ must be SIED (namely
\textit{stationary}, \textit{isotropic} and \textit{% everywhere-defined}).
This implies, in turn, that the functional form of the PDF is uniquely
prescribed at all times. In particular, it is found that necessarily the PDF
must coincide with an isotropic Gaussian distribution. The conclusion is
relevant for the investigation of the so-called homogenous, isotropic and
stationary turbulence.
