  A distributional route to Gaussianity, associated with the concept of
Conservative Mixing Transformations in ensembles of random vector-valued
variables, is proposed. This route is completely different from the additive
mechanism characterizing the application of Central Limit Theorem, as it is
based on the iteration of a random transformation preserving the ensemble
variance. Gaussianity emerges as a ``supergeneric'' property of ensemble
statistics, in the case the energy constraint is quadratic in the norm of the
variables. This result puts in a different light the occurrence of equilibrium
Gaussian distributions in kinetic variables (velocity, momentum), as it shows
mathematically that, in the absence of any other dynamic mechanisms, almost
Gaussian distributions stems from the low-velocity approximations of the
physical conservation principles. Whenever, the energy constraint is not
expressed in terms of quadratic functions (as in the relativistic case), the
Juttner distribution is recovered from CMT.
