  We discuss the non--linear growth of the kurtosis of the smoothed peculiar
velocity field (along an arbitrary direction), in an Einstein--de Sitter
universe, induced by Gaussian primordial density fluctuations. Applying the
perturbative theory, we show that, for different cosmological models, a
departure from the original Gaussian distribution is gravitationally induced
only on small scales ($\simlt 20-30$ Mpc). Models with scale--free power
spectrum $P(k)\propto k^{\,n}$, with $-1< n \leq 1$, are also considered. When
the fluid particles move according to the Zel'dovich approximation the
probability distribution of the peculiar velocity field remains unaltered
during the evolution.
