  We introduce a simple stochastic system able to generate anomalous diffusion
both for position and velocity. The model represents a viable description of
the Fermi's acceleration mechanism and it is amenable to analytical treatment
through a linear Boltzmann equation. The asymptotic probability distribution
functions (PDF) for velocity and position are explicitly derived. The diffusion
process is highly non-Gaussian and the time growth of moments is characterized
by only two exponents $\nu_x$ and $\nu_v$. The diffusion process is anomalous
(non Gaussian) but with a defined scaling properties i.e. $P(|{\bf x}|,t) =
1/t^{\nu_x}F_x(|{\bf x}|/t^{\nu_x})$ and similarly for velocity.
