  We investigate the distribution properties of the fractional L\'evy motion.
We consider separately the cases $0<H<1/2$ (short memory) and $1/2<H<1$ (long
memory), where $H$ is the Hurst parameter, and present the asymptotic behaviour
of the distribution density of the process. Some examples are provided, in
which it is shown that the behaviour of the density in the cases $0<H<1/2$ and
$1/2<H<1$ is completely different.
