  We discuss diffusion properties of a dynamical system, which is characterised
by long-tail distributions and finite correlations. The particle velocity has
the stable L\'evy distribution; it is assumed as a jumping process (the
kangaroo process) with a variable jumping rate. Both the exponential and the
algebraic form of the covariance -- defined for the truncated distribution --
are considered. It is demonstrated by numerical calculations that the
stationary solution of the master equation for the case of power-law
correlations decays with time, but a simple modification of the process makes
the tails stable. The main result of the paper is a finding that -- in contrast
to the velocity fluctuations -- the position variance may be finite. It rises
with time faster than linearly: the diffusion is anomalously enhanced. On the
other hand, a process which follows from a superposition of the
Ornstein-Uhlenbeck-L\'evy processes always leads to position distributions with
a divergent variance which means accelerated diffusion.
