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

Anomalous diffusion for a correlated process with long jumps