  We consider a particle system with weights and the scaling limits derived
from its occupation time. We let the particles perform independent recurrent
L\'evy motions and we assume that their initial positions and weights are given
by a Poisson point process. In the limit we obtain a number of recently
discovered stationary stable self-similar processes recently studied in [SAM1]
and [SAM2]. We also observe very different limit processes depending on whether
the function, whose occupation time is considered, integrates to zero or not.
