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

Infinite variance $H$-sssi processes as limits of particle systems