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

TL;DR

This paper studies the scaling limits of a particle system with weights performing independent Lévy motions, revealing convergence to stationary stable self-similar processes and dependence on the integral of the occupation function.

Contribution

It introduces a new particle system model and demonstrates its convergence to known stable self-similar processes, highlighting the impact of the occupation function's integral.

Findings

Limit processes are stationary stable self-similar processes.

Different limit behaviors occur depending on the occupation function's integral.

The model connects particle systems with advanced stochastic processes.

Abstract

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.