Point processes associated with stationary stable processes

Point processes induced by stationary symmetric α-stable (SαS) processes can have diverse behavior. We distinguish two cases, depending on whether the stationary SαS process is governed by a dissipative or conservative flow. In the case of dissipative flows, the process is a mixed moving average and the family of scaled point processes converge to a Poisson cluster process. In the case of a conservative flow, we give two examples showing how diverse the behavior can be.