Identification of periodic and cyclic fractional stable motions

Self-similar stable mixed moving average processes can be related to nonsingular flows through their minimal representations. Self-similar stable mixed moving averages related to dissipative flows have been studied, as well as processes associated with identity flows which are the simplest type of conservative flows. The focus here is on self-similar stable mixed moving averages related to periodic and cyclic flows. Periodic flows are conservative flows such that each point in the space comes back to its initial position in finite time, either positive or null. The flow is cyclic if the return time is positive. Self-similar mixed moving averages are called periodic, respectively cyclic, fractional stable motions if their minimal representations are generated by periodic, respectively cyclic, flows. These processes, however, are often defined by a nonminimal representation. We provide a way to detect whether they are periodic or cyclic even if their representation is nonminimal. By using these identification results, we obtain a more refined decomposition of self-similar mixed moving averages.