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Stable stationary processes related to cyclic flows. (English) Zbl 1054.60056
The authors study real- and complex-valued symmetric \(\alpha\)-stable stationary processes having an integral representation that is generated by periodic or cyclic flows. For general flows J. Rosiński [Ann. Probab. 23, No. 1, 464–473 (1995; Zbl 0831.60007) and ibid., No. 3, 1163–1187 (1995; Zbl 0836.60038)] obtained a unique decomposition in distribution of such processes into independent processes generated by a dissipative, respectively a conservative flow. Moreover, in the real-valued (resp. complex-valued) case, the conservative part can again be uniquely decomposed into a trivial (resp. harmonizable) process and an independent process having no trivial (resp. harmonizable) component.
The present paper refines the decomposition of Rosiński for special conservative flows, in which every space point comes back to its initial position after a finite (periodic flows) or positive and finite (cyclic flows) period in time. It is shown that the conservative part again can be decomposed into a trivial (resp. harmonizable) component and a cyclic component. Processes being representable in such a way are called stationary periodic processes, where the notion periodic refers to the flow. Among these, stationary cyclic processes fail to have a trivial (resp. harmonizable) part. In case of a minimal (nonredundant) representation, the authors prove that stationary periodic processes can only be generated by periodic flows, which fails for general representations. For general representations criteria others than the flow are provided to identify stationary periodic and cyclic processes among symmetric \(\alpha\)-stable stationary processes. For minimal representations, a further decomposition shows that the cyclic component still inhibits a nonperiodic conservative part which can be separated uniquely in distribution. Throughout the paper illustrative examples are given.

MSC:
60G52 Stable stochastic processes
60G10 Stationary stochastic processes
37A50 Dynamical systems and their relations with probability theory and stochastic processes
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