On measure-preserving transformations and doubly stationary symmetric stable processes
In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral representation which is itself stationary, and they gave an example of a stationary symmetric stable process which they claimed was not doubly stationary. Here we show that their process actually had a moving average representation, and hence was doubly stationary. We also characterize doubly stationary processes in terms of measure-preserving regular set isomorphisms and the existence of σ-finite invariant measures. One consequence of the characterization is that all harmonizable symmetric stable processes are doubly stationary. Another consequence is that there exist stationary symmetric stable processes which are not doubly stationary.
- 28D10: One-parameter continuous families of measure-preserving transformations
- 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
- 60G10: Stationary processes
- 60E07: Infinitely divisible distributions; stable distributions
- Department of Statistics and Applied Probability, University of California, Santa Barbara, California 93106-3110, U.S.A.
- Hugo Steinhaus Center for Stochastic Methods, Technical University of Wrocławi, 50-370 Wrocław, Poland
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