ON MEASURE-PRESERVING TRANSFORMATIONS AND DOUBLY STATIONARY SYMMETRICAL STABLE PROCESSES

In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral repre sentation which is itself stationary, and they gave an example of a st ationary symmetric stable process which they claimed was not doubly st ationary. Here we show that their process actually had a moving averag e representation, and hence was doubly stationary. We also characteriz e doubly stationary processes in terms of measure-preserving regular s et isomorphisms and the existence of sigma-finite invariant measures. One consequence of the characterization is that all harmonizable symme tric stable processes are doubly stationary Another consequence is tha t there exist stationary symmetric stable processes which are not doub ly stationary.