Ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes

{\abstract{\textwidth=4,5 in} A discrete time process, with law $\mu$, is quasi-exchangeable if for any finite permutation $\sigma$ of time indices, the law ${\mu }_{\sigma }$ of the resulting process is equivalent to $\mu$. For a quasi-exchangeable stationary process we prove mainly (1) that if the process is ergodic then it is isomorphic to a Bernoulli process and (2) that if the family of all Radon-Nikodym derivatives $\{\frac{d{\mu }_{\sigma }}{d\mu }\}$ is uniformly integrable then the process is a mixture of Bernoulli processes, which generalizes De Finetti’s Theorem. We give application of (1) to some determinantal processes. }