# Ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes

## TL;DR

This paper proves that ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes and extends De Finetti's theorem to mixtures of Bernoulli processes under uniform integrability conditions.

## Contribution

It establishes the isomorphism between ergodic quasi-exchangeable stationary processes and Bernoulli processes, and generalizes De Finetti's theorem for mixtures.

## Key findings

- Ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes.
- Processes with uniformly integrable Radon-Nikodym derivatives are mixtures of Bernoulli processes.
- Application to determinantal processes demonstrates the theoretical results.

## Abstract

{\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 $\{{d\mu_\sigma\over d\mu}\}$ 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. }