Exchangeability and the Cramér-Wold theorem

We address the problem of testing exchangeability and sign-invariant exchangeability based on one-dimensional projections. For that purpose, we first characterize those distributions that are (sign-invariant) exchangeable in terms of their one-dimensional projections. Based on these results, we propose new test procedures that are powerful, computationally efficient, and that circumvent the curse of dimensionality. We compare our procedure with some previous proposals in a small simulation study. Some of the results can be extended to the case of infinite-dimensional spaces (multivariate functional data).