Application of the Cram\'er-Wold theorem to testing for invariance under group actions

We address the problem of testing for the invariance of a probability measure under the action of a group of linear transformations. We propose a procedure based on consideration of one-dimensional projections, justified using a variant of the Cram\'er-Wold theorem. Our test procedure is powerful, computationally efficient, and dimension-independent, extending even to the case of infinite-dimensional spaces (multivariate functional data). It includes, as special cases, tests for exchangeability and sign-invariant exchangeability. We compare our procedure with some previous proposals in these cases, in a small simulation study. The paper concludes with two real-data examples.