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 circumvents the curse of dimensionality. 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. Our methods extend to the case of infinite-dimensional spaces (multivariate functional data). The paper concludes with two real-data examples.

翻译：我们处理在一组线性变换行动下对概率测量的误差进行测试的问题,我们根据对一维预测的考虑提出一个程序,并使用Cram\'er-Wold定理的变体进行论证。我们的测试程序是强大的、计算效率高的,并绕过维度的诅咒。它包括作为特殊案例的可交换性和标志性易换性的测试。我们用一个小型模拟研究将我们的程序与以前在这些情况下的一些提议进行比较。我们的方法扩大到无限空间(多变量功能数据)的情况。文件最后提供了两个真实数据实例。