A general approach for testing independence in Hilbert spaces
将投影相关检验方法推广到希尔伯特空间，用于检验随机向量的独立性，适用于高维甚至无限维数据，并保持检验的有效性和一致性。
We generalize the projection correlation idea for testing independence of random vectors which is known as a powerful method in multivariate analysis. A universal Hilbert space approach makes the new testing procedures useful in various cases and ensures the applicability to high or even infinite dimensional data. We prove that the new tests keep the significance level under the null hypothesis of independence exactly and can detect any alternative of dependence in the limit, in particular in settings where the dimensions of the observations is infinite or tend to infinity simultaneously with the sample size. Simulations demonstrate that the generalization does not impair the good performance of the approach and confirm our theoretical findings. Furthermore, we describe the implementation of the new approach and present a real data example for illustration.