In this article, we propose a test of independence for functional random variables modelled as elements of Hilbert spaces. First, we provide a general recipe for constructing measures of dependence among multiple random functions. These measures are non-negative, and under fairly general assumptions, they take the value zero only when the functions are independent. We consider one such measure based on the d-variable Hilbert-Schmidt Independence Criterion and propose a consistent estimator of this measure. Next, we construct a test for independence based on this estimator and establish its large sample consistency under general alternatives. Extensive simulation studies are carried out to compare the performance of the proposed test with some popular tests available in the literature.

中文翻译：

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希尔伯随机变量的独立性检验

在本文中，我们提出了对建模为希尔伯特空间元素的函数随机变量的独立性测试。首先，我们提供了一个通用方法来构建多个随机函数之间的依赖度量。这些度量是非负的，并且在相当普遍的假设下，它们仅在函数独立时取零值。我们考虑一种基于 d 变量 Hilbert-Schmidt 独立准则的此类度量，并提出该度量的一致估计量。接下来，我们基于该估计量构建独立性检验，并建立其在一般备选方案下的大样本一致性。进行了广泛的模拟研究，以将提议的测试的性能与文献中可用的一些流行测试进行比较。