In this article, we study the test for independence of two random elements $X$ and $Y$ lying in an infinite dimensional space ${\cal{H}}$ (specifically, a real separable Hilbert space equipped with the inner product $\langle ., .\rangle_{\cal{H}}$). In the course of this study, a measure of association is proposed based on the sup-norm difference between the joint probability density function of the bivariate random vector $(\langle l_{1}, X \rangle_{\cal{H}}, \langle l_{2}, Y \rangle_{\cal{H}})$ and the product of marginal probability density functions of the random variables $\langle l_{1}, X \rangle_{\cal{H}}$ and $\langle l_{2}, Y \rangle_{\cal{H}}$, where $l_{1}\in{\cal{H}}$ and $l_{2}\in{\cal{H}}$ are two arbitrary elements. It is established that the proposed measure of association equals zero if and only if the random elements are independent. In order to carry out the test whether $X$ and $Y$ are independent or not, the sample version of the proposed measure of association is considered as the test statistic after appropriate normalization, and the asymptotic distributions of the test statistic under the null and the local alternatives are derived. The performance of the new test is investigated for simulated data sets and the practicability of the test is shown for three real data sets related to climatology, biological science and chemical science.

翻译：本文研究无限维空间$\cal{H}$（具体为装备内积$\langle ., .\rangle_{\cal{H}}$的实可分希尔伯特空间）中两个随机元素$X$和$Y$的独立性检验问题。研究中提出了一种基于二元随机向量$(\langle l_{1}, X \rangle_{\cal{H}}, \langle l_{2}, Y \rangle_{\cal{H}})$的联合概率密度函数与随机变量$\langle l_{1}, X \rangle_{\cal{H}}$和$\langle l_{2}, Y \rangle_{\cal{H}}$边缘概率密度函数乘积之间上确界范数差异的关联度量，其中$l_{1}\in{\cal{H}}$和$l_{2}\in{\cal{H}}$为任意两个元素。证明该关联度量等于零当且仅当随机元素相互独立。为检验$X$与$Y$是否独立，经适当标准化后，将所提关联度量的样本版本作为检验统计量，并推导出原假设和局部备择假设下该检验统计量的渐近分布。通过模拟数据集评估新检验的性能，并利用气象学、生物科学和化学科学领域的三个真实数据集展示该检验的实用性。