We provide a unified framework for independence and mean independence tests based on the Hilbert-Schmidt independence criterion, extending some previous results in the literature to hold in general topological spaces. We also present a complete theoretical analysis of the test statistic asymptotic behavior when the observed sample corresponds to a partial sample path of some stationary and ergodic stochastic process under near epoch dependence assumptions. In particular, we explore the test statistic consistency and limit distribution under both fixed and local hypothesis. The finite sample performance of the test(s) is illustrated with a succinct simulation study involving functional data.

翻译：本文为基于希尔伯特-施密特独立性准则的独立性与均值独立性检验提供了统一框架，将先前文献中的部分结论推广至一般拓扑空间。当观测样本对应于满足近邻相依假设的某平稳遍历随机过程的部分样本路径时，我们给出了检验统计量渐近行为的完整理论分析。特别地，我们探讨了在固定假设与局部假设下检验统计量的一致性与极限分布。通过一项涉及函数型数据的简洁模拟研究，展示了该检验的有限样本表现。