Conditional independence is a fundamental concept in many areas of statistical research, including, for example, sufficient dimension reduction, causal inference, and statistical graphical models. In many modern applications, data arise in the form of random functions, making it important to determine whether two random functions are conditionally independent given a third. However, to the best of our knowledge, existing conditional independence tests in the literature apply only to multivariate data, and extensions to the functional setting are not available. To fill this gap, we develop a kernel-based test for conditional independence of random functions based on the conjoined conditional covariance operator (CCCO). We rigorously derive the asymptotic distribution of the CCCO estimator using a recently established sharpened convergence rate for the regression operator (Choi et al., 2026). Based on this result, we construct a test statistic using the spectral decomposition of the operator appearing in the asymptotic distribution. The proposed method is illustrated through applications to an activity and biometrics dataset and a macroeconomic dataset.

翻译：条件独立性是统计研究中许多领域的基本概念，例如充分降维、因果推断和统计图模型。在许多现代应用中，数据以随机函数的形式出现，这使得确定两个随机函数在给定第三个函数时是否条件独立变得至关重要。然而，据我们所知，现有文献中的条件独立性检验仅适用于多变量数据，且尚未扩展到函数型数据。为填补这一空白，我们基于联合条件协方差算子开发了一种用于随机函数条件独立性的核检验。我们利用近期建立的回归算子收敛速率（Choi 等，2026），严格推导了联合条件协方差算子估计量的渐近分布。基于这一结果，我们利用渐近分布中出现的算子的谱分解构造了一个检验统计量。所提出的方法通过应用于活动和生物特征数据集以及宏观经济数据集进行了展示。