We provide an epsilon-delta interpretation of Chatterjee's rank correlation by tracing its origin to a notion of local dependence between random variables. Starting from a primitive epsilon-delta construction, we show that rank-based dependence measures arise naturally as epsilon to zero limits of local averaging procedures. Within this framework, Chatterjee's rank correlation admits a transparent interpretation as an empirical realization of a local L1 residual. We emphasize that the probability integral transform plays no structural role in the underlying epsilon-delta mechanism, and is introduced only as a normalization step that renders the final expression distribution-free. We further consider a moment-based analogue obtained by replacing the absolute deviation with a squared residual. This L2 formulation is independent of rank transformations and, under a Gaussian assumption, recovers Pearson's coefficient of determination.

翻译：我们通过追踪Chatterjee秩相关源于随机变量间局部依赖的概念，为其提供了ε-δ解释。从原始的ε-δ构造出发，我们证明基于秩的依赖度量自然显现为局部平均过程在ε趋于零时的极限。在此框架下，Chatterjee秩相关可被透明地解释为局部L1残差的经验实现。我们强调概率积分变换在底层ε-δ机制中不具结构作用，仅作为使最终表达式无分布依赖的归一化步骤引入。我们进一步考虑通过用平方残差替代绝对偏差得到的矩基类比。该L2表述独立于秩变换，并在高斯假设下恢复了皮尔逊决定系数。