On the epsilon-delta Structure Underlying Chatterjee's Rank Correlation

Abstract

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.