{"ID":2829570,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.12363","arxiv_id":"2512.12363","title":"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.","short_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 proce...","url_abs":"https://arxiv.org/abs/2512.12363","url_pdf":"https://arxiv.org/pdf/2512.12363v1","authors":"[\"Zeusu Sato\"]","published":"2025-12-13T15:19:22Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\",\"stat.ME\"]","methods":"[]","has_code":false}