Asymptotic normality of the one-dimensional Integrated R^2 estimator under independence

Establish the central limit theorem for the one-dimensional estimator ν_n^{1-dim}(Y, X) of the integrated R^2 dependence measure: Prove that, when X and Y are independent and both have continuous distributions, the scaled statistic √n · ν_n^{1-dim}(Y, X) converges in distribution to a normal law N(0, π^2/3 − 3) as n → ∞.

Background

Section 6 introduces a simplified one-dimensional estimator ν_n^{1-dim}(Y, X) for the integrated R^2 dependence measure, obtained by ordering the sample by X, taking ranks of Y, and aggregating weighted indicator contributions across adjacent rank intervals (equation (eq:simpleEst)). Under independence with continuous margins, Proposition 6.2 derives the exact asymptotic mean 2/n and variance (π^2/3 − 3)/n for the estimator.

Motivated by the practical need for simple null distributions to enable efficient testing (similar in spirit to Chatterjee's correlation), the authors explicitly conjecture a central limit theorem for √n * ν_n^{1-dim}(Y, X) under independence. They note that existing techniques in the cited works do not apply due to non-local dependency structure in their statistic and suggest that a moment-method approach may succeed, though they do not carry out the calculations.