A Martingale Approach To Fluctuations of Rank Estimators in Sensitivity Analysis

Given a bivariate random pair $(X,Y)$, a natural problem is to estimate, from a single sample $(X_i,Y_i)_{1\le i\le n}$, quantities such as $\mathbb{E}\left[ \mathbb{E}[ Y\mid X ]^2 \right]$. More broadly, sensitivity indices are designed to quantify the possibly nonlinear influence of an input variable $X$ on an output variable $Y$. A classical example is the Sobol' index $$\frac{\mathrm{Var}(\mathbb{E}[Y\mid X])}{\mathrm{Var}(Y)} \in [0,1] \ .$$ Another important example is the Cram\'er--von Mises (CvM) index. Following the pioneering work of Chatterjee \cite{chatterjee2021new}, consistent rank-based estimators are now available for such quantities. In this paper, we prove sharp fluctuation results using martingale methods. Our framework yields a unified treatment of the univariate Sobol' index, a multivariate extension involving several functions of the same scalar input, and the CvM index. As a consequence, we recover, unify, and simplify results from Gamboa et al. \cite{gamboa2022global, gamboa2023erratum}, Lin--Han \cite{lin2022limit}, and Kroll \cite{kroll2024asymptotic}. In particular, we work under minimal regularity assumptions. Furthermore, while the Gaussian fluctuation phenomenon itself was already known, the novelty lies in the structure of the asymptotic variance: for the CvM index, we obtain, to the best of our knowledge, the first explicit formula, while for the Sobol' index, we derive a new expression with a more structured form.