The Optimal Rate in the Averaged Random-Marginal Central Limit Theorem for Log-Concave Measures

Abstract

Let $X$ be a centered isotropic log-concave random vector in $\mathbb{R}^n$. For $\theta\in S^{n-1}$, let $\mu_\theta$ be the law of $\langle X,\theta\rangle$, and let $\Theta$ be uniformly distributed on $S^{n-1}$, independently of $X$. We prove the sharp estimate \[ \textsf{E} W_1(\mu_\Theta,\gamma_1) \le \frac{C}{n}. \] Here the Wasserstein distance is computed after the direction is fixed and is then averaged over the sphere. No symmetry assumption is imposed. A product measure with centered exponential coordinates gives a matching lower bound of order $n^{-1}$. The proof separates the averaged-direction law from the fluctuation among fixed directions. For the first part, a Taylor expansion in the random radius retains a mean-zero cancellation and yields an $O(n^{-1})$ error. For the second, a weighted $L^2$ distance between distribution functions is converted into an exact spherical kernel depending only on $|x|^2$, $|y|^2$, and $\langle x,y\rangle$. Expanding this kernel in $\langle x,y\rangle$, we control its linear, quadratic, and cubic terms using the quadratic variance inequality $\operatorname{Var}(X^\top M X) \le 8\operatorname{Tr}(M^2)$, while fixed-order moment estimates control the remainder.