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 $\hat{I}¸\in S^{n-1}$, let $\hat{I}{¼}_{\hat{I}}¸$ be the law of $⟨X,\hat{I}¸⟩$, and let $\hat{I}˜$ be uniformly distributed on $S^{n-1}$, independently of $X$. We prove the sharp estimate \[ \textsf{E} W_1(μ_Θ,γ_1) \le \frac{C}{n}. \] 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 $⟨x,y⟩$. Expanding this kernel in $⟨x,y⟩$, we control its linear, quadratic, and cubic terms using the quadratic variance inequality $\mathrm{Var}(X^{\top }MX)\le 8\mathrm{Tr}(M^2)$, while fixed-order moment estimates control the remainder.