AI 中文总结
该研究针对中心化迷向对数凹随机向量，证明了平均随机边际分布与标准高斯分布间Wasserstein距离的$O(n^{-1})$最优上界，并给出匹配下界，通过分解平均分布与方向波动、结合泰勒展开和二次方差不等式完成证明。
AI 中文摘要
设$X$是$\u211d^n$中的中心化迷向对数凹随机向量。对于$θ∈S^{n-1}$，令$μ_θ$为$⟨X,θ⟩$的分布，令$Θ$独立于$X$且在$S^{n-1}$上均匀分布。我们证明了如下精确估计：$\textsf{E} W_1(μ_Θ,γ_1) \le \frac{C}{n}$。此处Wasserstein距离在方向固定后计算，随后在球面上取平均。我们未施加任何对称性假设。具有中心化指数坐标的乘积测度给出了阶为$n^{-1}$的匹配下界。证明将平均方向分布与固定方向间的波动分离开来。对于第一部分，随机半径的泰勒展开保留了零均值抵消性，得到$O(n^{-1})$的误差。对于第二部分，分布函数之间的加权$L^2$距离被转化为仅依赖于$|x|^2$、$|y|^2$和$⟨x,y⟩$的精确球面核。将该核按$⟨x,y⟩$展开后，我们利用二次方差不等式$\operatorname{Var}(X^\top M X) \le 8\operatorname{Tr}(M^2)$控制其线性项、二次项和三次项，而固定阶矩估计控制余项。
英文摘要
Let $X$ be a centered isotropic log-concave random vector in $\mathbb{R}^n$. For $θ\in S^{n-1}$, let $μ_θ$ be the law of $\langle X,θ\rangle$, and let $Θ$ 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 $\langle x,y\rangle$. 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.
