We show that under minimal assumptions on a random vector $X\in\mathbb{R}^d$ and with high probability, given $m$ independent copies of $X$, the coordinate distribution of each vector $(\langle X_i,\theta \rangle)_{i=1}^m$ is dictated by the distribution of the true marginal $\langle X,\theta \rangle$. Specifically, we show that with high probability, \[\sup_{\theta \in S^{d-1}} \left( \frac{1}{m}\sum_{i=1}^m \left|\langle X_i,\theta \rangle^\sharp - \lambda^\theta_i \right|^2 \right)^{1/2} \leq c \left( \frac{d}{m} \right)^{1/4},\] where $\lambda^{\theta}_i = m\int_{(\frac{i-1}{m}, \frac{i}{m}]} F_{ \langle X,\theta \rangle }^{-1}(u)\,du$ and $a^\sharp$ denotes the monotone non-decreasing rearrangement of $a$. Moreover, this estimate is optimal. The proof follows from a sharp estimate on the worst Wasserstein distance between a marginal of $X$ and its empirical counterpart, $\frac{1}{m} \sum_{i=1}^m \delta_{\langle X_i, \theta \rangle}$.

翻译：我们证明，在随机向量$X\in\mathbb{R}^d$的最小假设下，且以高概率成立，给定$m$个独立同分布的$X$样本，每个向量$(\langle X_i,\theta \rangle)_{i=1}^m$的坐标分布由真实边际分布$\langle X,\theta \rangle$的分布决定。具体而言，我们以高概率证明：\[\sup_{\theta \in S^{d-1}} \left( \frac{1}{m}\sum_{i=1}^m \left|\langle X_i,\theta \rangle^\sharp - \lambda^\theta_i \right|^2 \right)^{1/2} \leq c \left( \frac{d}{m} \right)^{1/4},\] 其中$\lambda^{\theta}_i = m\int_{(\frac{i-1}{m}, \frac{i}{m}]} F_{ \langle X,\theta \rangle }^{-1}(u)\,du$，且$a^\sharp$表示$a$的单调非递减重排。此外，该估计是最优的。证明源于$X$的边际分布与其经验分布$\frac{1}{m} \sum_{i=1}^m \delta_{\langle X_i, \theta \rangle}$之间的最坏Wasserstein距离的精确估计。