Sharp Fourth-Moment Wasserstein CLT Bounds

Abstract

We prove that for the normalized sum $W=\sum_{i=1}^n X_i/\sqrt{n}$ of independent, mean-zero, $d$-dimensional random vectors with $\operatorname{Cov}(W) =\Sigma=\operatorname{diag}(\sigma_1^2,\dots, \sigma_d^2)$ (the diagonal form of the covariance matrix is assumed without loss of generality), the 2-Wasserstein distance from $W$ to $N(0,\Sigma)$ is bounded by \[\sqrt{2\sum_{j=1}^d \min \left\{\sigma_j^2, \frac{\sum_{i=1}^n \mathbb{E} [|X_i'-X_i|^2(X'_{ij}-X_{ij})^2 ] }{n^2 \sigma_j^2} \right\}},\] where $|\cdot|$ denotes the Euclidean norm, $X_i'$ is an independent copy of $X_i$, and $X_{ij}$ ($X_{ij}'$ resp.) denotes the $j$th coordinate of the vector $X_i$ ($X_i'$ resp.). On the one hand, it gives the optimal rate $O(\beta\sqrt{d/n})$ if $|X_i|\leq \beta$ for all $i$. On the other hand, in the isotropic case where $\Sigma=I_d$, it gives the optimal fourth-moment bound $O\left(\sqrt{\sum_{i=1}^n \mathbb{E}|X_i|^4/n^2}\right)$. Moreover, the coordinatewise minima suggest a tradeoff between the na\"ive coupling and the normal coupling for each component. To prove the main result, we develop a new transportation inequality using a coupling via the F\"ollmer process with a general covariance matrix and the martingale property of its drift. The resulting bound is then controlled using a Bonis-type expansion together with Bessel's inequality. The method and the result extend to $p$-Wasserstein bounds. As a consequence, we obtain the Hilbert space 4-Wasserstein CLT bound $O(n^{-1/4}[\operatorname{tr}(Q^{1/2})]^{1/2})$, where, in the i.i.d. case, $Q=\mathbb{E}[|X_1'-X_1|^2(X_1'-X_1)^{\otimes2}]$ and the $O(n^{-1/4})$ rate is optimal given the dimension-free form of $Q$. We further strengthen it to a dimension-free strong approximation result with a near-optimal error rate through the KMT embedding construction.