A multivariate CLT in Wasserstein distance with near optimal convergence rate

Let $X_1, ldots , X_n$ be i.i.d. random vectors in $mathbb{R}^d$ with $|X_1| le beta$. Then, we show that $frac{1}{sqrt{n}}(X_1 + ldots + X_n)$ converges to a Gaussian in Wasserstein-2 distance at a rate of $Oleft(frac{sqrt{d} beta log n}{sqrt{n}} right)$, improving a result of Valiant and Valiant. The main feature of our theorem is that the rate of convergence is within $log n$ of optimal.