A high-dimensional CLT in $$\mathcal {W}_2$$ W 2 distance with near optimal convergence rate

Let (X_1,ldots ,X_n) be i.i.d. random vectors in (mathbb {R}^d) with (Vert X_1Vert le beta ). Then, we show that $$begin{aligned} frac{1}{sqrt{n}}left( X_1 + cdots + X_nright) end{aligned}$$converges to a Gaussian in quadratic transportation (also known as “Kantorovich” or “Wasserstein”) distance at a rate of (O left( 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 for (n, d rightarrow infty ).