A high-dimensional CLT in W2 distance with near optimal convergence rate

Let X-1, ... , X-n be i.i.d. random vectors in R-d with parallel to X-1 parallel to <= beta. Then, we show that 1/root n (X-1 + ... + X-n) converges to a Gaussian in quadratic transportation (also known as "Kantorovich" or "Wasserstein") distance at a rate of O(root d beta log n/root n), 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 -> infinity.