On the rate of convergence of the martingale central limit theorem in Wasserstein distances

For martingales with a wide range of integrability, we will quantify the rate of convergence of the central limit theorem via Wasserstein distances of order $r$, $1\le r\le 3$. Our bounds are in terms of Lyapunov’s coefficients and the ${\mathcal{L}}^{r\mathrm{/}2}$ fluctuation of the total conditional variances. We will show that our Wasserstein-1 bound is optimal up to a multiplicative constant.