Martingale central limit theorems in $p$-Wasserstein distance

We obtain multivariate martingale central limit theorems in $p$-Wasserstein distance with respect to the $\ell_r$ norm in $\mathbb{R}^d$ for $p\geq 1$ and $r\in [1,\infty]$, which generalize the results for $p=1$ and $r=2$ in the literature. As corollaries, we obtain the Yurinskii coupling and Cramér-type moderate deviation results. We also provide an illustrative application to the stochastic gradient descent algorithm. To prove our main results, we combine Lindeberg's swapping argument with a new Gaussian convolution inequality controlling the $p$-Wasserstein distance between a Gaussian convolved with a perturbation and the Gaussian with matching mean and covariance matrix. The latter is obtained by developing the recent line of research on $p$-Wasserstein bounds.