Martingale central limit theorems in $p$-Wasserstein distance

Abstract

We obtain multivariate martingale central limit theorems in $p$-Wasserstein distance with respect to the ${\ell }_r$ norm in ${\mathbb{R}}^d$ for $p\ge 1$ 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.

37 pages. Preliminary version