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Martingale central limit theorems in p-Wasserstein distance
Abstract
We obtain multivariate martingale central limit theorems in p-Wasserstein distance with respect to the ℓr norm in ℝd for p≥ 1 and r∈ [1,∞], 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.
Related
- Global central limit theorems for Markov chains
- Global Central Limit Theorems for Markov chains with normal transition operator
- Central limit theorem for Wasserstein projection - the case of convex order
- Functional central limit theorems for Laguerre tessellations
- Central Limit Theorems for Finitary Factors of iid Processes
BibTeX
@misc{fang2026martingale,
title = {{Martingale central limit theorems in $p$-Wasserstein distance}},
author = {Xiao Fang and Yuta Koike and Zi-Yao Su},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11557},
eprint = {2609.11557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11557},
}
RIS
TY - UNPB
TI - Martingale central limit theorems in p-Wasserstein distance
AU - Xiao Fang
AU - Yuta Koike
AU - Zi-Yao Su
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11557
UR - https://arxiv.org/abs/2609.11557
AB - We obtain multivariate martingale central limit theorems in p-Wasserstein distance with respect to the ℓ_r norm in ℝ^d for p≥ 1 and r∈ [1,∞], 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.
ER -