The central limit theorem is one of the most fundamental results in probability and has been successfully extended to locally dependent data and strongly-mixing random fields. In this paper, we establish its rate of convergence for transport distances, namely for arbitrary $p\ge1$ we obtain an upper bound for the Wassertein-$p$ distance for locally dependent random variables and strongly mixing stationary random fields. Our proofs adapt the Stein dependency neighborhood method to the Wassertein-$p$ distance and as a by-product we establish high-order local expansions of the Stein equation for dependent random variables. Finally, we demonstrate how our results can be used to obtain tail bounds that are asymptotically tight, and decrease polynomially fast, for the empirical average of weakly dependent random variables.

翻译：中心极限定理是概率论中最基本的结果之一，已被成功推广至局部依赖数据和强混合随机场。本文研究该定理在传输距离意义下的收敛速率：对任意$p\ge1$，我们给出了局部依赖随机变量与强混合平稳随机场的Wasserstein-$p$距离上界。证明过程将Stein依赖邻域方法推广至Wasserstein-$p$距离，并由此建立了依赖随机变量Stein方程的高阶局部展开。最后，我们展示了如何利用这些结果获得弱依赖随机变量经验均值的尾部界，该尾部界具有渐近紧性且以多项式速度衰减。