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 Wasserstein-$p$ distance for locally dependent random variables and strongly mixing stationary random fields. Our proofs adapt the Stein dependency neighborhood method to the Wasserstein-$p$ 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，并已成功推广至局部相依数据和强混合随机场。本文建立了其关于传输距离的收敛速率，即对任意$p\ge1$，获得了局部相依随机变量和强混合平稳随机场的Wasserstein-$p$距离的上界。我们的证明将Stein依赖邻域方法推广至Wasserstein-$p$距离，并作为副产品，建立了相依随机变量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为弱相依随机变量的经验均值获得渐近紧且多项式快速衰减的尾界。