The central limit theorem (CLT) is one of the most fundamental results in probability; and establishing its rate of convergence has been a key question since the 1940s. For independent random variables, a series of recent works established optimal error bounds under the Wasserstein-p distance (with p>=1). In this paper, we extend those results to locally dependent random variables, which include m-dependent random fields and U-statistics. Under conditions on the moments and the dependency neighborhoods, we derive optimal rates in the CLT for the Wasserstein-p distance. Our proofs rely on approximating the empirical average of dependent observations by the empirical average of i.i.d. random variables. To do so, we expand the Stein equation to arbitrary orders by adapting the Stein's dependency neighborhood method. Finally we illustrate the applicability of our results by obtaining efficient tail bounds.

翻译：中心极限定理（CLT）是概率论中最基本的成果之一，自20世纪40年代以来，其收敛速度的确定一直是一个关键问题。对于独立随机变量，近期一系列工作建立了Wasserstein-p距离（p≥1）下的最优误差界。本文将这些结果推广至局部依赖随机变量，包括m依赖随机场和U统计量。在矩条件及依赖邻域假设下，我们推导出Wasserstein-p距离下CLT的最优收敛速率。证明过程通过用独立同分布随机变量的经验平均值逼近相依观测值的经验平均值来实现。为此，我们通过改进Stein的依赖邻域方法，将Stein方程展开至任意阶。最后，通过获取高效尾部界，我们展示了结果的应用性。