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：中心极限定理是概率论中最基础的结果之一；自20世纪40年代以来，确定其收敛速度一直是一个关键问题。对于独立随机变量，最近一系列研究在Wasserstein-p距离（p≥1）下建立了最优误差界。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。在矩条件和依赖邻域假设下，我们推导了Wasserstein-p距离下中心极限定理的最优速率。证明方法依赖于用独立同分布随机变量的经验平均近似依赖观测值的经验平均。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，通过获得高效尾界，我们展示了所得结果的适用性。