论文标题
中心极限定理中的WASSERSTEIN-P边界在弱依赖性下
Wasserstein-p Bounds in the Central Limit Theorem Under Weak Dependence
论文作者
论文摘要
中心极限定理是概率中最基本的结果之一，已成功扩展到局部依赖的数据和强烈混合的随机字段。在本文中，我们建立了其用于运输距离的收敛速度，即，对于任意$ p \ ge1 $，我们为Wasserstein-$ p $距离获得了局部依赖的随机变量的上限，并强烈混合了固定的随机场。我们的证明将Stein依赖性邻居方法适应Wasserstein-$ p $距离，作为副产品，我们为依赖性随机变量建立了Stein方程的高阶局部扩展。最后，我们证明了如何使用我们的结果来获得渐近紧密的尾部边界，并快速降低多项式降低，对于弱依赖的随机变量的经验平均值。
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.
