论文标题
大维中心极限定理，凸组和球上具有第四段误差界限
Large-dimensional Central Limit Theorem with Fourth-moment Error Bounds on Convex Sets and Balls
论文作者
论文摘要
我们证明了$ \ Mathbb {r}^d $的$ n $独立随机向量的大维高斯近似，以及凸组和欧几里得球上的第四摩尔错误界限。我们表明，与经典的第三矩界相比，我们的界限几乎依赖于$ n $，并且可以改善对尺寸$ d $的依赖。对于中心的球，我们获得了一个额外的错误限制，该错误对$ n $具有次优的依赖性，但是当且仅当$ d = o（n）$时，才能恢复高斯近似值的已知结果。我们讨论了引导程序的应用程序。我们使用Stein的方法证明了我们的主要结果。
We prove the large-dimensional Gaussian approximation of a sum of $n$ independent random vectors in $\mathbb{R}^d$ together with fourth-moment error bounds on convex sets and Euclidean balls. We show that compared with classical third-moment bounds, our bounds have near-optimal dependence on $n$ and can achieve improved dependence on the dimension $d$. For centered balls, we obtain an additional error bound that has a sub-optimal dependence on $n$, but recovers the known result of the validity of the Gaussian approximation if and only if $d=o(n)$. We discuss an application to the bootstrap. We prove our main results using Stein's method.
