论文标题
Stein方法的高维中心限制定理
High-dimensional Central Limit Theorems by Stein's Method
论文作者
论文摘要
对于具有Stein内核的随机矢量，或接收可交换对耦合的随机向量，我们获得了$ d $二维正常近似的明确错误界限，或者是独立随机变量的非线性统计量，或者是$ n $ n $依赖的随机矢量。我们假设近似正态分布具有非偏差协方差矩阵。即使尺寸$ d $比样本量$ n $大得多，错误范围也会消失。我们使用Stein方法中的Götze（1991）的方法证明了我们的主要结果，以及对Anderson，Hall and Titterington（1998）的估计值的修改以及Bhattacharya和Rao（1976）的平滑不平等。对于具有log-conconcave密度的$ n $独立和相同分布的各向同性随机向量的总和，我们获得了一个错误绑定，该错误绑定到最佳的$ \ log n $ factor。我们还讨论了多个Wiener-Itô积分的申请。
We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a sum of $n$ locally dependent random vectors. We assume the approximating normal distribution has a non-singular covariance matrix. The error bounds vanish even when the dimension $d$ is much larger than the sample size $n$. We prove our main results using the approach of Götze (1991) in Stein's method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of $n$ independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a $\log n$ factor. We also discuss an application to multiple Wiener-Itô integrals.
