论文标题
通过非线性交换对正常近似
Normal approximation via non-linear exchangeable pairs
论文作者
论文摘要
我们为Stein的可交换对方法提出了一种新的功能分析方法，该方法不需要手头来满足任何近似线性回归属性。我们利用该理论来得出在瓦斯坦距离距离某些功能的正常和伽马近似的抽象界限。此外，我们通过三种可以应用的情况说明了这种方法的相关性：独立随机变量的功能，有限的人群统计数据和有限组的功能。在独立案例中，尤其是对于对称$ u $ $统计的情况下，我们证明了这种方法在哪些情况下产生的界限比现有文献中的范围更好。最后，我们将结果应用于CLT中的Wasserstein界限，以基于$ n $ i.i.d.的几何随机图中的子图计数。欧几里得空间中的点以及皮尔森统计数据的正常近似值。
We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.
