Survey on the fourth moment theorem, Stein’s method and related topics
Survey on the fourth moment theorem, Stein’s method and related topics
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四阶矩定理、斯坦因方法及相关主题综述
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
S. Kusuoka
中科院分区：
文献类型：
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作者：
S. Kusuoka
四阶矩定理最初是由Nualart和Peccati[11]提出的。该定理给出了属于Wiener混沌水平的随机变量序列收敛于标准正态分布的一些等价条件。该定理最令人惊讶的部分是：如果序列的方差收敛到1，则收敛到标准正态分布等价于序列的四阶矩收敛到3。此后，Nualart和Ortiz-Latorre[10]给出了另一个等价条件，并在他们的文章中作了更明确的证明。受Nualart和Ortiz-Latorre[10]的启发，Nourdin和Peccati[6]发现了一种新的方法来估计标准正态分布与其他分布之间的距离，以及中心伽马分布与其他分布之间的距离。该方法是Stein方法和Malliavin微积分的结合。Nourdin和Peccati的方法使我们能够用另一种方式证明四阶矩定理的一部分。现在考虑四阶矩定理和斯坦界的应用和其他版本。在这次演讲中，我们主要回顾了四阶矩定理和Stein方法，并对进一步的结果和相关主题作了简要的回顾。现在我们给出一些有用的信息。由Nourdin和Peccati编写的教科书[7]最近出版了。这本书涵盖了从这个主题的基本工具到四阶矩定理和由Stein方法获得的密度估计。有关这一主题的最新结果可在以下网页上找到：http://www.iecn.u-nancy.fr/nourdin/steinmarliavin.htm该网页上列出了许多文献(如课堂讲稿、文章)。
The fourth moment theorem was originally introduced by Nualart and Peccati [11]. The theorem gives some equivalent conditions for a sequence of random variables belonging to a level of Wiener chaos to convergent to the standard normal distribution. The most surprising part of the theorem is that; if the variances of the sequence converge to 1, then the convergence to the standard normal distribution is equivalent to the convergence of the fourth moments of the sequence to 3. After that, Nualart and Ortiz-Latorre [10] gave another equivalent condition and made a clearer proof in their paper. Stimulated by Nualart and Ortiz-Latorre [10], Nourdin and Peccati [6] discovered a new method to estimate distances between the standard normal distribution and other distributions, and between the centered Gamma distributions and other distributions. The method is a combination of Stein’s method and Malliavin calculus. Nourdin and Peccati’s method enables us to prove a part of the fourth moment theorem in another way. Now applications and other versions of the fourth moment theorem and Stein’s bound are considered. In this talk, we review the fourth moment theorem and Stein’s method mainly, give a short review of further results and related topics. Now we give some useful information. A textbook [7] written by Nourdin and Peccati was published recently. This book covers from the elementary tools for this topic to the fourth moment theorem and the density estimates obtained by Stein’s method. The latest results on this topic are found on the webpage: http://www.iecn.u-nancy.fr/ nourdin/steinmalliavin.htm Many of literatures (e.g. lecture notes, articles) are listed up on this webpage.
影响因子：
4
作者：
D. Nualart
通讯作者：
D. Nualart