对部分广义四矩定理的再认识

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-11-01 DOI:10.3150/20-BEJ1310

Dandan Jiang, Z. Bai

{"title":"对部分广义四矩定理的再认识","authors":"Dandan Jiang, Z. Bai","doi":"10.3150/20-BEJ1310","DOIUrl":null,"url":null,"abstract":"This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described in “Generalized Four Moment Theorem (G4MT) and its Application to CLT for Spiked Eigenvalues of High-dimensional Covariance Matrices”. Since the G4MT proposed in that paper requires both the matrices X and Y satisfying the assumption maxt,s|uts|2E{|x11|4I(|x11|<n)−μ}→0 with the same μ which maybe restrictive in real applications, we proposed a new G4MT, called PG4MT, without proof. After the manuscript posed in ArXiv, the authors received high interests in the proof of PG4MT through private communications and find the PG4MT more general than G4MT, it is necessary to give a detailed proof of it. Moreover, it is found that the PG4MT derives a CLT of spiked eigenvalues of sample covariance matrices which covers the work in Bai and Yao (J. Multivariate Anal. 106 (2012) 167–177) as a special case.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Partial generalized four moment theorem revisited\",\"authors\":\"Dandan Jiang, Z. Bai\",\"doi\":\"10.3150/20-BEJ1310\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described in “Generalized Four Moment Theorem (G4MT) and its Application to CLT for Spiked Eigenvalues of High-dimensional Covariance Matrices”. Since the G4MT proposed in that paper requires both the matrices X and Y satisfying the assumption maxt,s|uts|2E{|x11|4I(|x11|<n)−μ}→0 with the same μ which maybe restrictive in real applications, we proposed a new G4MT, called PG4MT, without proof. After the manuscript posed in ArXiv, the authors received high interests in the proof of PG4MT through private communications and find the PG4MT more general than G4MT, it is necessary to give a detailed proof of it. Moreover, it is found that the PG4MT derives a CLT of spiked eigenvalues of sample covariance matrices which covers the work in Bai and Yao (J. Multivariate Anal. 106 (2012) 167–177) as a special case.\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2021-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3150/20-BEJ1310\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3150/20-BEJ1310","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 3

摘要

这是“广义四矩定理（G4MT）及其在高维协方差矩阵尖峰特征值CLT中的应用”中提到和描述的部分广义4矩定理（PG4MT）的补充证明。由于该文提出的G4MT要求矩阵X和Y都满足假设maxt，s|uts|2E{|x11|4I（|x11|＜n）-μ}→0，这在实际应用中可能是限制性的，我们提出了一种新的G4MT，称为PG4MT，没有证据。在ArXiv提出手稿后，作者通过私人通信对PG4MT的证明产生了高度兴趣，并发现PG4MT比G4MT更通用，有必要对其进行详细的证明。此外，发现PG4MT导出了样本协方差矩阵的尖峰特征值的CLT，该CLT涵盖了Bai和Yao（J.Multivariate Anal.106（2012）167–177）作为特例的工作。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Partial generalized four moment theorem revisited

This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described in “Generalized Four Moment Theorem (G4MT) and its Application to CLT for Spiked Eigenvalues of High-dimensional Covariance Matrices”. Since the G4MT proposed in that paper requires both the matrices X and Y satisfying the assumption maxt,s|uts|2E{|x11|4I(|x11|

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

京公网安备 11010802042870号

Book学术文献互助群
群号：481959085