[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-268549-105":3,"detail-sidebar-cat-0-en-105":73,"doc-detail-268549-en":123},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":66,"head_meta":68,"extra_data":70,"updated_unix":72},105,"en","limiting-laws-for-divergent-spiked-eigenvalues-and-largest-non-spiked-eigenvalue-of-sample-covariance-matrices-abstracts-and-results","LIMITING LAWS FOR DIVERGENT SPIKED EIGENVALUES AND LARGEST NON-SPIKED EIGENVALUE OF SAMPLE COVARIANCE MATRICES - Abstracts and results","","Asymptotic distributions are studied for spiked eigenvalues and for the largest nonspiked eigenvalue of a sample covariance matrix under a general covariance model where spiked eigenvalues diverge, while the remaining eigenvalues stay bounded but are otherwise arbitrary. A limiting normal distribution is proved for spiked sample eigenvalues, with mean depending on both population spikes and nonspikes and variance depending on population eigenvectors. A limiting Tracy-Widom law is established for the largest nonspiked eigenvalue, alongside results on estimating the number of spikes and leading eigenvector convergence.",{"@graph":14,"@context":65},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/limiting-laws-for-divergent-spiked-eigenvalues-and-largest-non-spiked-eigenvalue-of-sample-covariance-matrices-abstracts-and-results/268549/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/limiting-laws-for-divergent-spiked-eigenvalues-and-largest-non-spiked-eigenvalue-of-sample-covariance-matrices-abstracts-and-results/268549.png","ImageObject",300,407,{"name":42,"@type":43},"Cipher","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-28","2026-09-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",11,{"@type":57,"mainEntity":58},"FAQPage",[59],{"name":60,"@type":61,"acceptedAnswer":62},"What additional aspects are considered beyond eigenvalue limits?","Question",{"text":63,"@type":64},"The paper examines estimation of the number of spikes and the convergence of leading eigenvectors, and develops a central limit theorem for certain dependent random quadratic forms.","Answer","https://schema.org",{"og:url":32,"og:type":67,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":69,"canonical":32},"index,follow",{"doc_id":71,"site_id":7},268549,1790263801,{"code":4,"msg":74,"data":75},"success",[76,80,84,88,93,98,103,107,112,115,119],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":77,"show_sort_weight":78,"slug":79},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":81,"show_sort_weight":82,"slug":83},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Exam",70,"exam",{"id":89,"doc_module":4,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},5,"Comic",60,"comic",{"id":94,"doc_module":4,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},6,"Technology",50,"technology",{"id":99,"doc_module":4,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},7,"Healthcare",40,"healthcare",{"id":104,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":105,"slug":106},8,30,"research-report",{"id":108,"doc_module":4,"doc_module_name":25,"category_name":109,"show_sort_weight":110,"slug":111},9,"Religion & Spirituality",20,"religion-spirituality",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":113,"show_sort_weight":110,"slug":114},"World Cup","world-cup",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":116,"slug":118},10,"Lifestyle","lifestyle",{"id":120,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":89,"slug":122},19,"General","general",{"code":4,"msg":74,"data":124},{"doc_id":71,"user_id":125,"nickname":42,"user_avatar":126,"doc_module":4,"category_id":104,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":127,"file_id":128,"file_url":129,"file_type":130,"file_size":131,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":132,"language":133,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":12,"update_tm":137,"read_time":138,"preview_page_count":4},687208528416,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","arXiv : 17 11 .00217v2 [math . ST] 6 Nov 2017  \nLIMITING LAWS FOR DIVERGENT SPIKED EIGENVALUESAND LARGEST NON-SPIKED EIGENVALUE OF SAMPLE COVARIANCE MATRICES  \nT. TONY CAI, XIAO HAN, AND GUANGMING PAN  \nAbstract. We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance matrix model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy-Widom law for the largest nonspiked sample eigenvalue is obtained.  \nEstimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a Central Limit Theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.  \nKEYWORDS: Extreme eigenvalues, factor model, principal component analysis, sample covariance matrix, spiked covariance matrix model, Tracy-Widom distribution.  \n1. Introduction  \nCovariance matrix plays a fundamental role in multivariate analysis and highdimensional statistics. There has been signi􀀌cant recent interest in studying the properties of the leading eigenvalues and eigenvectors of the sample covariance matrix, especially in the high-dimensional setting. See, for example, [2, 10 , 12 , 13 , 21 , 26–29 , 31] . These problems are not only of interest in their own right they also have close connections to important statistical problems such as principal component analysis and testing for the covariance structure of high-dimensional data.  \nPrincipal component analysis (PCA) is a widely used technique in multivariate analysis for a range of purposes, including dimension reduction, data visualization, clustering, and feature extraction [1 , 20] . PCA is particularly well suited for the settings where the signal of interest lies in a much lower dimensional subspace and it  \nDate: November 7, 2017 .  \n2010 Mathematics Subject Classi􀀌cation. 62H25, 60B20, 60F05, 62H10 .  \nKey words and phrases. Central Limit Theorem, Tracy-Widom distribution, Extreme Eigenvalues, Sample Covariance Matrix.  \nTony Cai was supported in part by NSF Grant DMS-1712735 and NIH Grant R01 GM-123056 . Guangming Pan was supported in part by by a MOE Tier 2 grant 2014-T2-2-060 and by a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore.  \n2 T. TONY CAI, XIAO HAN, AND GUANGMING PAN  \nhas been applied in a broad range of 􀀌elds such as genomics, image recognition, data compression, and 􀀌nancial econometrics. For example, widely used factor models in 􀀌nancial econometrics typically assume that a small number of unknown common factors drive the asset returns [15] . In PCA, the leading eigenvalues and eigenvectors of the population covariance matrix need to be estimated from data and are conventionally estimated by their empirical counterparts. It is thus important to understand the spectral properties of the sample covariance matrix.  \n1.1. The Problem. To be concrete, consider the data matrix Y = 􀀀X where X = (x1 , ··· , xn) is a (p+l) × n random matrix whose entries are independent with zero mean and unit variance and 􀀀 is a p × (p+l) deterministic matrix with l/p → 0. Let 􀀆 = 􀀀􀀀⊺ be the population covariance matrix. The sample covariance matrix is de􀀌ned as  \n(1.1) Sn = 1nYY⊺ = 1n􀀀XX⊺ 􀀀⊺ .  \nDenote the singular value decomposition (SVD) of 􀀀 by (1 .2) 􀀀 = V􀀃 ~~1~~2 U,  \nwhere V and U are p × p and p × (p + l) orthogonal matrices respectively (VV⊺ = UU⊺ = I), and 􀀃 is a diagonal matrix consisting in descending","cbCaifgwhHRX1Tqu","https://ap.wps.com/l/cbCaifgwhHRX1Tqu","pdf",507693,47,"English","# 1. Introduction\n## 1.1. The Problem","[{\"question\":\"What additional aspects are considered beyond eigenvalue limits?\",\"answer\":\"The paper examines estimation of the number of spikes and the convergence of leading eigenvectors, and develops a central limit theorem for certain dependent random quadratic forms.\"}]","LIMITING LAWS FOR DIVERGENT SPIKED EIGENVALUES AND LARGEST NON-SPIKED EIGENVALUE OF SAMPLE COVARIANCE MATRICES - Abstracts and results | PDF",1789432221,118]