{"id":"https://openalex.org/W2103543826","doi":"https://doi.org/10.1137/s0895479801385116","title":"Convergence Rates of Spectral Distributions of Large Sample Covariance Matrices","display_name":"Convergence Rates of Spectral Distributions of Large Sample Covariance Matrices","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W2103543826","doi":"https://doi.org/10.1137/s0895479801385116","mag":"2103543826"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479801385116","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479801385116","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058289590","display_name":"Zhidong Bai","orcid":"https://orcid.org/0000-0002-5300-5513"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Z. D. Bai","raw_affiliation_strings":["National University of Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"National University of Singapore","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Baiqi Miao","orcid":null},"institutions":[{"id":"https://openalex.org/I126520041","display_name":"University of Science and Technology of China","ror":"https://ror.org/04c4dkn09","country_code":"CN","type":"education","lineage":["https://openalex.org/I126520041","https://openalex.org/I19820366"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Baiqi Miao","raw_affiliation_strings":["Department of Statistics and Finance, University Science and Technology of China, Hefei, Anhui, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics and Finance, University Science and Technology of China, Hefei, Anhui, China","institution_ids":["https://openalex.org/I126520041"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101512985","display_name":"Jianfeng Yao","orcid":"https://orcid.org/0000-0003-3147-2775"},"institutions":[{"id":"https://openalex.org/I4210161663","display_name":"Institut de recherche math\u00e9matique de Rennes","ror":"https://ror.org/05y76vp22","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I2800285241","https://openalex.org/I28221208","https://openalex.org/I3124938446","https://openalex.org/I4210117091","https://openalex.org/I4210141950","https://openalex.org/I4210146044","https://openalex.org/I4210161663","https://openalex.org/I4399657933","https://openalex.org/I56067802","https://openalex.org/I56067802"]},{"id":"https://openalex.org/I56067802","display_name":"Universit\u00e9 de Rennes","ror":"https://ror.org/015m7wh34","country_code":"FR","type":"education","lineage":["https://openalex.org/I56067802"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Jian-feng Yao","raw_affiliation_strings":["IRMAR, Universit\u00e9 de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"IRMAR, Universit\u00e9 de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France","institution_ids":["https://openalex.org/I4210161663","https://openalex.org/I56067802"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.9093,"has_fulltext":false,"cited_by_count":50,"citation_normalized_percentile":{"value":0.87360441,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"25","issue":"1","first_page":"105","last_page":"127"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9911999702453613,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.9100966453552246},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.6875945329666138},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.6445308327674866},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5987129807472229},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5444262623786926},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5069625377655029},{"id":"https://openalex.org/keywords/spectral-power-distribution","display_name":"Spectral power distribution","score":0.504456639289856},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.4948579668998718},{"id":"https://openalex.org/keywords/sample-mean-and-sample-covariance","display_name":"Sample mean and sample covariance","score":0.48563137650489807},{"id":"https://openalex.org/keywords/sample-size-determination","display_name":"Sample size determination","score":0.4631480872631073},{"id":"https://openalex.org/keywords/moment","display_name":"Moment (physics)","score":0.45077869296073914},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3783498704433441},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.3361378014087677},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.32296186685562134}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9100966453552246},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.6875945329666138},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.6445308327674866},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5987129807472229},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5444262623786926},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5069625377655029},{"id":"https://openalex.org/C94360122","wikidata":"https://www.wikidata.org/wiki/Q1967557","display_name":"Spectral power distribution","level":2,"score":0.504456639289856},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.4948579668998718},{"id":"https://openalex.org/C181243257","wikidata":"https://www.wikidata.org/wiki/Q1693522","display_name":"Sample mean and sample covariance","level":3,"score":0.48563137650489807},{"id":"https://openalex.org/C129848803","wikidata":"https://www.wikidata.org/wiki/Q2564360","display_name":"Sample size determination","level":2,"score":0.4631480872631073},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.45077869296073914},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3783498704433441},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.3361378014087677},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32296186685562134},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/s0895479801385116","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479801385116","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:scholarbank.nus.edu.sg:10635/105072","is_oa":false,"landing_page_url":"http://scholarbank.nus.edu.sg/handle/10635/105072","pdf_url":null,"source":{"id":"https://openalex.org/S7407052290","display_name":"National University of Singapore","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Scopus","raw_type":"Article"},{"id":"pmh:oai:hub.hku.hk:10722/132627","is_oa":false,"landing_page_url":"http://hdl.handle.net/10722/132627","pdf_url":null,"source":{"id":"https://openalex.org/S4377196271","display_name":"The HKU Scholars Hub (University of Hong Kong)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I889458895","host_organization_name":"University of Hong Kong","host_organization_lineage":["https://openalex.org/I889458895"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W1482572935","https://openalex.org/W1526402967","https://openalex.org/W1971523416","https://openalex.org/W2018848000","https://openalex.org/W2021217937","https://openalex.org/W2022800595","https://openalex.org/W2022811481","https://openalex.org/W2023426016","https://openalex.org/W2043905980","https://openalex.org/W2055791817","https://openalex.org/W2060581589","https://openalex.org/W2070017958","https://openalex.org/W2089794108","https://openalex.org/W2136118086","https://openalex.org/W2328187444","https://openalex.org/W4243997607","https://openalex.org/W4244751130"],"related_works":["https://openalex.org/W4293088233","https://openalex.org/W2496077116","https://openalex.org/W3152660226","https://openalex.org/W3082212156","https://openalex.org/W4243140484","https://openalex.org/W1998278676","https://openalex.org/W2299603463","https://openalex.org/W2126916073","https://openalex.org/W4252184478","https://openalex.org/W3209383151"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,28],"improve":[4],"known":[5],"results":[6],"on":[7],"the":[8,25,32,38,43,65,68,76,80,85,109],"convergence":[9,81,88],"rates":[10,77],"of":[11,14,19,52],"spectral":[12,34],"distributions":[13],"large-dimensional":[15],"sample":[16,45],"covariance":[17],"matrices":[18],"size":[20,46],"p":[21],"\u00d7":[22],"n.":[23],"Using":[24],"Stieltjes":[26],"transform,":[27],"first":[29],"prove":[30],"that":[31,67,108],"expected":[33],"distribution":[35,41],"converges":[36],"to":[37,91,121],"limiting":[39],"Marcenko--Pastur":[40],"with":[42],"dimension":[44],"ratio":[47],"y=yn=p/n":[48],"at":[49],"a":[50,71],"rate":[51,110],"O(n-":[53],"1/2":[54],")":[55,94,116],"if":[56],"y":[57,100,118],"keeps":[58],"away":[59,102],"from":[60,103],"0":[61],"and":[62,84,95],"1,":[63],"under":[64],"assumption":[66],"entries":[69],"have":[70],"finite":[72],"eighth":[73],"moment.":[74],"Furthermore,":[75],"for":[78],"both":[79],"in":[82,111],"probability":[83],"almost":[86],"sure":[87],"are":[89],"shown":[90],"be":[92],"Op(n-2/5":[93],"oa.s.(n-2/5+\\eta":[96],"),":[97],"respectively,":[98],"when":[99,117],"is":[101,106,114,119],"1.":[104,122],"It":[105],"interesting":[107],"all":[112],"senses":[113],"O(n-1/8":[115],"close":[120]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":3}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2016-06-24T00:00:00"}