{"id":"https://openalex.org/W7212286198","doi":"https://doi.org/10.48550/arxiv.2609.11688","title":"The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants","display_name":"The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212286198","doi":"https://doi.org/10.48550/arxiv.2609.11688"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.11688","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11688","pdf_url":"https://arxiv.org/pdf/2609.11688","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.11688","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5074058373","display_name":"Songhao Liu","orcid":"https://orcid.org/0009-0005-6448-5937","observed_orcids":["https://orcid.org/0009-0005-6448-5937"]},"institutions":[{"id":"https://openalex.org/I27357992","display_name":"Dalian University of Technology","ror":"https://ror.org/023hj5876","country_code":"CN","type":"education","lineage":["https://openalex.org/I27357992"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liu, Song-Hao","raw_affiliation_strings":["School of Mathematical Sciences , Dalian University of Technology , Dalian , Liaoning , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Dalian University of Technology , Dalian , Liaoning , China","institution_ids":["https://openalex.org/I27357992"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111660010","display_name":"Qi-Man Shao","orcid":"https://orcid.org/0000-0002-8916-3943","observed_orcids":["https://orcid.org/0000-0002-8916-3943"]},"institutions":[{"id":"https://openalex.org/I3045169105","display_name":"Southern University of Science and Technology","ror":"https://ror.org/049tv2d57","country_code":"CN","type":"education","lineage":["https://openalex.org/I3045169105"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shao, Qi-Man","raw_affiliation_strings":["Department of Statistics and Data Science , Shenzhen International Center for Mathematics , Southern University of Science and Technology , Shenzhen , Guangdong , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics and Data Science , Shenzhen International Center for Mathematics , Southern University of Science and Technology , Shenzhen , Guangdong , China","institution_ids":["https://openalex.org/I3045169105"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090889180","display_name":"Jing\u2010yu Xu","orcid":"https://orcid.org/0000-0002-1058-2257","observed_orcids":["https://orcid.org/0000-0002-1058-2257"]},"institutions":[{"id":"https://openalex.org/I3045169105","display_name":"Southern University of Science and Technology","ror":"https://ror.org/049tv2d57","country_code":"CN","type":"education","lineage":["https://openalex.org/I3045169105"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xu, Jing-Yu","raw_affiliation_strings":["Department of Statistics and Data Science , Southern University of Science and Technology , Shenzhen , Guangdong , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics and Data Science , Southern University of Science and Technology , Shenzhen , Guangdong , China","institution_ids":["https://openalex.org/I3045169105"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.9951000213623047,"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":0.9951000213623047,"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/T13500","display_name":"Probability and Statistical Research","score":0.00139999995008111,"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/T11166","display_name":"Analytic Number Theory Research","score":0.0005000000237487257,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/logarithmic-law","display_name":"logarithmic law","score":0.7480000257492065},{"id":"https://openalex.org/keywords/central-limit-theorem","display_name":"central limit theorem","score":0.7049999833106995},{"id":"https://openalex.org/keywords/berry-esseen-bounds","display_name":"Berry-Esseen bounds","score":0.8820000290870667},{"id":"https://openalex.org/keywords/uniform-integrability","display_name":"uniform integrability","score":0.7239999771118164},{"id":"https://openalex.org/keywords/fourth-moment-condition","display_name":"fourth moment condition","score":0.3630000054836273},{"id":"https://openalex.org/keywords/kolmogorov-distance","display_name":"Kolmogorov distance","score":0.7480000257492065},{"id":"https://openalex.org/keywords/normal-approximation","display_name":"Gaussian approximation","score":0.43700000643730164},{"id":"https://openalex.org/keywords/independent-random-matrices","display_name":"independent random matrices","score":0.4490000009536743},{"id":"https://openalex.org/keywords/convergence-to-normal-distribution","display_name":"convergence to normal distribution","score":0.37299999594688416},{"id":"https://openalex.org/keywords/log-concave-distributions","display_name":"log-concave distributions","score":0.4300000071525574},{"id":"https://openalex.org/keywords/logarithmic-sobolev-inequality","display_name":"logarithmic Sobolev inequality","score":0.3610000014305115},{"id":"https://openalex.org/keywords/log-concave-measures","display_name":"log-concave measures","score":0.3109999895095825}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8472999930381775},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.722100019454956},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6843000054359436},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.6044999957084656},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5478000044822693},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5281000137329102},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5216000080108643},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.516700029373169},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.5146999955177307},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4814000129699707},{"id":"https://openalex.org/C139907963","wikidata":"https://www.wikidata.org/wiki/Q198740","display_name":"Law of the iterated logarithm","level":3,"score":0.4323999881744385},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3910999894142151},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3880000114440918},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.3808000087738037},{"id":"https://openalex.org/C65092005","wikidata":"https://www.wikidata.org/wiki/Q7885107","display_name":"Uniform limit theorem","level":2,"score":0.37790000438690186},{"id":"https://openalex.org/C185767445","wikidata":"https://www.wikidata.org/wiki/Q207952","display_name":"Law of large numbers","level":3,"score":0.3434999883174896},{"id":"https://openalex.org/C129301229","wikidata":"https://www.wikidata.org/wiki/Q741865","display_name":"Uniform continuity","level":3,"score":0.30140000581741333},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2838999927043915},{"id":"https://openalex.org/C104528550","wikidata":"https://www.wikidata.org/wiki/Q506265","display_name":"Triangular matrix","level":3,"score":0.2680000066757202},{"id":"https://openalex.org/C61062188","wikidata":"https://www.wikidata.org/wiki/Q835065","display_name":"Second moment of area","level":2,"score":0.2524999976158142},{"id":"https://openalex.org/C130594498","wikidata":"https://www.wikidata.org/wiki/Q2028293","display_name":"Iterated logarithm","level":3,"score":0.2515000104904175},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.25130000710487366},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.11688","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11688","pdf_url":"https://arxiv.org/pdf/2609.11688","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.11688","doi":"https://doi.org/10.48550/arxiv.2609.11688","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.11688","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.11688","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11688","pdf_url":"https://arxiv.org/pdf/2609.11688","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G2356940958","display_name":null,"funder_award_id":"12031005","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8812420149","display_name":null,"funder_award_id":"DUT25RC","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212286198.pdf","grobid_xml":"https://content.openalex.org/works/W7212286198.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W1497390081","https://openalex.org/W2079962798","https://openalex.org/W7140096702"],"related_works":["https://openalex.org/W1975204349","https://openalex.org/W2079962798","https://openalex.org/W3215523286","https://openalex.org/W4321460431","https://openalex.org/W2963818301","https://openalex.org/W3197853149","https://openalex.org/W4387022277","https://openalex.org/W2588381893","https://openalex.org/W2178538693","https://openalex.org/W1497390081"],"abstract_inverted_index":{"Let":[0],"$A=(A_n)_{n\\ge2}$":[1],"be":[2],"a":[3],"triangular":[4],"array":[5],"of":[6,105],"random":[7,17],"matrices,":[8],"where":[9],"$A_n=(a_{ij})_{1\\le":[10],"i,j\\le":[11,123,141],"n}$":[12,67],"is":[13,68,94],"an":[14],"$n\\times":[15],"n$":[16],"matrix":[18],"with":[19],"independent":[20],"real":[21],"entries":[22,75,148],"satisfying":[23],"$\\mathbb":[24,28],"E":[25,47,125,143],"a_{ij}=0$":[26],"and":[27,30,35,116,146,165,172,188],"Ea_{ij}^2=1$,":[29],"put":[31],"$\\mathcal":[32],"L_n=\\log|\\det":[33],"A_n|$":[34],"\\[":[36],"W_n^{\\mathrm":[37,43],"d}(A_n):=\\frac{\\mathcal":[38],"L_n":[39],"-":[40],"\\frac12\\log(n-1)!}{\\sqrt{\\frac12\\log":[41],"n}},\\quad":[42],"e}(A_n):=":[44],"\\frac{\\mathcal":[45],"L_n-\\mathbb":[46],"\\mathcal":[48,58,84],"L_n}{\\sqrt{\\frac12\\log":[49],"n}}.":[50],"\\]":[51],"We":[52,109],"prove":[53],"that":[54,118],"$W_n^{\\mathrm":[55,81],"d}(A_n)":[56],"\\Rightarrow":[57,83],"N(0,1)$,":[59],"whenever":[60,86],"the":[61,74,87,103,112,147,168,181,189],"family":[62,88],"$\\left\\{\\frac{|a_{ij}|^{4}}{\\sqrt{\\log(e+|a_{ij}|)}}":[63],"\\right\\}_{n\\geq":[64],"2;1\\leq":[65,90],"i,j\\leq":[66,91],"uniformly":[69,77,95,150],"integrable.":[70,96],"If,":[71],"in":[72],"addition,":[73],"have":[76,149],"bounded":[78,151],"densities,":[79,152],"then":[80,128,153],"e}(A_n)":[82],"N(0,1)$":[85],"$\\left\\{\\frac{|a_{ij}|^{4}}{\\log(e+|a_{ij}|)}\\right\\}_{n\\geq":[89],"n}":[92],"$":[93],"These":[97],"two":[98],"conditions":[99],"are":[100,175],"optimal":[101],"at":[102],"level":[104],"universal":[106],"moment":[107],"assumptions.":[108],"further":[110],"establish":[111],"corresponding":[113],"Berry--Esseen":[114,190],"bounds,":[115],"show":[117],"for":[119],"$0<\u03b4\\le\\tfrac12$,":[120],"if":[121,139],"$\\sup\\limits_{n}\\max\\limits_{1\\le":[122,140],"n}\\mathbb":[124,142],"\\frac{|a_{ij}|^4}":[126,144],"{\\{\\log(e+|a_{ij}|)\\}^{1/2-\u03b4}}<\\infty$,":[127],"\\begin{align*}":[129,154],"d_{\\mathrm":[130,155],"K}(W_n^{\\mathrm":[131,156],"d}(A_n),\\mathcal":[132],"N(0,1))\\le":[133,158],"C(\\log":[134,159],"n)^{-\u03b4}.":[135],"\\end{align*}":[136,161],"For":[137],"$0<\u03b3\\le1$,":[138],"{\\{\\log(e+|a_{ij}|)\\}^{1-\u03b3}}<\\infty$":[145],"e}(A_n),\\mathcal":[157],"n)^{-\u03b3}.":[160],"When":[162],"$\u03b4=":[163],"1/2$":[164],"$\u03b3=":[166],"1$,":[167],"bounds":[169],"$(\\log":[170,173],"n)^{-1/2}$":[171],"n)^{-1}$":[174],"optimal,":[176],"respectively.":[177],"Our":[178],"results":[179],"improve":[180],"earlier":[182],"Central":[183],"Limit":[184],"Theorem":[185],"by":[186,192],"\\cite{BaoPanZhou2015}":[187],"bound":[191],"\\cite{NguyenVu2014}.":[193]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-12T00:00:00"}
