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Tail Bounds on the Spectral Norm of Sub-Exponential Random Matrices 次指数随机矩阵谱范数的尾界
4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-11-01 DOI: 10.1142/s2010326323500132
Guozheng Dai, Zhonggen Su, Hanchao Wang
{"title":"Tail Bounds on the Spectral Norm of Sub-Exponential Random Matrices","authors":"Guozheng Dai, Zhonggen Su, Hanchao Wang","doi":"10.1142/s2010326323500132","DOIUrl":"https://doi.org/10.1142/s2010326323500132","url":null,"abstract":"Let $X$ be an $ntimes n$ symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of $X$ with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of $X$ with sub-Exponential entries. Our method relies upon a crucial ingredient of a novel chaining argument that essentially involves both the particular structure of the sets used for the chaining and the distribution of coordinates of a point on the unit sphere.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135272140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Finite Rank Perturbations of Heavy-Tailed Wigner Matrices 重尾Wigner矩阵的有限秩摄动
4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-10-27 DOI: 10.1142/s2010326323500119
Simona Diaconu
{"title":"Finite Rank Perturbations of Heavy-Tailed Wigner Matrices","authors":"Simona Diaconu","doi":"10.1142/s2010326323500119","DOIUrl":"https://doi.org/10.1142/s2010326323500119","url":null,"abstract":"One-rank perturbations of Wigner matrices have been closely studied: let [Formula: see text] with [Formula: see text] symmetric, [Formula: see text] i.i.d. with centered standard normal distributions, and [Formula: see text] It is well known [Formula: see text] the largest eigenvalue of [Formula: see text] has a phase transition at [Formula: see text]: when [Formula: see text] [Formula: see text] whereas for [Formula: see text] [Formula: see text] Under more general conditions, the limiting behavior of [Formula: see text] appropriately normalized, has also been established: it is normal if [Formula: see text] or the convolution of the law of [Formula: see text] and a Gaussian distribution if [Formula: see text] is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions [Formula: see text] heavy-tailed with index [Formula: see text] the fluctuations are shown to be universal and dependent on [Formula: see text] but not on [Formula: see text] whereas a subfamily of the edge case [Formula: see text] displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether [Formula: see text] is localized, each presenting a continuous phase transition at [Formula: see text] respectively. These results build on our previous work which analyzes the asymptotic behavior of [Formula: see text] in the aforementioned subfamily.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Strong limit theorem for largest entry of large-dimensional random tensor 大维随机张量最大入口的强极限定理
4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-10-12 DOI: 10.1142/s2010326323500120
Xue Ding
{"title":"Strong limit theorem for largest entry of large-dimensional random tensor","authors":"Xue Ding","doi":"10.1142/s2010326323500120","DOIUrl":"https://doi.org/10.1142/s2010326323500120","url":null,"abstract":"Suppose that [Formula: see text] are i.i.d. copies of random vector [Formula: see text]. Let [Formula: see text] then the random tensor product constructed by [Formula: see text] is defined by [Formula: see text] In this paper, we obtain the strong limit theorems of the largest entry of large-dimensional random tensor product [Formula: see text] under two high-dimensional settings the polynomial rate and the exponential rate. The conclusions are established under weaker moment condition than the exist papers and the relationship between [Formula: see text] and [Formula: see text] is more flexible.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135923734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rank 1 perturbations in random matrix theory — A review of exact results 随机矩阵理论中的秩1扰动——精确结果综述
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-08-19 DOI: 10.1142/s2010326323300012
Peter J. Forrester
{"title":"Rank 1 perturbations in random matrix theory — A review of exact results","authors":"Peter J. Forrester","doi":"10.1142/s2010326323300012","DOIUrl":"https://doi.org/10.1142/s2010326323300012","url":null,"abstract":"<p>A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> perturbation. Considered in this review are the additive rank <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> perturbation of the Hermitian Gaussian ensembles, the multiplicative rank <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> perturbation of the Wishart ensembles, and rank <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> perturbations of Hermitian and unitary matrices giving rise to a two-dimensional support for the eigenvalues. The focus throughout is on exact formulas, which are typically the result of various integrable structures. The simplest is that of a determinantal point process, with others relating to partial differential equations implied by a formulation in terms of certain random tridiagonal matrices. Attention is also given to eigenvector overlaps in the setting of a rank <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> perturbation.</p>","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Limiting spectral distribution of stochastic block model 随机块段模型的极限谱分布
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-07-18 DOI: 10.1142/s2010326323500089
Giap Van Su, May-Ru Chen, Mei-Hui Guo, Hao-Wei Huang
{"title":"Limiting spectral distribution of stochastic block model","authors":"Giap Van Su, May-Ru Chen, Mei-Hui Guo, Hao-Wei Huang","doi":"10.1142/s2010326323500089","DOIUrl":"https://doi.org/10.1142/s2010326323500089","url":null,"abstract":"<p>The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span> communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the size of the matrix tends to infinity. We also obtain an upper bound for the operator norm of such random matrices by means of the Stieltjes transform and random matrix theory.</p>","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Corrigendum: Sampling distributions of optimal portfolio weights and characteristics in small and large dimensions 更正:最优投资组合权重和特征的小、大维度抽样分布
4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-07-01 DOI: 10.1142/s2010326323920016
Taras Bodnar, Holger Dette, Nestor Parolya, Erik Thorsén
{"title":"Corrigendum: Sampling distributions of optimal portfolio weights and characteristics in small and large dimensions","authors":"Taras Bodnar, Holger Dette, Nestor Parolya, Erik Thorsén","doi":"10.1142/s2010326323920016","DOIUrl":"https://doi.org/10.1142/s2010326323920016","url":null,"abstract":"Random Matrices: Theory and ApplicationsOnline Ready Free AccessCorrigendum: Sampling distributions of optimal portfolio weights and characteristics in small and large dimensionsis erratum ofSampling distributions of optimal portfolio weights and characteristics in small and large dimensionsTaras Bodnar, Holger Dette, Nestor Parolya, and Erik ThorsénTaras BodnarDepartment of Mathematics, Stockholm University, Stockholm, Sweden, Holger DetteDepartment of Mathematics, Ruhr University Bochum, D-44870 Bochum, Germany, Nestor ParolyaDelft Institute of Applied Mathematics, Delft University of Technology, Delft, The NetherlandsCorresponding author., and Erik ThorsénDepartment of Mathematics, Stockholm University, Stockholm, Swedenhttps://doi.org/10.1142/S2010326323920016Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetailsRelated articlesSampling distributions of optimal portfolio weights and characteristics in small and large dimensions7 Jun 2021Random Matrices: Theory and Applications Recommended Online Ready Metrics History Received 16 April 2023 Accepted 31 May 2023 Published: 11 July 2023 PDF download","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135509284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Matrix deviation inequality for ℓp-norm 的矩阵偏差不等式ℓp-范数
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-06-15 DOI: 10.1142/s2010326323500077
Yuan-Chung Sheu, Te-Chun Wang
{"title":"Matrix deviation inequality for ℓp-norm","authors":"Yuan-Chung Sheu, Te-Chun Wang","doi":"10.1142/s2010326323500077","DOIUrl":"https://doi.org/10.1142/s2010326323500077","url":null,"abstract":"<p>Motivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, <i>High-Dimensional Probability: An Introduction with Applications in Data Science</i>, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018), doi:10.1017/9781108231596 of Theorem 11.1.5], we show that this property holds for the <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span>-norm with <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo>≤</mo><mi>p</mi><mo>&lt;</mo><mi>∞</mi></math></span><span></span> and i.i.d. ensemble sub-Gaussian matrices, i.e. random matrices with i.i.d. mean-zero, unit variance, sub-Gaussian entries. As a consequence of our result, we establish the Johnson–Lindenstrauss lemma from <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span><span></span>-space to <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span><span></span>-space for all i.i.d. ensemble sub-Gaussian matrices.</p>","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Random matrices associated to Young diagrams 与杨氏图相关的随机矩阵
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2023-01-31 DOI: 10.1142/s2010326323500090
F. D. Cunden, M. Ligabò, Tommaso Monni
{"title":"Random matrices associated to Young diagrams","authors":"F. D. Cunden, M. Ligabò, Tommaso Monni","doi":"10.1142/s2010326323500090","DOIUrl":"https://doi.org/10.1142/s2010326323500090","url":null,"abstract":"We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalisation of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes-Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko-Pastur and Dykema-Haagerup measures of square and triangular random matrices, respectively. We find a further factorisation of the moments in terms of two complex-valued random variables that generalises the factorisation of the Marcenko-Pastur law as product of independent uniform and arcsine random variables.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81910918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The moderate deviation principles of likelihood ratio tests under alternative hypothesis 备择假设下似然比检验的适度偏差原则
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2022-12-15 DOI: 10.1142/s201032632350003x
Yansong Bai, Yong Zhang
{"title":"The moderate deviation principles of likelihood ratio tests under alternative hypothesis","authors":"Yansong Bai, Yong Zhang","doi":"10.1142/s201032632350003x","DOIUrl":"https://doi.org/10.1142/s201032632350003x","url":null,"abstract":"","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89827004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic Properties of GEE with Diverging Dimension of Covariates 协变量维数发散的GEE的渐近性质
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2022-10-10 DOI: 10.1142/s2010326323500016
Qibing Gao, Chunhua Zhu, Yi Yao
{"title":"Asymptotic Properties of GEE with Diverging Dimension of Covariates","authors":"Qibing Gao, Chunhua Zhu, Yi Yao","doi":"10.1142/s2010326323500016","DOIUrl":"https://doi.org/10.1142/s2010326323500016","url":null,"abstract":"","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77148415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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