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On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Frechet Variation 二阶非交换概率空间及有界Frechet变分函数的解析结构
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2022-01-11 DOI: 10.1142/s2010326322500447
Mario Díaz, J. Mingo
{"title":"On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Frechet Variation","authors":"Mario Díaz, J. Mingo","doi":"10.1142/s2010326322500447","DOIUrl":"https://doi.org/10.1142/s2010326322500447","url":null,"abstract":"In this paper we propose a new approach to the central limit theorem (CLT), based on functions of bounded Féchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on: a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and, as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g., the CLT for the continuously differentiable linear statistics of block Gaussian matrices. In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, G2. Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in G2.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91319633","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
Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices 重叠随机Wishart矩阵谱的三维高斯起伏
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-12-27 DOI: 10.1142/s2010326322500484
Jeffrey Kuan, Zhengye Zhou
{"title":"Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices","authors":"Jeffrey Kuan, Zhengye Zhou","doi":"10.1142/s2010326322500484","DOIUrl":"https://doi.org/10.1142/s2010326322500484","url":null,"abstract":"In [DP18], the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic evolution, the fluctuations asymptotically converge to a three-dimensional Gaussian field, which has an explicit contour integral formula. This is analogous to the result of [Bor14] for stochastic Wigner matrices.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78106932","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
Limiting Eigenvalue Behavior of a Class of Large Dimensional Random Matrices Formed From a Hadamard Product 一类由Hadamard积构成的大维随机矩阵的极限特征值行为
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-12-08 DOI: 10.1142/s2010326322500502
J. W. Silverstein
{"title":"Limiting Eigenvalue Behavior of a Class of Large Dimensional Random Matrices Formed From a Hadamard Product","authors":"J. W. Silverstein","doi":"10.1142/s2010326322500502","DOIUrl":"https://doi.org/10.1142/s2010326322500502","url":null,"abstract":"This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices 1 N (Dn ◦Xn)(Dn ◦Xn)∗, studied in Girko 2001. Here, Xn = (xij) is an n×N random matrix consisting of independent complex standardized random variables, Dn = (dij), n × N , has nonnegative entries, and ◦ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of Xn and Dn which are different from those in Girko (2001), which include a Lindeberg condition on the entries of Dn ◦Xn, as well as a bound on the average of the rows and columns of Dn ◦ Dn. The present paper separates the assumptions needed on Xn and Dn. It assumes a Lindeberg condition on the entries of Xn, along with a tigntness-like condition on the entries of Dn,","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77030687","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
Some strong convergence theorems for eigenvalues of general sample covariance matrices 一般样本协方差矩阵特征值的几个强收敛定理
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-11-27 DOI: 10.1142/s2010326322500290
Yanqing Yin
{"title":"Some strong convergence theorems for eigenvalues of general sample covariance matrices","authors":"Yanqing Yin","doi":"10.1142/s2010326322500290","DOIUrl":"https://doi.org/10.1142/s2010326322500290","url":null,"abstract":"The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form [Formula: see text], where [Formula: see text] is a [Formula: see text] nonrandom matrix and [Formula: see text] is an [Formula: see text] matrix consisting of i.i.d standard complex entries. [Formula: see text] as [Formula: see text] while [Formula: see text] can be arbitrary but no smaller than [Formula: see text]. We first prove that under some mild assumptions, with probability 1, for all large [Formula: see text], there will be no eigenvalues in any closed interval contained in an open interval which is outside the supports of the limiting distributions for all sufficiently large [Formula: see text]. Then we get the strong convergence result for the extreme eigenvalues as an extension of Bai-Yin law.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78130877","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
Central Limit Theorem for Linear Spectral Statistics of Block-Wigner-type Matrices 块wigner型矩阵线性谱统计量的中心极限定理
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-10-23 DOI: 10.1142/s2010326323500065
Zheng-G Wang, Jianfeng Yao
{"title":"Central Limit Theorem for Linear Spectral Statistics of Block-Wigner-type Matrices","authors":"Zheng-G Wang, Jianfeng Yao","doi":"10.1142/s2010326323500065","DOIUrl":"https://doi.org/10.1142/s2010326323500065","url":null,"abstract":"Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures, and establish a CLT for the corresponding linear spectral statistics via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type matrices. Further, we show that for certain estimator of such renormalized adjacency matrices, which will be no longer Wigner-type but share long-range non-decaying weak correlations among the entries, the linear spectral statistics of such estimators will still share the same limiting behavior as those of the block-Wigner-type matrices, thus enabling hypothesis testing about stochastic block model.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90246255","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
On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series 高维时间序列块相关矩阵特征值分布的渐近性
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-10-22 DOI: 10.1142/s2010326322500241
P. Loubaton, X. Mestre
{"title":"On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series","authors":"P. Loubaton, X. Mestre","doi":"10.1142/s2010326322500241","DOIUrl":"https://doi.org/10.1142/s2010326322500241","url":null,"abstract":"We consider linear spectral statistics built from the block-normalized correlation matrix of a set of [Formula: see text] mutually independent scalar time series. This matrix is composed of [Formula: see text] blocks. Each block has size [Formula: see text] and contains the sample cross-correlation measured at [Formula: see text] consecutive time lags between each pair of time series. Let [Formula: see text] denote the total number of consecutively observed windows that are used to estimate these correlation matrices. We analyze the asymptotic regime where [Formula: see text] while [Formula: see text], [Formula: see text]. We study the behavior of linear statistics of the eigenvalues of this block correlation matrix under these asymptotic conditions and show that the empirical eigenvalue distribution converges to a Marcenko–Pastur distribution. Our results are potentially useful in order to address the problem of testing whether a large number of time series are uncorrelated or not.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75408773","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
Spectral measure of empirical autocovariance matrices of high dimensional Gaussian stationary processes 高维高斯平稳过程经验自协方差矩阵的谱测度
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-10-16 DOI: 10.1142/s2010326322500538
A. Bose, W. Hachem
{"title":"Spectral measure of empirical autocovariance matrices of high dimensional Gaussian stationary processes","authors":"A. Bose, W. Hachem","doi":"10.1142/s2010326322500538","DOIUrl":"https://doi.org/10.1142/s2010326322500538","url":null,"abstract":"Consider the empirical autocovariance matrix at a given non-zero time lag based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measures in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices, at first the probabilistic behavior of the small singular values of the shifted versions of the autocovariance matrix are obtained. This is then used to infer about the large sample behaviour of the empirical spectral measure of the autocovariance matrices at any lag. Matrix orthogonal polynomials on the unit circle play a crucial role in our study.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79718390","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: 3
Author index Volume 10 (2021) 作者索引第10卷(2021)
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-10-01 DOI: 10.1142/s201032632199001x
{"title":"Author index Volume 10 (2021)","authors":"","doi":"10.1142/s201032632199001x","DOIUrl":"https://doi.org/10.1142/s201032632199001x","url":null,"abstract":"","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76864588","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
Characteristic polynomials of random truncations: moments, duality and asymptotics 随机截断的特征多项式:矩、对偶性和渐近性
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-09-21 DOI: 10.1142/s2010326322500496
A. Serebryakov, N. Simm, Guillaume Dubach
{"title":"Characteristic polynomials of random truncations: moments, duality and asymptotics","authors":"A. Serebryakov, N. Simm, Guillaume Dubach","doi":"10.1142/s2010326322500496","DOIUrl":"https://doi.org/10.1142/s2010326322500496","url":null,"abstract":"We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73031153","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
Universal Scaling Limits of the Symplectic Elliptic Ginibre Ensemble 辛椭圆型Ginibre系综的普遍标度极限
IF 0.9 4区 数学
Random Matrices-Theory and Applications Pub Date : 2021-08-12 DOI: 10.1142/S2010326322500472
Sunggyu Byun, M. Ebke
{"title":"Universal Scaling Limits of the Symplectic Elliptic Ginibre Ensemble","authors":"Sunggyu Byun, M. Ebke","doi":"10.1142/S2010326322500472","DOIUrl":"https://doi.org/10.1142/S2010326322500472","url":null,"abstract":". We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behaviour of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel-Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82390215","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: 15
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