No eigenvalues outside the support of the limiting spectral distribution of quaternion sample covariance matrices

In this paper, we consider the spectral properties of quaternion sample covariance matrices. Let [Formula: see text], where [Formula: see text] is the square root of a [Formula: see text] quaternion Hermitian non-negative definite matrix [Formula: see text] and [Formula: see text] is a [Formula: see text] matrix consisting of i.i.d. standard quaternion entries. Under the framework of random matrix theory, i.e. [Formula: see text] as [Formula: see text], we prove that if the fourth moment of the entries is finite, then there will almost surely be no eigenvalues that appear in any closed interval outside the support of the limiting distribution as [Formula: see text].

Download Full-text

Some strong convergence theorems for eigenvalues of general sample covariance matrices

Random Matrices Theory and Application ◽

10.1142/s2010326322500290 ◽

2021 ◽

Author(s):

Yanqing Yin

Keyword(s):

General Population ◽

Strong Convergence ◽

Spectral Properties ◽

Closed Interval ◽

Open Interval ◽

Convergence Result ◽

Covariance Matrices ◽

Sample Covariance Matrices ◽

Sample Covariance ◽

Class Of Matrices

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.

Download Full-text

Spectral Properties of Sample Covariance Matrices

Theory of Probability and Its Applications ◽

10.1137/1140091 ◽

1996 ◽

Vol 40 (4) ◽

pp. 777-786 ◽

Cited By ~ 6

Author(s):

V. I. Serdobolskii

Keyword(s):

Spectral Properties ◽

Covariance Matrices ◽

Sample Covariance Matrices ◽

Sample Covariance

Download Full-text

Multivariate statistics

10.1093/oxfordhb/9780198744191.013.28 ◽

2018 ◽

Author(s):

Paul Zinn-Justin ◽

Jean-Bernard Zuber

Keyword(s):

Matrix Theory ◽

Wishart Distribution ◽

Covariance Matrices ◽

Unknown Parameters ◽

Sample Covariance Matrices ◽

Large N ◽

Sample Covariance ◽

Parametric Statistics ◽

Components Analysis

This article considers some classical and more modern results obtained in random matrix theory (RMT) for applications in statistics. In the classic paradigm of parametric statistics, data are generated randomly according to a probability distribution indexed by parameters. From this data, which is by nature random, the properties of the deterministic (and unknown) parameters may be inferred. The ability to infer properties of the unknown Σ (the population covariance matrix) will depend on the quality of the estimator. The article first provides an overview of two spectral statistical techniques, principal components analysis (PCA) and canonical correlation analysis (CCA), before discussing the Wishart distribution and normal theory. It then describes extreme eigenvalues and Tracy–Widom laws, taking into account the results obtained in the asymptotic setting of ‘large p, large n’. It also analyses the results for the limiting spectra of sample covariance matrices..

Download Full-text

Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

Journal of Theoretical Probability ◽

10.1007/s10959-021-01153-x ◽

2022 ◽

Author(s):

G. L. Zitelli

Keyword(s):

Lévy Processes ◽

Matrix Theory ◽

Free Probability ◽

Levy Processes ◽

Covariance Matrices ◽

Sample Covariance Matrices ◽

Sample Covariance ◽

Probability Spaces ◽

Spectral Distributions ◽

Non Gaussian

AbstractWe prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When the underlying Lévy process is non-Gaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operator-valued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operator-valued $$^*$$ ∗ -probability spaces, where the limits of sample covariance matrices play the role of non-commutative Lévy processes whose increments are free with amalgamation.

Download Full-text

On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

Random Matrices Theory and Application ◽

10.1142/s2010326322500307 ◽

2021 ◽

Author(s):

Alicja Dembczak-Kołodziejczyk ◽

Anna Lytova

Keyword(s):

Random Matrix Theory ◽

Random Matrix ◽

Spectral Distribution ◽

Matrix Theory ◽

Random Variable ◽

Covariance Matrices ◽

Convergence In Probability ◽

Stat Phys ◽

Sample Covariance ◽

Arbitrary Space

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175(2) (2019) 384–401, MR3968860].

Download Full-text

Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices

Random Matrices Theory and Application ◽

10.1142/s2010326320500069 ◽

2019 ◽

Vol 09 (03) ◽

pp. 2050006 ◽

Cited By ~ 1

Author(s):

Giorgio Cipolloni ◽

László Erdős

Keyword(s):

Covariance Matrices ◽

Gaussian Field ◽

Linear Statistics ◽

Sample Covariance Matrices ◽

Linear Eigenvalue Statistics ◽

Sample Covariance ◽

Eigenvalue Statistics ◽

Matrix Formula ◽

The Difference ◽

The Individual

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix [Formula: see text] and its minor [Formula: see text]. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of [Formula: see text] and [Formula: see text]. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.

Download Full-text