Strong convergence of ESD for large quaternion sample covariance matrices and correlation matrices when p/n → 0

In this paper, we study the strong convergence of empirical spectral distribution (ESD) of the large quaternion sample covariance matrices and correlation matrices when the ratio of the population dimension [Formula: see text] to sample size [Formula: see text] tends to zero. We prove that the ESD of renormalized quaternion sample covariance matrices converges almost surely to the semicircle law.

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Limit Properties of the Largest Entries of High-Dimensional Sample Covariance and Correlation Matrices

Mathematical Problems in Engineering ◽

10.1155/2021/9635202 ◽

2021 ◽

Vol 2021 ◽

pp. 1-8

Author(s):

Xue Ding

Keyword(s):

Sample Size ◽

Covariance Matrices ◽

High Dimensional ◽

Extreme Statistics ◽

Correlation Matrices ◽

Underlying Distribution ◽

Sample Covariance Matrices ◽

Sample Covariance ◽

Sample Correlation

In this paper, we consider the limit properties of the largest entries of sample covariance matrices and the sample correlation matrices. In order to make the statistics based on the largest entries of the sample covariance matrices and the sample correlation matrices more applicable in high-dimensional tests, the identically distributed assumption of population is removed. Under some moment’s assumption of the underlying distribution, we obtain that the almost surely limit and asymptotical distribution of the extreme statistics as both the dimension p and sample size n tend to infinity.

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Weak Convergence of the Empirical Spectral Distribution of High-Dimensional Band Sample Covariance Matrices

Journal of Theoretical Probability ◽

10.1007/s10959-017-0751-7 ◽

2017 ◽

Vol 31 (3) ◽

pp. 1273-1302

Author(s):

Kamil Jurczak

Keyword(s):

Weak Convergence ◽

Spectral Distribution ◽

Covariance Matrices ◽

High Dimensional ◽

Empirical Spectral Distribution ◽

Sample Covariance Matrices ◽

Sample Covariance

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Convergence of eigenvector empirical spectral distribution of sample covariance matrices

The Annals of Statistics ◽

10.1214/19-aos1832 ◽

2020 ◽

Vol 48 (2) ◽

pp. 953-982 ◽

Cited By ~ 1

Author(s):

Haokai Xi ◽

Fan Yang ◽

Jun Yin

Keyword(s):

Spectral Distribution ◽

Covariance Matrices ◽

Empirical Spectral Distribution ◽

Sample Covariance Matrices ◽

Sample Covariance

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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.

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