Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices

@article{Cai2020LimitingLF,
  title={Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices},
  author={T. Tony Cai and Xiao Han and Guangming Pan},
  journal={The Annals of Statistics},
  year={2020}
}
We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance matrix model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic… Expand

Tables from this paper

Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices
We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when theExpand
Generalized four moment theorem and an application to CLT for spiked eigenvalues of high-dimensional covariance matrices
We consider a more generalized spiked covariance matrix $\Sigma$, which is a general non-definite matrix with the spiked eigenvalues scattered into a few bulks and the largest ones allowed to tend toExpand
The limits of the sample spiked eigenvalues for a high-dimensional generalized Fisher matrix and its applications.
A generalized spiked Fisher matrix is considered in this paper. We establish a criterion for the description of the support of the limiting spectral distribution of high-dimensional generalizedExpand
Asymptotics and fluctuations of largest eigenvalues of empirical covariance matrices associated with long memory stationary processes
Large covariance matrices play a fundamental role in the multivariate analysis and high-dimensional statistics. Since the pioneer’s works of Marcenko and Pastur (1967), the asymptotic behavior of theExpand
Generalized Four Moment Theorem with an application to the CLT for the spiked eigenvalues of high-dimensional general Fisher-matrices.
The universality for the local spiked eigenvalues is a powerful tool to deal with the problems of the asymptotic law for the bulks of spiked eigenvalues of high-dimensional generalized FisherExpand
Limiting laws for extreme eigenvalues of large-dimensional spiked Fisher matrices with a divergent number of spikes
TLDR
In the regime of high dimension where both n and T are proportional to p, this work investigates the limiting laws for extreme (spiked) eigenvalues of the sample (spiking) Fisher matrix when the number of spikes is divergent and these spikes are unbounded. Expand
Central Limit Theory for Linear Spectral Statistics of Normalized Separable Sample Covariance Matrix
This paper focuses on the separable covariance matrix when the dimension p and the sample size n grow to infinity but the ratio p/n tends to zero. The separable sample covariance matrix can beExpand
CLT for LSS of sample covariance matrices with unbounded dispersions
Under the high-dimensional setting that data dimension and sample size tend to infinity proportionally, we derive the central limit theorem (CLT) for linear spectral statistics (LSS) ofExpand
Limiting laws and consistent estimation criteria for fixed and diverging number of spiked eigenvalues
In this paper, we study limiting laws and consistent estimation criteria for the extreme eigenvalues in a spiked covariance model of dimension $p$. Firstly, for fixed $p$, we propose a generalizedExpand
Non-Asymptotic Properties of Spectral Decomposition of Large Gram-Type Matrices and Applications
Gram-type matrices and their spectral decomposition are of central importance for numerous problems in statistics, applied mathematics, physics, and machine learning. In this paper, we carefullyExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 44 REFERENCES
Asymptotics of empirical eigenstructure for high dimensional spiked covariance.
TLDR
These results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013) and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Expand
ASYMPTOTICS OF SAMPLE EIGENSTRUCTURE FOR A LARGE DIMENSIONAL SPIKED COVARIANCE MODEL
This paper deals with a multivariate Gaussian observation model where the eigenvalues of the covariance matrix are all one, except for a finite number which are larger. Of interest is the asymptoticExpand
Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices
We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go toExpand
Central limit theorems for eigenvalues in a spiked population model
In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope withExpand
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
AbstractWe compute the limiting distributions of the largest eigenvalue of a complex Gaussian samplecovariance matrix when both the number of samples and the number of variables in each samplebecomeExpand
Surprising Asymptotic Conical Structure in Critical Sample Eigen-Directions
The aim of this paper is to establish several deep theoretical properties of principal component analysis for multiple-component spike covariance models. Our new results reveal a surprisingExpand
Eigenvalues of large sample covariance matrices of spiked population models
We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sampleExpand
Finite sample approximation results for principal component analysis: a matrix perturbation approach
Principal component analysis (PCA) is a standard tool for dimensional reduction of a set of $n$ observations (samples), each with $p$ variables. In this paper, using a matrix perturbation approach,Expand
Optimal estimation and rank detection for sparse spiked covariance matrices
TLDR
The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariances matrices. Expand
A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix
For the test of sphericity, Ledoit and Wolf [Ann. Statist. 30 (2002) 1081-1102] proposed a statistic which is robust against high dimensionality. In this paper, we consider a natural generalizationExpand
...
1
2
3
4
5
...