scholarly journals Second-Order Moment Convergence Rates for Spectral Statistics of Random Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Junshan Xie
Keyword(s):  
Random Matrices ◽  
Complete Convergence ◽  
Convergence Rates ◽  
Covariance Matrices ◽  
Second Order ◽  
Order Moment ◽  
Precise Asymptotics ◽  
Moment Convergence ◽  
Sample Covariance ◽  
Spectral Statistics

This paper considers the precise asymptotics of the spectral statistics of random matrices. Following the ideas of Gut and Spătaru (2000) and Liu and Lin (2006) on the precise asymptotics of i.i.d. random variables in the context of the complete convergence and the second-order moment convergence, respectively, we will establish the precise second-order moment convergence rates of a type of series constructed by the spectral statistics of Wigner matrices or sample covariance matrices.

scholarly journals Convergence rates in precise asymptotics for a kind of complete moment convergence

2017 ◽  
Vol 17 (02) ◽  
pp. 1750015
Author(s):  
Lingtao Kong ◽  
Hongshuai Dai
Keyword(s):  
Complete Convergence ◽  
Convergence Rates ◽  
Complete Moment Convergence ◽  
Precise Asymptotics ◽  
Moment Convergence ◽  
Special Case

Liu and Lin (Statist. Probab. Lett. 2006) introduced a kind of complete moment convergence which includes complete convergence as a special case. In this paper, we study the convergence rates of the precise asymptotics for complete moment convergence introduced by Liu and Lin (2006) and get the corresponding convergence rates.

scholarly journals Bootstrapping spectral statistics in high dimensions

Biometrika ◽  
2019 ◽  
Vol 106 (4) ◽  
pp. 781-801 ◽  
Author(s):  
Miles E Lopes ◽  
Andrew Blandino ◽  
Alexander Aue
Keyword(s):  
Parametric Bootstrap ◽  
Covariance Matrices ◽  
Asymptotic Formulas ◽  
High Dimensional ◽  
High Dimensions ◽  
Full Data ◽  
Sample Covariance ◽  
Practical Standpoint ◽  
Spectral Statistics ◽  
Low Dimensional

Summary Statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics, and they play a central role in multivariate testing. Although bootstrap methods are an established approach to approximating the laws of spectral statistics in low-dimensional problems, such methods are relatively unexplored in the high-dimensional setting. The aim of this article is to focus on linear spectral statistics as a class of prototypes for developing a new bootstrap in high dimensions, a method we refer to as the spectral bootstrap. In essence, the proposed method originates from the parametric bootstrap and is motivated by the fact that in high dimensions it is difficult to obtain a nonparametric approximation to the full data-generating distribution. From a practical standpoint, the method is easy to use and allows the user to circumvent the difficulties of complex asymptotic formulas for linear spectral statistics. In addition to proving the consistency of the proposed method, we present encouraging empirical results in a variety of settings. Lastly, and perhaps most interestingly, we show through simulations that the method can be applied successfully to statistics outside the class of linear spectral statistics, such as the largest sample eigenvalue and others.

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