Asymptotic properties of eigenmatrices of a large sample covariance matrix

Z. D. Bai; H. X. Liu; W. K. Wong

doi:10.1214/10-AAP748

Asymptotic properties of eigenmatrices of a large sample covariance matrix

Z. D. Bai^*, H. X. Liu, W. K. Wong

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › peer-review

21 Citations (Scopus)

Abstract

Let Sn = 1/n X_nX_n where X_n = {X _ij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Y_n(t₁, t ₂,σ)=√p(x_n(t₁) ^*(Sn +σI)^-1x_n(t₂)-x _n(t₁)*x_n(t₂)m _n(σ)) in which σ > 0 and m_n(σ)= ∫dF_yn(x)/x+σ where F_yn(x) is the Marčenko-Pastur law with parameter y_n = p/n; which converges to a positive constant as n → ∞ and x_n(t₁) and x_n(t₂) are unit vectors in ℂp, having indices t ₁ and t₂, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Y_n(t₁, t₂, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S _n is asymptotically close to that of a Haar-distributed unitary matrix.

Original language	English
Pages (from-to)	1994-2015
Number of pages	22
Journal	Annals of Applied Probability
Volume	21
Issue number	5
DOIs	https://doi.org/10.1214/10-AAP748
Publication status	Published - Oct 2011

Scopus Subject Areas

Statistics and Probability
Statistics, Probability and Uncertainty

User-Defined Keywords

Central limit theorems
Haar distribution
Linear spectral statistics
Marčenko-Pastur law
Random matrix
Sample covariance matrix
Semicircular law

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10.1214/10-AAP748

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title = "Asymptotic properties of eigenmatrices of a large sample covariance matrix",

abstract = "Let Sn = 1/n XnXn where Xn = {X ij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1, t 2,σ)=√p(xn(t1) *(Sn +σI)-1xn(t2)-x n(t1)*xn(t2)m n(σ)) in which σ > 0 and mn(σ)= ∫dFyn(x)/x+σ where Fyn(x) is the Mar{\v c}enko-Pastur law with parameter yn = p/n; which converges to a positive constant as n → ∞ and xn(t1) and xn(t2) are unit vectors in ℂp, having indices t 1 and t2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1, t2, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S n is asymptotically close to that of a Haar-distributed unitary matrix.",

keywords = "Central limit theorems, Haar distribution, Linear spectral statistics, Mar{\v c}enko-Pastur law, Random matrix, Sample covariance matrix, Semicircular law",

author = "Bai, {Z. D.} and Liu, {H. X.} and Wong, {W. K.}",

year = "2011",

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AU - Liu, H. X.

AU - Wong, W. K.

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N2 - Let Sn = 1/n XnXn where Xn = {X ij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1, t 2,σ)=√p(xn(t1) *(Sn +σI)-1xn(t2)-x n(t1)*xn(t2)m n(σ)) in which σ > 0 and mn(σ)= ∫dFyn(x)/x+σ where Fyn(x) is the Marčenko-Pastur law with parameter yn = p/n; which converges to a positive constant as n → ∞ and xn(t1) and xn(t2) are unit vectors in ℂp, having indices t 1 and t2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1, t2, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S n is asymptotically close to that of a Haar-distributed unitary matrix.

AB - Let Sn = 1/n XnXn where Xn = {X ij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1, t 2,σ)=√p(xn(t1) *(Sn +σI)-1xn(t2)-x n(t1)*xn(t2)m n(σ)) in which σ > 0 and mn(σ)= ∫dFyn(x)/x+σ where Fyn(x) is the Marčenko-Pastur law with parameter yn = p/n; which converges to a positive constant as n → ∞ and xn(t1) and xn(t2) are unit vectors in ℂp, having indices t 1 and t2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1, t2, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S n is asymptotically close to that of a Haar-distributed unitary matrix.

KW - Central limit theorems

KW - Haar distribution

KW - Linear spectral statistics

KW - Marčenko-Pastur law

KW - Random matrix

KW - Sample covariance matrix

KW - Semicircular law

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SN - 1050-5164

VL - 21

SP - 1994

EP - 2015

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 5

ER -

Asymptotic properties of eigenmatrices of a large sample covariance matrix

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