The Annals of Probability

CLT for linear spectral statistics of large-dimensional sample covariance matrices

Z. D. Bai and Jack W. Silverstein

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Abstract

Let $B_n=(1/N)T_n^{1/2}X_nX_n^*T_n^{1/2}$ where $X_n=(X_{ij})$ is $n\times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. The limiting behavior, as $n\to\infty$ with $n/N$ approaching a positive constant, of functionals of the eigenvalues of $B_n$, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of $B_n$, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be $1/n$ by proving, after proper scaling, that they form a tight sequence. Moreover, if $\expp X^2_{11}=0$ and $\expp|X_{11}|^4=2$, or if $X_{11}$ and $T_n$ are real and $\expp X_{11}^4=3$, they are shown to have Gaussian limits.

Article information

Source
Ann. Probab., Volume 32, Number 1A (2004), 553-605.

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1078415845

Digital Object Identifier
doi:10.1214/aop/1078415845

Mathematical Reviews number (MathSciNet)
MR2040792

Zentralblatt MATH identifier
1063.60022

Subjects
Primary: 15A52 60F05: Central limit and other weak theorems
Secondary: 62H99: None of the above, but in this section

Keywords
Linear spectral statistics random matrix empirical distribution function of eigenvalues Stieltjes transform

Citation

Bai, Z. D.; Silverstein, Jack W. CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 (2004), no. 1A, 553--605. doi:10.1214/aop/1078415845. https://projecteuclid.org/euclid.aop/1078415845


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