## The Annals of Probability

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

#### 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

#### 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

#### References

• Bai, Z. D. (1999). Methodologies in spectral analysis of large dimensional random matrices, A review. Statist. Sinica 9 611--677.
• Bai, Z. D. and Saranadasa, H. (1996). Effect of high dimension comparison of significance tests for a high dimensional two sample problem. Statist. Sinica 6 311--329.
• Bai, Z. D. and Silverstein, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large dimensional random matrices. Ann. Probab. 26 316--345.
• Bai, Z. D. and Silverstein, J. W. (1999). Exact separation of eigenvalues of large dimensional sample covariance matrices. Ann. Probab. 27 1536--1555.
• Bai, Z. D. and Yin, Y. Q. (1993). Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21 1275--1294.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
• Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19--42.
• Dempster, A. P. (1958). A high dimensional two sample significance test. Ann. Math. Statist. 29 995--1010.
• Diaconis, P. and Evans, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615--2633.
• Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151--204.
• Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix J. Multivariate Anal. 12 1--38.
• Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb. 1 457--483.
• Silverstein, J. W. (1985). The limiting eigenvalue distribution of a multivariate F matrix. SIAM J. Math. Anal. 16 641--646.
• Silverstein, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. J. Multivariate Anal. 55 331--339.
• Silverstein, J. W. and Choi, S. I. (1995). Analysis of the limiting spectral distribution of large dimensional random matrices. J. Multivariate Anal. 54 295--309.
• Silverstein, J. W. and Combettes, P. L. (1992). Signal detection via spectral theory of large dimensional random matrices. IEEE Trans. Signal Process. 40 2100--2105.
• Sinai, Ya. and Soshnikov, A. (1998). Central limit theorem for traces of large symmetric matrices with independent matrix elements. Bol. Soc. Brasil Mat. (N.S.) 29 1--24.
• Soshnikov, A. (2000). The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28 1353--1370.
• Titchmarsh, E. C. (1939). The Theory of Functions, 2nd ed. Oxford Univ. Press.
• Yin, Y. Q. (1986). Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20 50--68.
• Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 509--521.
• Yin, Y. Q. and Krishnaiah, P. R. (1983). A limit theorem for the eigenvalues of product of two random matrices. J. Multivariate Anal. 13 489--507.