## The Annals of Statistics

### Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix

#### Abstract

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of sample covariance matrix and the Marčenko–Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely, requiring finite 8th moment of the underlying distribution.

#### Article information

Source
Ann. Statist., Volume 41, Number 5 (2013), 2572-2607.

Dates
First available in Project Euclid: 19 November 2013

https://projecteuclid.org/euclid.aos/1384871346

Digital Object Identifier
doi:10.1214/13-AOS1154

Mathematical Reviews number (MathSciNet)
MR3161438

Zentralblatt MATH identifier
1285.15018

#### Citation

Xia, Ningning; Qin, Yingli; Bai, Zhidong. Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix. Ann. Statist. 41 (2013), no. 5, 2572--2607. doi:10.1214/13-AOS1154. https://projecteuclid.org/euclid.aos/1384871346

#### References

• [1] Anderson, T. W. (1963). Asymptotic theory for principal component analysis. Ann. Inst. Statist. Math. 34 122–148.
• [2] Bai, Z. D. (1993). Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices. Ann. Probab. 21 625–648.
• [3] Bai, Z. D. (1993). Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices. Ann. Probab. 21 649–672.
• [4] Bai, Z. D., Miao, B. Q. and Pan, G. M. (2007). On asymptotics of eigenvectors of large sample covariance matrix. Ann. Probab. 35 1532–1572.
• [5] Bai, Z. D. and Silverstein, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316–345.
• [6] Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605.
• [7] Bai, Z. D. and Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York.
• [8] Bai, Z. D. and Xia, N. (2013). Functional CLT of eigenvectors for large sample covariance matrices. Statist. Papers. To appear.
• [9] Cai, T., Ma, Z. M. and Wu, Y. H. (2013). Sparse PCA: Optimal rates and adaptive estimation. Available at arXiv:1211.1309.
• [10] Erdős, L., Schlein, B. and Yau, H.-T. (2009). Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 815–852.
• [11] Götze, F. and Tikhomirov, A. (2004). Rate of convergence in probability to the Marchenko–Pastur law. Bernoulli 10 503–548.
• [12] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
• [13] Knowles, A. and Yin, J. (2013). Eigenvector distribution of Wigner matrices. Probab. Theory Related Fields 155 543–582.
• [14] Loève, M. (1977). Probability Theory. I, 4th ed. Springer, New York. Graduate Texts in Mathematics 45.
• [15] Ma, Z. M. (2013). Sparse principal component analysis and iterative thresholding. Ann. Statist. 41 772–801.
• [16] Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb. 1 457–483.
• [17] Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 1617–1642.
• [18] Pillai, N. S. and Yin, J. (2013). Universality of covariance matrices. Available at arXiv:1110.2501v6.
• [19] Silverstein, J. W. (1981). Describing the behavior of eigenvectors of random matrices using sequences of measures on orthogonal groups. SIAM J. Math. Anal. 12 274–281.
• [20] Silverstein, J. W. (1984). Some limit theorems on the eigenvectors of large-dimensional sample covariance matrices. J. Multivariate Anal. 15 295–324.
• [21] Silverstein, J. W. (1989). On the eigenvectors of large-dimensional sample covariance matrices. J. Multivariate Anal. 30 1–16.
• [22] Silverstein, J. W. (1990). Weak convergence of random functions defined by the eigenvectors of sample covariance matrices. Ann. Probab. 18 1174–1194.
• [23] Silverstein, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 331–339.
• [24] Silverstein, J. W. and Bai, Z. D. (1995). On the empirical distribution of eigenvalues of a class of large-dimensional random matrices. J. Multivariate Anal. 54 175–192.
• [25] Tao, T. and Vu, V. (2011). Universal properties of eigenvectors. Available at arXiv:1103.2801v2.
• [26] Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Related Fields 78 509–521.