Comparison between two types of large sample covariance matrices

Pan, Guangming

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 655-677.

Soit $\left\{X_{ij}\right\}$, $i,j=1,2,...$, un tableau à double entrées, les $X_{ij}$ étant des variables aléatoires réelles indépendantes et identiquement distribuées (i.i.d.) et où $𝐄X_{11}=\mu$, $𝐄|X_{11}{-\mu |}^2=1$ et $𝐄|X_{11}{|}^4<\infty$. Considérons les matrices de covariances empiriques suivantes (avec/sans centrage empirique): $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ et $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, avec $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ et ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$, où ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$ est une matrice déterministe définie positive. Nous démontrons que, sous le régime asymptotique $n\to \infty$ et $p/n$ converge vers une constante positive, le théorème central limite pour la statistique $𝒮$ est différent de celui concernant la statistique $𝐒$. En outre, nous montrons que cette différence de comportement n’est pas observée pour le comportement moyen des vecteurs propres.

Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^4<\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.

Classification : 15A52, 60F15, 62E20, 60F17
Mots clés : central limit theorems, eigenvectors and eigenvalues, sample covariance matrix, Stieltjes transform, strong convergence

[1] T. W. Anderson. An Introduction to Multivariate Statistical Analysis, 2nd edition. Wiley, New York, 1984. | MR | Zbl

[2] Z. D. Bai and J. W. Silverstein. No eigenvalues outside the support of the limiting spectral distribution of large dimensional random matrices. Ann. Probab. 26 (1998) 316-345. | MR | Zbl

[3] Z. D. Bai and J. W. Silverstein. CLT for linear spectral statistics of large dimensional sample covariance matrices. Ann. Probab. 32 (2004) 553-605. | MR | Zbl

[4] Z. D. Bai, B. Q. Miao and G. M. Pan. On asymptotics of eigenvectors of large sample covariance matrix. Ann. Probab. 35 (2007) 1532-1572. | MR | Zbl

[5] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edition. Springer, New York, 2010. | MR | Zbl

[6] Z. D. Bai and G. M. Pan. Limiting behavior of eigenvectors of large Wigner matrices. J. Stat. Phys. 146 (2012) 519-549. | MR | Zbl

[7] F. Benaych-Georges. Eigenvectors of Wigner matrices: Universality of global fluctuations. Available at http://arxiv.org/abs/1104.1219.

[8] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl

[9] C. Donati-Martin and A. Rouault. Truncations of Haar unitary matrices, traces and bivariate Brownian bridge. RMTA 1(1) (2012) 1150007. | MR | Zbl

[10] I. Johnstone. On the distribution of the largest eigenvalue in principal component analysis. Ann. Statist. 29 (2001) 295-327. | MR | Zbl

[11] T. F. Jiang. The limiting distriubtions of eigenvalues of sample correlation matrices. Sankhya 66 (2004) 35-48. | MR | Zbl

[12] D. Jonsson. Some limit theorems of the eigenvalues of sample convariance matrix. J. Multivariate Anal. 12 (1982) 1-38. | MR | Zbl

[13] O. Ledoit and S. Peche. Eigenvectors of some large sample covariance matrix ensembles. Probab. Theory Related Fields 151 (2011) 233-264. | MR | Zbl

[14] V. A. Marčenko and L. A. Pastur. Distribution for some sets of random matrices. Math. USSR-Sb. 1 (1967) 457-483. | Zbl

[15] G. M. Pan and W. Zhou. Central limit theorem for Hotelling’s $T^2$ statistic under large dimension. Ann. Appl. Probab. 21 (2011) 1860-1910. | MR | Zbl

[16] G. M. Pan and W. Zhou. Central limit theorem for signal-to-interference ratio of reduced rank linear receiver. Ann. Appl. Probab. 18 (2008) 1232-1270. | MR | Zbl

[17] G. M. Pan, Q. M. Shao and W. Zhou. Universality of sample covariance matrices: CLT of the smoothed empirical spectral distribution. Preprint.

[18] J. W. Silverstein. On the eigenvectors of large dimensional sample covariance matrices. J. Multivariate Anal. 30 (1989) 1-16. | MR | Zbl

[19] J. W. Silverstein. Strong convergence of the limiting distribution of the eigenvalues of large dimensional random matrices. J. Multivariate Anal. 55 (1995) 331-339. | MR | Zbl

[20] J. W. Silverstein. Weak convergence of random functions defined by the eigenvectors of sample covariance matrices. Ann. Probab. 18 (1990) 1174-1194. | MR | Zbl

[21] T. Tao and V. Vu. Random matrices: Universal properties of eigenvectors. Available at arXiv:1103.2801v2. | MR | Zbl

[22] H. Xiao and W. Zhou. On the limit of the smallest eigenvalue of some sample covariance matrix. J. Theoret. Probab. 23 (2010) 1-20. | MR | Zbl

[23] Y. Q. Yin, Z. D. Bai and P. R. Krishanaiah. On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 (1988) 509-521. | MR | Zbl