The Annals of Applied Probability

Asymptotic properties of eigenmatrices of a large sample covariance matrix

Z. D. Bai, H. X. Liu, and W. K. Wong

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Abstract

Let Sn = 1/n XnXn* where Xn = {Xij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_{n}(\mathbf{t}_{1},\mathbf{t}_{2},\sigma)=\sqrt{p}({\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}(S_{n}+\sigma I)^{-1}{\mathbf{x}}_{n}(\mathbf{t}_{2})-{\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}{\mathbf{x}}_{n}(\mathbf{t}_{2})m_{n}(\sigma))$ in which σ > 0 and mn(σ) = ∫d Fyn(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 ${\mathbb{C}}^{p}$, having indices t1 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 Sn is asymptotically close to that of a Haar-distributed unitary matrix.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 5 (2011), 1994-2015.

Dates
First available in Project Euclid: 25 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1319576615

Digital Object Identifier
doi:10.1214/10-AAP748

Mathematical Reviews number (MathSciNet)
MR2884057

Zentralblatt MATH identifier
1234.15013

Subjects
Primary: 15A52
Secondary: 60F05: Central limit and other weak theorems 15A18: Eigenvalues, singular values, and eigenvectors

Keywords
Random matrix central limit theorems linear spectral statistics sample covariance matrix Haar distribution Marčenko–Pastur law semicircular law

Citation

Bai, Z. D.; Liu, H. X.; Wong, W. K. Asymptotic properties of eigenmatrices of a large sample covariance matrix. Ann. Appl. Probab. 21 (2011), no. 5, 1994--2015. doi:10.1214/10-AAP748. https://projecteuclid.org/euclid.aoap/1319576615


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References

  • Anderson, T. W. (1951). The asymptotic distribution of certain characteristic roots and vectors. In Proceedings of the Second Berkeley Sympos. Math. Statist. Probab. 1950 103–130. Univ. of California Press, Berkeley and Los Angeles.
    Zentralblatt MATH: 0045.07302
  • Anderson, T. W. (1989). The asymptotic distribution of characteristic roots and vectors in multivariate components of variance. In Contributions to Probability and Statistics 177–196. Springer, New York.
  • Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611–677.
    Zentralblatt MATH: 0949.60077
  • 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.
    Zentralblatt MATH: 0937.60017
    Digital Object Identifier: doi:10.1214/aop/1022855421
    Project Euclid: euclid.aop/1022855421
  • Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605.
    Zentralblatt MATH: 1063.60022
    Digital Object Identifier: doi:10.1214/aop/1078415845
    Project Euclid: euclid.aop/1078415845
  • Bai, Z., Liu, H. and Wong, W.-K. (2009). Enhancement of the applicability of Markowitz’s portfolio optimization by utilizing random matrix theory. Math. Finance 19 639–667.
    Zentralblatt MATH: 1185.91155
    Digital Object Identifier: doi:10.1111/j.1467-9965.2009.00383.x
  • 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.
    Zentralblatt MATH: 1162.15012
    Digital Object Identifier: doi:10.1214/009117906000001079
    Project Euclid: euclid.aop/1181334252
  • Billingsley, P. (1995). Convergence of Probability Measures, 3rd ed. Wiley, New York.
    Zentralblatt MATH: 0172.21201
  • Donsker, M. D. (1951). An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc. 1951 12.
    Zentralblatt MATH: 0042.37602
  • Evans, J. and Tse, D. N. C. (2000). Large system performance of linear multiuser receivers in multipath fading channels. IEEE Trans. Inform. Theory 46 2059–2078.
    Mathematical Reviews (MathSciNet): MR1781367
    Zentralblatt MATH: 0999.94509
    Digital Object Identifier: doi:10.1109/18.868478
  • Loève, M. (1978). Probability Theory. II, 4th ed. Graduate Texts in Mathematics 46. Springer, New York.
  • Marčenko, V. A. and Pastur, L. A. (1967). Distribution for some sets of random matrices. Mathematics of the USSR-Sbornik 1 457–483.
  • Markowitz, H. M. (1952). Portfolio selection. Journal of Finance 7 77–91.
  • Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investments. Cowles Foundation for Research in Economics at Yale University, Monograph 16. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR0103768
  • Markowitz, H. M. (1991). Portfolio Selection: Efficient Diversification of Investment. Blackwell, Cambridge, MA.
  • 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.
    Zentralblatt MATH: 0465.60017
    Digital Object Identifier: doi:10.1137/0512025
  • Silverstein, J. W. (1984). Some limit theorems on the eigenvectors of large-dimensional sample covariance matrices. J. Multivariate Anal. 15 295–324.
    Mathematical Reviews (MathSciNet): MR768500
    Zentralblatt MATH: 0553.60011
    Digital Object Identifier: doi:10.1016/0047-259X(84)90054-X
  • Silverstein, J. W. (1989). On the eigenvectors of large-dimensional sample covariance matrices. J. Multivariate Anal. 30 1–16.
    Zentralblatt MATH: 0678.60011
    Digital Object Identifier: doi:10.1016/0047-259X(89)90084-5
  • Silverstein, J. W. (1990). Weak convergence of random functions defined by the eigenvectors of sample covariance matrices. Ann. Probab. 18 1174–1194.
    Zentralblatt MATH: 0708.62051
    Digital Object Identifier: doi:10.1214/aop/1176990741
    Project Euclid: euclid.aop/1176990741
  • Wachter, K. W. (1978). The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6 1–18.
    Zentralblatt MATH: 0374.60039
    Digital Object Identifier: doi:10.1214/aop/1176995607
    Project Euclid: euclid.aop/1176995607
  • Yin, Y. Q. (1986). Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20 50–68.
    Zentralblatt MATH: 0614.62060
    Digital Object Identifier: doi:10.1016/0047-259X(86)90019-9

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