The Annals of Applied Probability

Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix

Chen Wang, Baisuo Jin, Z. D. Bai, K. Krishnan Nair, and Matthew Harding

Full-text: Open access

Abstract

The auto-cross covariance matrix is defined as

\[\mathbf{M}_{n}=\frac{1}{2T}\sum_{j=1}^{T}(\mathbf{e}_{j}\mathbf{e}_{j+\tau}^{*}+\mathbf{e}_{j+\tau}\mathbf{e}_{j}^{*}),\] where $\mathbf{e}_{j}$’s are $n$-dimensional vectors of independent standard complex components with a common mean 0, variance $\sigma^{2}$, and uniformly bounded $2+\eta$th moments and $\tau$ is the lag. Jin et al. [Ann. Appl. Probab. 24 (2014) 1199–1225] has proved that the LSD of $\mathbf{M}_{n}$ exists uniquely and nonrandomly, and independent of $\tau$ for all $\tau\ge1$. And in addition they gave an analytic expression of the LSD. As a continuation of Jin et al. [Ann. Appl. Probab. 24 (2014) 1199–1225], this paper proved that under the condition of uniformly bounded fourth moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of $\mathbf{M}_{n}$ for all large $n$. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of $\mathbf{M}_{n}$ are also obtained.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3624-3683.

Dates
Received: September 2014
Revised: December 2014
First available in Project Euclid: 1 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1443703784

Digital Object Identifier
doi:10.1214/14-AAP1092

Mathematical Reviews number (MathSciNet)
MR3404646

Zentralblatt MATH identifier
1328.60088

Subjects
Primary: 60F15: Strong theorems 15A52 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles

Keywords
Auto-cross covariance dynamic factor analysis Marčenko–Pastur law limiting spectral distribution order detection random matrix theory strong limit of extreme eigenvalues Stieltjes transform

Citation

Wang, Chen; Jin, Baisuo; Bai, Z. D.; Nair, K. Krishnan; Harding, Matthew. Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix. Ann. Appl. Probab. 25 (2015), no. 6, 3624--3683. doi:10.1214/14-AAP1092. https://projecteuclid.org/euclid.aoap/1443703784


Export citation

References

  • Bai, Z. D. (1993). Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices. Ann. Probab. 21 625–648.
  • Bai, Z. D., Miao, B. Q. and Rao, C. R. (1991). Estimation of directions of arrival of signals: Asymptotic results. In Advances in Spectrum Analysis and Array Processing, Vol. I (S. Haykin, ed.) 327–347. Prentice Hall, West Nyack, NY.
  • 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.
  • Bai, Z. D. and Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York.
  • Bai, Z. D. and Silverstein, J. W. (2012). No eigenvalues outside the support of the limiting spectral distribution of information-plus-noise type matrices. Random Matrices Theory Appl. 1 1150004, 44.
  • Bai, Z. D. and Wang, C. (2015). A note on the limiting spectral distribution of a symmetrized auto-cross covariance matrix. Statist. Probab. Lett. 96 333–340.
  • Bai, Z. D. and Yao, J.-f. (2008). Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. Henri Poincaré Probab. Stat. 44 447–474.
  • Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382–1408.
  • Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19–42.
  • Jin, B., Wang, C., Bai, Z. D., Nair, K. K. and Harding, M. (2014). Limiting spectral distribution of a symmetrized auto-cross covariance matrix. Ann. Appl. Probab. 24 1199–1225.
  • Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • Li, Z., Wang, Q. and Yao, J. F. (2014). Identifying the number of factors from singular values of a large sample auto-covariance matrix. Preprint. Available at arXiv:1410.3687v2.
  • Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. 72 (114) 507–536.
  • Paul, D. and Silverstein, J. W. (2009). No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix. J. Multivariate Anal. 100 37–57.
  • Rao, C. R. and Rao, M. B. (1998). Matrix Algebra and Its Applications to Statistics and Econometrics. World Scientific, River Edge, NJ.
  • 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.