The Annals of Statistics

The Tracy–Widom law for the largest eigenvalue of F type matrices

Xiao Han, Guangming Pan, and Bo Zhang

Full-text: Open access


Let ${\mathbf{A}}_{p}=\frac{{\mathbf{Y}}{\mathbf{Y}}^{*}}{m}$ and ${\mathbf{B}}_{p}=\frac{{\mathbf{X}}{\mathbf{X}}^{*}}{n}$ be two independent random matrices where ${\mathbf{X}}=(X_{ij})_{p\times n}$ and ${\mathbf{Y}}=(Y_{ij})_{p\times m}$ respectively consist of real (or complex) independent random variables with $\mathbb{E}X_{ij}=\mathbb{E}Y_{ij}=0$, $\mathbb{E}|X_{ij}|^{2}=\mathbb{E}|Y_{ij}|^{2}=1$. Denote by $\lambda_{1}$ the largest root of the determinantal equation $\det(\lambda{\mathbf{A}}_{p}-{\mathbf{B}}_{p})=0$. We establish the Tracy–Widom type universality for $\lambda_{1}$ under some moment conditions on $X_{ij}$ and $Y_{ij}$ when $p/m$ and $p/n$ approach positive constants as $p\rightarrow\infty$.

Article information

Ann. Statist., Volume 44, Number 4 (2016), 1564-1592.

Received: June 2015
Revised: December 2015
First available in Project Euclid: 7 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 34K25: Asymptotic theory
Secondary: 60F05: Central limit and other weak theorems 62H10: Distribution of statistics

Tracy–Widom distribution largest eigenvalue sample covariance matrix F matrix


Han, Xiao; Pan, Guangming; Zhang, Bo. The Tracy–Widom law for the largest eigenvalue of F type matrices. Ann. Statist. 44 (2016), no. 4, 1564--1592. doi:10.1214/15-AOS1427.

Export citation


  • [1] Bai, Z. and Silverstein, J. W. (2006). Spectral Analysis of Large Dimensional Random Matrices, 1st ed. Springer, New York.
  • [2] Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382–1408.
  • [3] Bao, Z., Pan, G. and Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. Ann. Statist. 43 382–421.
  • [4] Bao, Z. G., Pan, G. M. and Zhou, W. (2013). Local density of the spectrum on the edge for sample covariance matrices with general population. Preprint. Available at
  • [5] Dharmawansa, P., Johnstone, I. M. and Onatski, A. (2014). Local asymptotic normality of the spectrum of high-dimensional spiked F-ratios. Available at
  • [6] El Karoui, N. (2007). Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. Ann. Probab. 35 663–714.
  • [7] Erdős, L., Knowles, A. and Yau, H. (2013). Averaging fluctuations in resolvents of random band matrices. Ann. Henri Poincaré 14 1837–1926.
  • [8] Erdős, L., Schlein, B. and Yau, H. (2009). Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 641–655.
  • [9] Erdős, L., Yau, H. and Yin, J. (2012). Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229 1435–1515.
  • [10] Féral, D. and Péché, S. (2009). The largest eigenvalues of sample covariance matrices for a spiked population: Diagonal case. J. Math. Phys. 50 073302, 33.
  • [11] Fujikoshi, Y., Ulyanov, V. V. and Shimizu, R. (2010). Multivariate Statistics: High-Dimensional and Large-Sample Approximations. Wiley, Hoboken, NJ.
  • [12] Han, X., Pan, G. and Zhang, B. (2016). Supplement to “The Tracy–Widom law for the largest eigenvalue of F type matrices.” DOI:10.1214/15-AOS1427SUPP.
  • [13] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
  • [14] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • [15] Johnstone, I. M. (2008). Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence. Ann. Statist. 36 2638–2716.
  • [16] Johnstone, I. M. (2009). Approximate null distribution of the largest root in multivariate analysis. Ann. Appl. Stat. 3 1616–1633.
  • [17] Knowles, A. and Yin, J. (2015). Anisotropic local laws for random matrices. Available at arXiv:1410.3516v3.
  • [18] Ledoit, O. and Wolf, M. (2002). Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann. Statist. 30 1081–1102.
  • [19] Lee, J. O. and Schnelli, K. (2014). Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Available at arXiv:1409.4979v1.
  • [20] Levanon, N. (1988). Radar Principles. Wiley, New York.
  • [21] Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Sb. Math. 4 457–483.
  • [22] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • [23] Nadakuditi, R. R. and Silverstein, J. W. Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples. IEEE J. Sel. Top. Signal Process. 4 468–480.
  • [24] 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.
  • [25] Pillai, N. S. and Yin, J. (2014). Universality of covariance matrices. Ann. Appl. Probab. 24 935–1001.
  • [26] Soshnikov, A. (2002). A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108 1033–1056.
  • [27] Tao, T. and Vu, V. (2011). Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 127–204.
  • [28] Tao, T. and Vu, V. (2012). Random covariance matrices: Universality of local statistics of eigenvalues. Ann. Probab. 40 1285–1315.
  • [29] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • [30] Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754.
  • [31] Vinogradova, J., Couillet, R. and Hachem, W. (2013). Statistical inference in large antenna arrays under unknown noise pattern. IEEE Trans. Signal Process. 61 5633–5645.
  • [32] Wachter, K. W. (1980). The limiting empirical measure of multiple discriminant ratios. Ann. Statist. 8 937–957.
  • [33] Wang, K. (2012). Random covariance matrices: Universality of local statistics of eigenvalues up to the edge. Random Matrices Theory Appl. 1 1150005, 24.
  • [34] Wang, Q. and Yao, J. (2015). Extreme eigenvalues of large-dimensional spiked Fisher matrices with application. Available at
  • [35] Yao, J. F., Zheng, S. R. and Bai, Z. D. (2015). Large Sample Covariance Matrices and High-Dimemnsional Data Analysis. Cambridge Univ. Press, Cambridge.
  • [36] Zeng, Y. H. and Liang, Y. C. (2009). Eigenvalue-based spectrum sensing algorithms for cognitive radio. IEEE Transactions Communications 57 1784–1793.
  • [37] Zhang, L. X. (2006). Spectral Analysis of Large Dimensional Random Matrices. Ph.D. Thesis, National University of Singapore.
  • [38] Zheng, S. (2012). Central limit theorems for linear spectral statistics of large dimensional $F$-matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 444–476.

Supplemental materials