A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. (English) Zbl 1108.62014

Summary: It has been recently shown that if \(X\) is an \(n\times N\) matrix whose entries are i.i.d. standard complex Gaussian and \(l_1\) is the largest eigenvalue of \(X^*X\), there exist sequences \(m_{n,N}\) and \(s_{n,N}\) such that \((l_1-m_{n,N})/s_{n,N}\) converges in distribution to \(W_2\), the Tracy-Widom law appearing in the study of the Gaussian unitary ensemble [see C. A. Tracy and H. Widom, Commun. Math. Phys. 159, 151–174 (1994; Zbl 0789.35152); J. Stat. Phys. 92, No. 5–6, 809–835 (1998; Zbl 0942.60099)]. This probability law has a density which is known and computable. The cumulative distribution function of \(W_2\) is denoted \(F_2\).
We show that, under the assumption that \(n/N\to\gamma\in (0,\infty)\), we can find a function \(M\), continuous and nonincreasing, and sequences \(\widetilde{\mu}_{n,N}\) and \(\widetilde{\sigma}_{n,N}\) such that, for all real \(s_0\), there exists an integer \(N(s_0,\gamma)\) for which, if \((n\wedge N)\geq N(s_0,\gamma)\), we have, with \(l_{n,N}= (l_1- \widetilde{\mu}_{n,N})/ \widetilde{\sigma}_{n,N}\), \[ \forall s\geq s_0, \quad (n\wedge N)^{2/3} |P(l_{n,N}\leq s)- F_2(s)|\leq M(s_0) \exp(-s). \] The surprisingly good \(2/3\) rate and qualitative properties of the bounding function help explain the fact that the limiting distribution \(W_2\) is a good approximation to the empirical distribution of \(l_{n,N}\) in simulations, an important fact from the point of view of (e.g., statistical) applications.


62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
15B52 Random matrices (algebraic aspects)
62H10 Multivariate distribution of statistics
15A18 Eigenvalues, singular values, and eigenvectors


Full Text: DOI arXiv


[1] Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis , 3rd ed. Wiley, New York. · Zbl 1039.62044
[2] Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605. · Zbl 1063.60022
[3] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697. · Zbl 1086.15022
[4] Baik, J. and Silverstein, J. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382–1408. · Zbl 1220.15011
[5] Dieng, M. (2005). Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations. Int. Math. Res. Not. 37 2263–2287. · Zbl 1093.60009
[6] Dozier, B. and Silverstein, J. (2007). On the empirical distribution of eigenvalues of large dimensional information-plus-noise type matrices. J. Multivariate Anal. · Zbl 1115.60035
[7] El Karoui, N. (2003). On the largest eigenvalue of Wishart matrices with identity covariance when \(n,p\) and \(p/n\rightarrow\infty\). Technical report, Dept. Statistics, Stanford Univ. Available at http://arxiv.org/abs/math/0309355.
[8] Forrester, P. J. (1993). The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 709–728. · Zbl 1043.82538
[9] Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators . Birkhäuser, Basel. · Zbl 0946.47013
[10] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476. · Zbl 0969.15008
[11] Johnstone, I. (2001). On the distribution of the largest eigenvalue in principal component analysis. Ann. Statist. 29 295–327. · Zbl 1016.62078
[12] Johnstone, I. (2006). Canonical correlation analysis and Jacobi ensembles: Tracy–Widom limits and rates of convergence. Manuscript.
[13] Lax, P. D. (2002). Functional Analysis . Wiley, New York. · Zbl 1009.47001
[14] Olver, F. (Web). Airy and related functions. Digital Library of Mathematical Functions. Available at http://dlmf.nist.gov/Contents/AI/. · Zbl 0957.01035
[15] Olver, F. W. J. (1974). Asymptotics and Special Functions . Academic Press, New York–London. · Zbl 0303.41035
[16] Reed, M. and Simon, B. (1972). Methods of Modern Mathematical Physics. I. Functional Analysis . Academic Press, New York. · Zbl 0459.46001
[17] Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics. IV. Analysis of Operators . Academic Press, New York. · Zbl 0459.46001
[18] Seiler, E. and Simon, B. (1975). On finite mass renormalizations in the two-dimensional Yukawa model. J. Math. Phys. 16 2289–2293.
[19] Szegő, G. (1975). Orthogonal Polynomials , 4th ed. Amer. Math. Soc., Providence, RI. · Zbl 0305.42011
[20] Tracy, C. and Widom, H. (1994). Level-spacing distribution and the Airy kernel. Comm. Math. Phys. 159 151–174. · Zbl 0789.35152
[21] Tracy, C. and Widom, H. (1998). Correlation functions, cluster functions and spacing distributions for random matrices. J. Statist. Phys. 92 809–835. · Zbl 0942.60099
[22] Widom, H. (1999). On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Statist. Phys. 94 347–363. · Zbl 0935.60090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.