## Annals of Probability

### A rate of convergence result for the largest eigenvalue of complex white Wishart matrices

Noureddine El Karoui

#### Abstract

It has been recently shown that if X is an n×N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X*X, there exist sequences mn,N and sn,N such that (l1mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F2.

In this paper we show that, under the assumption that n/Nγ∈(0, ∞), we can find a function M, continuous and nonincreasing, and sequences μ̃n,N and σ̃n,N such that, for all real s0, there exists an integer N(s0, γ) for which, if (nN)≥N(s0, γ), we have, with ln,N=(l1μ̃n,N)/σ̃n,N,

∀ ss0 (nN)2/3|P(ln,Ns)−F2(s)|≤M(s0)exp(−s).

The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (e.g., statistical) applications.

#### Article information

Source
Ann. Probab., Volume 34, Number 6 (2006), 2077-2117.

Dates
First available in Project Euclid: 13 February 2007

https://projecteuclid.org/euclid.aop/1171377438

Digital Object Identifier
doi:10.1214/009117906000000502

Mathematical Reviews number (MathSciNet)
MR2294977

Zentralblatt MATH identifier
1108.62014

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

#### Citation

El Karoui, Noureddine. A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. 34 (2006), no. 6, 2077--2117. doi:10.1214/009117906000000502. https://projecteuclid.org/euclid.aop/1171377438

#### References

• Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, New York.
• Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605.
• 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.
• Baik, J. and Silverstein, J. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382–1408.
• Dieng, M. (2005). Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations. Int. Math. Res. Not. 37 2263–2287.
• Dozier, B. and Silverstein, J. (2007). On the empirical distribution of eigenvalues of large dimensional information-plus-noise type matrices. J. Multivariate Anal. To appear.
• 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.
• Forrester, P. J. (1993). The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 709–728.
• Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators. Birkhäuser, Basel.
• Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
• Johnstone, I. (2001). On the distribution of the largest eigenvalue in principal component analysis. Ann. Statist. 29 295–327.
• Johnstone, I. (2006). Canonical correlation analysis and Jacobi ensembles: Tracy–Widom limits and rates of convergence. Manuscript.
• Lax, P. D. (2002). Functional Analysis. Wiley, New York.
• Olver, F. (Web). Airy and related functions. Digital Library of Mathematical Functions. Available at http://dlmf.nist.gov/Contents/AI/.
• Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York–London.
• Reed, M. and Simon, B. (1972). Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York.
• Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York.
• Seiler, E. and Simon, B. (1975). On finite mass renormalizations in the two-dimensional Yukawa model. J. Math. Phys. 16 2289–2293.
• Szegő, G. (1975). Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI.
• Tracy, C. and Widom, H. (1994). Level-spacing distribution and the Airy kernel. Comm. Math. Phys. 159 151–174.
• Tracy, C. and Widom, H. (1998). Correlation functions, cluster functions and spacing distributions for random matrices. J. Statist. Phys. 92 809–835.
• Widom, H. (1999). On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Statist. Phys. 94 347–363.