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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.

MSC:

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

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References:

[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
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