Abstract
Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X*X(X t X) converges to the Tracy–Widom law as n,p (the dimensions of X) tend to ∞ in some ratio n/p→γ>0. We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n−p=O(p 1/3). We also prove that λ max≤(n 1/2+p 1/2)2+O(p 1/2 log(p)) (a.e.) for general γ>0.
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REFERENCES
R. J. Muirhead, Aspects of Multivariate Statistical Theory (Wiley, New York, 1982).
S. Wilks, Mathematical Statistics (Wiley, New York, 1967).
Z. D. Bai and Y. Q. Yin, Convergence of the semicircle law, Ann. Probab. 16:863-875 (1988).
Z. D. Bai, J. W. Silverstein, and Y. Q. Yin, A note on the largest eigenvalue of a large dimensional sample covariance matrix, J. Mult. Anal. 26:166-168 (1988).
Z. D. Bai, Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices, Ann. Probab. 21:649-672 (1993).
Z. D. Bai and J. W. Silverstein, No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices, Ann. Probab. 26:316-345 (1998).
Z. D. Bai, Methodologies in spectral analysis of large dimensional random matrices. A review, Statist. Sinica 9:611-677 (1999).
B. V. Bronk, Exponential ensemble for random matrices, J. Math. Phys. 6:228-237 (1965).
A. Edelman, The distribution and moments of the smallest eigenvalue of a random matrix of Wishart type, Linear Algebra Appl. 159:55-80 (1991).
P. J. Forrester, The spectral edge of random matrix ensembles, Nucl. Phys. B. 402:709-728 (1994).
P. J. Forrester, T. Nagao, and G. Honner, Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nucl. Phys. B. 553:601-643 (1999).
S. Geman, A limit theorem for the norm of random matrices, Ann. Probab. 8:252-261 (1980).
A. Guionnet, Large deviation upper bounds and central limit theorems for band matrices and non-commutative functionals of Gaussian large random matrices, preprint (2000).
U. Grenader and J. Silverstein, Spectral analysis of networks with random topologies, SIAM J. Appl. Math. 32:499-519 (1977).
A. T. James, Distribution of matrices variates and latent roots derived from normal samples, Ann. Math. Statist. 35:475-501-237 (1965).
I.M. Johnstone, On the distribution of the largest principal component, Ann. Statist. 29, (2001).
V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sbornik 1:457-483 (1967).
Y. Q. Yin, Z. D. Bai, and P. R. Krishnaiah, On the limit of the largest eigenvalue of a large dimensional sample covariance matrix, Probab. Theory Related Fields 78:509-521 (1988).
K. W. Wachter, The strong limits of random matrix spectra for sample matrices of independent elements, Ann. Probab. 6:1-18 (1978).
J. Silverstein, On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix, J. Multivariate Anal. 30:307-311 (1989).
K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209: 437-476, (2000).
C. A. Tracy and H. Widom, Level-spacing distribution and Airy kernel, Comm. Math. Phys. 159:151-174 (1994).
C. A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177:727-754 (1996).
D. Aldous and P. Diaconis, Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36:413-432 (1999).
P. Deift, Integrable Systems and Combinatorial Theory, Notices Amer. Math. Soc. 47: 631-640 (2000).
A. Soshnikov, Determinantal random point fields, Russian Math. Surveys 55:923-975 (2000).
A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys. 207:697-733 (1999).
L. A. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86:109-147 (1997).
P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52:1335-1425, (1999).
P. Bleher and A. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. 150:185-266 (2000).
K. Johansson, Universality of local spacing distributions in certain Hermitian Wigner matrices, Comm. Math. Phys. 215:683-705 (2001).
E. Brézin and A. Zee, Universality of the correlations between eigenvalues of large random matrices, Nuclear Phys. B. 402:613-627 (1993).
E. Kanzieper, V. Frelikher, Spectra of large matrices: a method of study, in Diffusive Waves in Complex Media (Kluwer, Dodrecht, 1999), pp. 165-211.
M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, New York, 1991).
G. Szego, Orthogonal Polynomials, 3rd ed. (American Mathematical Society, 1967).
C. A. Tracy and H. Widom, Correlation functions, cluster functions, and spacing distribution for random matrices, J. Statist. Phys. 92:809-835 (1998).
H. Widom, On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Statist. Phys. 94:347-363 (1999).
Ya. Sinai, A. Soshnikov, Central limit theorem for traces of large random symmetric matrices, Bol. Soc. Brasil. Mat. (N.S.) 29:1-24 (1998).
Ya. Sinai and A. Soshnikov, A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices, Funct. Anal. Appl. 32:114-131 (1998).
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Soshnikov, A. A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices. Journal of Statistical Physics 108, 1033–1056 (2002). https://doi.org/10.1023/A:1019739414239
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DOI: https://doi.org/10.1023/A:1019739414239