Abstract
Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β analogue, proving a conjecture of Baik et al. (Ann Probab 33:1643–1697, 2005). We also treat shifted mean Gaussian orthogonal and β ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrödinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β = 2,4, yielding in particular a new and simple proof of the Painlevé representations for these Tracy–Widom distributions.
References
Anderson G., Guionnet A., Zeitouni O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009)
Anderson T.W.: Asymptotic theory for principal component analysis. Ann. Math. Stat. 34, 122–148 (1963)
Anderson T.W.: An Introduction to Multivariate Statistical Analysis. 3rd edn. Wiley-Interscience, New York (2003)
Bai Z.D.: Methodologies in spectral analysis of large-dimensional random matrices a review. Stat. Sin. 9, 611–677 (1999)
Bai Z.D., Silverstein J.W.: No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26, 316–345 (1998)
Bai Z.D., Silverstein J.W.: Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27, 1536–1555 (1999)
Bai Z., Yao J.-F.: Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. Henri Poincaré Probab. Stat. 44, 447–474 (2008)
Baik J.: Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133, 205–235 (2006)
Baik J., Ben Arous G., Péché S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33, 1643–1697 (2005)
Baik J., Rains E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–541 (2000)
Baik J., Rains E.M.: The asymptotics of monotone subsequences of involutions. Duke Math. J. 109, 205–281 (2001)
Baik J., Silverstein J.W.: Eigenvalues of large sample covariance matrices of spiked population models. J. Multivar. Anal. 97, 1382–1408 (2006)
Bassler K.E., Forrester P.J., Frankel N.E.: Edge effects in some perturbations of the Gaussian unitary ensemble. J. Math. Phys. 51, 123305 (2010)
Ben Arous G., Corwin I.: Current fluctuations for TASEP: a proof of the Prähofer-Spohn conjecture. Ann. Probab. 39, 104–138 (2011)
Benaych-Georges, F., Nadakuditi, R.R.: The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. arXiv:0910.2120v2 (2009)
Bloemendal, A.: In preparation (2012)
Bloemendal, A., Sutton, B.D.: In preparation (2012)
Deift P.A., Zhou X.: Asymptotics for the Painlevé II equation. Commun. Pure Appl. Math. 48, 277–337 (1995)
Desrosiers, P., Forrester, P.J.: Asymptotic correlations for Gaussian and Wishart matrices with external source. Int. Math. Res. Not. 2006, Art. ID 27395 (2006)
Dumitriu I., Edelman A.: Matrix models for beta ensembles. J. Math. Phys. 43, 5830–5847 (2002)
Edelman A., Sutton B.D.: From random matrices to stochastic operators. J. Stat. Phys. 127, 1121–1165 (2007)
El Karoui, N.: On the largest eigenvalue of Wishart matrices with identity covariance when n, p, and p/n → ∞. arXiv:math/0309355v1 (2003)
El Karoui N.: Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. Ann. Probab. 35, 663–714 (2007)
Ethier S.N., Kurtz T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)
Féral D., Péché S.: The largest eigenvalue of rank one deformation of large Wigner matrices. Commun. Math. Phys. 272, 185–228 (2007)
Féral D., Péché S.: The largest eigenvalues of sample covariance matrices for a spiked population: diagonal case. J. Math. Phys. 50, 073302 (2009)
Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Y.: Painlevé Transcendents: The Riemann-Hilbert Approach. American Mathematical Society, Providence (2006)
Forrester P.J.: The spectrum edge of random matrix ensembles. Nuclear Phys. B 402, 709–728 (1993)
Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)
Forrester, P.J.: Probability densities and distributions for spiked Wishart β-ensembles. arXiv:1101. 2261v1 (2011)
Geman S.: A limit theorem for the norm of random matrices. Ann. Probab. 8, 252–261 (1980)
Halmos P.: Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea, New York (1957)
Harding M.: Explaining the single factor bias of arbitrage pricing models in finite samples. Econ. Lett. 99, 85–88 (2008)
Hastings S.P., McLeod J.B.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73, 31–51 (1980)
Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)
Johnstone I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29, 295–327 (2001)
Johnstone, I.M.: High dimensional statistical inference and random matrices. International Congress of Mathematicians, vol. I, pp. 307–333. Eur. Math. Soc., Zürich (2007)
Krishnapur, M., Rider, B., Virág, B.: In preparation (2012)
Marčenko V.A., Pastur L.A.: Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S) 72(114), 507–536 (1967)
Mo, M.Y.: The rank 1 real Wishart spiked model. arXiv:1101.5144v1 (2011)
Muirhead R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
Onatski A.: The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices. Ann. Appl. Probab. 18, 470–490 (2008)
Patterson N., Price A.L., Reich D.: Population structure and eigenanalysis. PLoS Genet. 2, e190 (2006)
Paul D.: Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Stat. Sin. 17, 1617–1642 (2007)
Péché S.: The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Relat. Fields 134, 127–173 (2006)
Péché S.: Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Relat. Fields 143, 481–516 (2009)
Ramírez J.A., Rider B., Virág B.: Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Am. Math. Soc. 24, 919–944 (2011)
Savchuk A.M., Shkalikov A.A.: Sturm-Liouville operators with singular potentials. Math. Notes 66, 897–912 (1999)
Silverstein J.W., Bai Z.D.: On the empirical distribution of eigenvalues of a class of large-dimensional random matrices. J. Multivar. Anal. 54, 175–192 (1995)
Soshnikov A.: A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108, 1033–1056 (2002)
Sutton, B.D.: The stochastic operator approach to random matrix theory. PhD thesis, Massachusetts Institute of Technology (2005)
Telatar E.: Capacity of multi-antenna Gaussian channels. Eur. Trans. Telecom. 10, 585–595 (1999)
Tracy C.A., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)
Tracy C.A., Widom H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)
Trotter H.F.: Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegő. Adv. Math. 54, 67–82 (1984)
Wang, D.: Spiked models in Wishart ensemble. PhD thesis, Brandeis University. arXiv:0804.0889v1 (2008)
Weidmann J.: Strong operator convergence and spectral theory of ordinary differential operators. Univ. Iagel. Acta Math. 34, 153–163 (1997)
Yin Y.Q., Bai Z.D., Krishnaiah P.R.: On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Relat. Fields 78, 509–521 (1988)
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Bloemendal, A., Virág, B. Limits of spiked random matrices I. Probab. Theory Relat. Fields 156, 795–825 (2013). https://doi.org/10.1007/s00440-012-0443-2
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DOI: https://doi.org/10.1007/s00440-012-0443-2
Mathematics Subject Classification
- 60B20
- Random matrices (probabilistic aspects)