The Annals of Statistics

Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence

Iain M. Johnstone

Full-text: Open access


Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B)−1B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that m and n grow in proportion to p. We show that after centering and scaling, the distribution is approximated to second-order, O(p−2/3), by the Tracy–Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.

Article information

Ann. Statist., Volume 36, Number 6 (2008), 2638-2716.

First available in Project Euclid: 5 January 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H10: Distribution of statistics
Secondary: 62E20: Asymptotic distribution theory 15A52

Canonical correlation analysis characteristic roots Fredholm determinant Jacobi polynomials largest root Liouville–Green multivariate analysis of variance random matrix theory Roy’s test soft edge Tracy–Widom distribution


Johnstone, Iain M. Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence. Ann. Statist. 36 (2008), no. 6, 2638--2716. doi:10.1214/08-AOS605.

Export citation


  • Absil, P.-A., Edelman, A. and Koev, P. (2006). On the largest principal angle between random subspaces. Linear Algebra Appl. 414 288–294.
  • Adler, M., Forrester, P. J., Nagao, T. and van Moerbeke, P. (2000). Classical skew-orthogonal polynomials and random matrices. J. Statist. Phys. 99 141–170.
  • Aubin, J.-P. (1979). Applied Functional Analysis. Wiley, New York.
  • Baik, J., Borodin, A., Deift, P. and Suidan, T. (2006). A model for the bus system in Cuernavaca (Mexico). J. Phys. A 39 8965–8975.
  • Bosbach, C. and Gawronski, W. (1999). Strong asymptotics for Jacobi polnomials with varying weights. Methods Appl. Anal. 6 39–54.
  • Carteret, H. A., Ismail, M. E. H. and Richmond, B. (2003). Three routes to the exact asymptotics for the one-dimensional quantum walk. J. Phys. A 36 8775–8795.
  • Chen, C. W. (1971). On some problems in canonical correlation analysis. Biometrika 58 399–400.
  • Chen, L.-C. and Ismail, M. E. H. (1991). On asymptotics of Jacobi polynomials. SIAM J. Math. Anal. 22 1442–1449.
  • Collins, B. (2005). Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Related Fields 133 315–344.
  • Constantine, A. G. (1963). Some noncentral distribution problems in multivariate analysis. Ann. Math. Statist. 34 1270–1285.
  • Deift, P. (2007). Universality for mathematical and physical systems. In Proceedings of the International Congress of Mathematicians I 125–152. Eur. Math. Soc., Zürich.
  • Deift, P. and Gioev, D. (2007). Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Comm. Pure Appl. Math. 60 867–910.
  • Deift, P., Gioev, D., Kriecherbauer, T. and Vanlessen, M. (2007). Universality for orthogonal and symplectic Laguerre-type ensembles. J. Statist. Phys. 129 949–1053.
  • Dumitriu, I. and Koev, P. (2008). Distributions of the extreme eigenvalues of Beta-Jacobi random matrices. SIAM J. Matrix Anal. Appl. 30 1–6.
  • Dunster, T. M. (1999). Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 281–316.
  • Dyson, F. J. (1970). Correlations between eigenvalues of a random matrix. Comm. Math. Phys. 19 235–250.
  • El Karoui, N. (2006). A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. 34 2077–2117.
  • Forrester, P. J. (2004). Log-gases and Random matrices. Book manuscript. Available at
  • Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs 18. Amer. Math. Soc., Providence, RI.
  • Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators. Birkhäuser, Basel.
  • Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations, 3rd ed. Johns Hopkins Univ. Press.
  • Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series, and Products, corrected and enlarged edition. Translated from Russian. ed. A. Jeffrey. Academic Press, New York.
  • James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 475–501.
  • Jiang, T. (2008). Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles. Probab. Theory Related Fields. DOI: 10.1007/s00440-008-0146-x.
  • Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • Johnstone, I. M. (2007). High-dimensional statistical inference and random matrices. In Proceedings of the International Congress of Mathematicians I 307–333. Eur. Math. Soc., Zürich.
  • Johnstone, I. M. (2009). Approximate null distribution of the largest root in multivariate analysis. Ann. Appl. Statist. To appear.
  • Khatri, C. (1972). On the exact finite series distribution of the smallest or the largest root of matrices in three situations. J. Multivariate Anal. 2 201–207.
  • Koev, P. (n.d.). Computing multivariate statistics. Manuscript in preparation.
  • Koev, P. and Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comput. 75 833–846.
  • Krbalek, M. and Seba, P. (2000). The statistical properties of the city transport in Cuernavaca (Mexico) and random matrix ensembles. J. Phys. A Math. Gen. 33 L229–L234. Available at
  • Kuijlaars, A. B. J., McLaughlin, K. T.-R., Van Assche, W. and Vanlessen, M. (2004). The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [−1, 1]. Adv. in Math. 188 337–398.
  • Ma, Z. (n.d.). Accuracy of the Tracy–Widom limit for the largest eigenvalue of white Wishart matrices. Draft manuscript, Dept. Statistics, Stanford Univ. Available at 0810.1329.
  • Mahoux, G. and Mehta, M. L. (1991). A method of integration over matrix variables. IV. J. Phys. I (France) 1 1093–1108.
  • Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
  • Morrison, D. F. (2005). Multivariate Statistical Methods, 4th ed. Thomson, Belmont, CA.
  • Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • Nagao, T. and Forrester, P. J. (1995). Asymptotic correlations at the spectrum edge of random matrices. Nuclear Phys. B 435 401–420.
  • Nagao, T. and Wadati, M. (1993). Eigenvalue distribution of random matrices at the spectrum edge. J. Phys. Soc. Japan 62 3845–3856.
  • Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, London.
  • Riesz, F. and Sz.-Nagy, B. (1955). Functional Analysis. Ungar, New York.
  • Seiler, E. and Simon, B. (1975). An inequality for determinants. Proc. Natl. Acad. Sci. 72 3277–3288.
  • Simon, B. (1977). Notes on infinite determinants of Hilbert space operators. Adv. in Math. 24 244–273.
  • Szegö, G. (1967). Orthogonal Polynomials, Colloquium Publications 23 3rd ed. Amer. Math. Soc., Providence, RI.
  • Timm, N. H. (1975). Multivariate Analysis, with Applications in Education and Psychology. Brooks/Cole Publishing Cop. [Wadsworth], Monterey, CA.
  • Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754.
  • Tracy, C. A. and Widom, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Statist. Phys. 92 809–835.
  • Tracy, C. A. and Widom, H. (2005). Matrix kernels for the Gaussian orthogonal and symplectic ensembles. Ann. Inst. Fourier (Grenoble) 55 2197–2207.
  • Tulino, A. and Verdu, S. (2004). Random Matrix Theory and Wireless Communications. Now Publishers, Hanover, MA.
  • Wachter, K. W. (1980). The limiting empirical measure of multiple discriminant ratios. Ann. Statist. 8 937–957.
  • Widom, H. (1999). On the relation between orthogonal, symplectic and unitary ensembles. J. Statist. Phys. 94 347–363.
  • Wong, R. and Zhao, Y.-Q. (2004). Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 2569–2586.