Let (Formula presented.) be random eigenvalues coming from the beta-Laguerre ensemble with parameter p, which is a generalization of the real, complex and quaternion Wishart matrices of parameter (n,p). In the case that the sample size n is much smaller than the dimension of the population distribution p, a common situation in modern data, we approximate the beta-Laguerre ensemble by a beta-Hermite ensemble, which is a generalization of the real, complex and quaternion Wigner matrices. As corollaries, when n is much smaller than p, we show that the largest and smallest eigenvalues of the complex Wishart matrix are asymptotically independent; we obtain the limiting distribution of the condition numbers as a sum of two i.i.d. random variables with a Tracy–Widom distribution, which is much different from the exact square case that (Formula presented.) by Edelman (SIAM J Matrix Anal Appl 9:543–560, 1988); we propose a test procedure for a spherical hypothesis test. By the same approximation tool, we obtain the asymptotic distribution of the smallest eigenvalue of the beta-Laguerre ensemble. In the second part of the paper, under the assumption that $$n$$n is much smaller than p in a certain scale, we prove the large deviation principles for three basic statistics: the largest eigenvalue, the smallest eigenvalue and the empirical distribution of (Formula presented.), where the last large deviation is derived by using a non-standard method.
Bibliographical noteFunding Information:
Dr. Alice Guionnet told the open problem to the first author on the large deviations for the eigenvalues of the Wishart matrices when the sample size is much smaller than the dimension of data at the “France-China Summer School and Conference: Random Matrix Theory and High-dimensional Statistics, Changchun, China, 2011.” The problem is studied in Theorem 4. We thank her very much. We also thank the associate editor and referees for their precious comments. The research of Tiefeng Jiang was supported in part by NSF FRG Grant DMS-0449365 and NSF Grant DMS-1208982
© 2013, Springer Science+Business Media New York.
- Condition number
- Empirical distribution of eigenvalues
- Laguerre ensemble
- Large deviation
- Largest eigenvalue
- Smallest eigenvalue
- Tracy–Widom distribution
- Variation norm
- Wigner ensemble