Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations

作者：Jiang Tiefeng; Li Danning^*

来源：Journal of Theoretical Probability, 2015, 28(3): 804-847.

DOI：10.1007/s10959-013-0519-7

• Laguerre ensemble
• Wigner ensemble
• Variation norm
• Large deviation
• Largest eigenvalue
• Smallest eigenvalue
• Empirical distribution of eigenvalues
• Tracy-Widom distribution
• Condition number

Let be random eigenvalues coming from the beta-Laguerre ensemble with parameter , which is a generalization of the real, complex and quaternion Wishart matrices of parameter In the case that the sample size is much smaller than the dimension of the population distribution , 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 is much smaller than 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 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 is much smaller than 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 , where the last large deviation is derived by using a non-standard method.

• 出版日期2015-9