Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations

Let $$\lambda _1, \ldots , \lambda _n$$ λ 1 , … , λ n be random eigenvalues coming from the beta-Laguerre ensemble with parameter $$p$$ p , which is a generalization of the real, complex and quaternion Wishart matrices of parameter $$(n,p).$$ ( n , p ) . In the case that the sample size $$n$$ n is much smaller than the dimension of the population distribution $$p$$ 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 $$n$$ n is much smaller than $$p,$$ 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 $$n=p$$ n = p 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 $$p$$ 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 $$\lambda _1, \ldots , \lambda _n$$ λ 1 , … , λ n , where the last large deviation is derived by using a non-standard method.