Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations

By Tiefeng Jiang, Danning Li. 2013

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Let $$\lambda _1, \ldots , \lambda _n$$λ1,…,λn 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 $$\lambda _1, \ldots , \lambda _n$$λ1,…,λn, where the last large deviation is derived by using a non-standard method.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1c5a82550cf056542c4f6c15af366ae4314b5d64