The Beta-MANOVA Ensemble with General Covariance

We find the joint generalized singular value distribution and largest generalized singular value distributions of the β -MANOVA ensemble with positive diagonal covariance, which is general. This has been done for the continuous β > 0 case for identity covariance (in eigenvalue form), a...

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Bibliographic Details
Main Authors: Dubbs, Alexander Joseph, Edelman, Alan
Other Authors: Massachusetts Institute of Technology. Department of Mathematics
Format: Article
Published: World Scientific Pub Co Pte Lt 2018
Subjects:
Online Access:http://hdl.handle.net/1721.1/116005
https://orcid.org/0000-0001-7676-3133
Description
Summary:We find the joint generalized singular value distribution and largest generalized singular value distributions of the β -MANOVA ensemble with positive diagonal covariance, which is general. This has been done for the continuous β > 0 case for identity covariance (in eigenvalue form), and by setting the covariance to I in our model we get another version. For the diagonal covariance case, it has only been done for β = 1, 2, 4 cases (real, complex, and quaternion matrix entries). This is in a way the first second-order β-ensemble, since the sampler for the generalized singular values of the β-MANOVA with diagonal covariance calls the sampler for the eigenvalues of the β-Wishart with diagonal covariance of Forrester and Dubbs-Edelman-Koev-Venkataramana. We use a conjecture of MacDonald proven by Baker and Forrester concerning an integral of a hypergeometric function and a theorem of Kaneko concerning an integral of Jack polynomials to derive our generalized singular value distributions. In addition we use many identities from Forrester’s Log-Gases and Random Matrices. We supply numerical evidence that our theorems are correct.