For a broad class of bi-unitarily invariant random matrix ensembles, the large-n limit of the joint density of one eigenradius and k singular values at the hard edge is expressed through the limiting kernel of the singular value process.
On eigenvalue distributions of large auto-covariance matrices
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abstract
In this article, we establish a limiting distribution for eigenvalues of a class of auto-covariance matrices. The same distribution has been found in the literature for a regularized version of these auto-covariance matrices. The original non-regularized auto-covariance matrices are non invertible which introduce supplementary diffculties for the study of their eigenvalues through Girko's Hermitization scheme. The key result in this paper is a new polynomial lower bound for the least singular value of the resolvent matrices associated to a rank-defective quadratic function of a random matrix with independent and identically distributed entries. Another improvement in the paper is that the lag of the auto-covariance matrices can grow to infinity with the matrix dimension.
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Hard edge asymptotics of correlation functions between singular values and eigenvalues
For a broad class of bi-unitarily invariant random matrix ensembles, the large-n limit of the joint density of one eigenradius and k singular values at the hard edge is expressed through the limiting kernel of the singular value process.