  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.
