On eigenvalue distributions of large autocovariance matrices

Abstract

In this article, we establish a limiting distribution for eigenvalues of a class of autocovariance matrices. The same distribution has been found in the literature for a regularized version of these autocovariance matrices. The original nonregularized autocovariance matrices are noninvertible, thus introducing supplementary difficulties for the study of their eigenvalues through Girko’s Hermitization scheme. The key result in this paper is a new polynomial lower bound for a specific family of least singular values associated to a rank-defective quadratic function of a random matrix with independent and identically distributed entries. Another innovation from the paper is that the lag of the autocovariance matrices can grow to infinity with the matrix dimension.