On eigenvalue distributions of large auto-covariance matrices

2020/11/18 by Jianfeng Yao, Yao, Jianfeng, Wangjun Yuan +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2011.09165

openalex publication_date 2020/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Smallest singular value and limit eigenvalue distribution of a class of\n non-Hermitian random matrices with statistical application
• Limiting Empirical Spectral Distribution for Products of Rectangular Matrices

Related

• Limiting Laws for Divergent Spiked Eigenvalues and Largest Non-spiked Eigenvalue of Sample Covariance Matrices
• Random covariance matrices: Universality of local statistics of eigenvalues up to the edge
• On singular value distribution of large dimensional auto-covariance matrices
• Phase transition of the largest eigenvalue for non-null complex sample covariance matrices
• Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population