LIMITING SPECTRAL DISTRIBUTION OF A SYMMETRIZED AUTO-CROSS COVARIANCE MATRIX

• Published in: The Annals of applied probability 2014-06, Vol.24 (3), p.1199-1225
• Main Authors: Jin, Baisuo, Wang, Chen, Bai, Z. D., Nair, K. Krishnan, Harding, Matthew
• Format: Article
• Language: English
• Subjects: 15A52; 60F05; 60F15; 60F17; 62H25; Auto-cross covariance; Covariance matrices; Eigenvalues; Equation roots; factor analysis; Fourier transformations; limiting spectral distribution; Marčenko–Pastur law; Mathematical models; Matrices; Matrix; Modeling; order detection; Probability distribution; random matrix theory; Random variables; Spectral theory; Spectroscopy; Spectrum analysis; Stieltjes transform; Theorems; Time series; Time series models; Vertices
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/13-AAP945

This paper studies the limiting spectral distribution (LSD) of a symmetrized auto-cross covariance matrix. The auto-cross covariance matrix is defined as ${M_\tau } = \frac{1}{{2T}}\sum\nolimits_{j = 1}^T {\left( {{e_j}e_{j + \tau }^* + {e_j}{ + _\tau }e_j^*} \right)} $ where ej is an N dimensional vectors of independent standard complex components with properties stated in Theorem 1.1, and τ is the lag. M0 is well studied in the literature whose LSD is the Marčenko-Pastur (MP) Law. The contribution of this paper is in determining the LSD of Mτ where τ > 1. It should be noted that the LSD of the Mτ does not depend on τ. This study arose from the investigation of and plays an key role in the model selection of any large dimensional model with a lagged time series structure, which is central to large dimensional factor models and singular spectrum analysis.