LIMITING SPECTRAL DISTRIBUTION OF A SYMMETRIZED AUTO-CROSS COVARIANCE MATRIX

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 = 1/2T Sigma(T)(j=1) (e(j) e(j+tau)*+ e(j+tau)e(j)*) where e(j) is an N dimensional vectors of independent standard complex components with properties stated in Theorem 1.1, and tau is the lag. M-0 is well studied in the literature whose LSD is the Mareenko-Pastur (MP) Law. The contribution of this paper is in determining the LSD of M-tau where tau >= 1. It should be noted that the LSD of the M-tau does not depend on tau. 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.