Limiting spectral distribution of a symmetrized auto-cross covariance matrix

Abstract

This paper studies the limiting spectral distribution (LSD) of a symmetrized auto-cross covariance matrix. The auto-cross covariance matrix is defined as Mτ= 1/2T ∑Tj=1(eje*j+τ+ej+τe*j), 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 theMarčenko-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. ©Institute of Mathematical Statistics, 2014.