Limiting spectral distribution of a symmetrized auto-cross covariance matrix

By Baisuo Jin∗,§, Chen Wang¶ Z. D. Bai†,¶,∥ K. Krishnan Nair∗∗ and Matthew Harding‡,∗∗ University of Science and Technology of China§, National University of Singapore,¶ Northeast Normal University, ∥ and Stanford University∗∗ 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 ∑T j=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 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 raised from the investigation and plays an key role in the model selection of any large dimensional model with a lagged time series structure which are central to large dimensional factor models and singular spectrum analysis. ∗Research of this author was supported by NSF China Young Scientist Grant 11101397 †Research of this author was supported by NSF China 11171057 as well as by Program for Changjiang Scholars and Innovative Research Team in University ‡The research of this author was supported by Stanford Presidential Fund for Innovation in International Studies AMS 2000 subject classifications: Primary 60F15, 15A52, 62H25; secondary 60F05, 60F17

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