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June 2014 Limiting spectral distribution of a symmetrized auto-cross covariance matrix
Baisuo Jin, Chen Wang, Z. D. Bai, K. Krishnan Nair, Matthew Harding
Ann. Appl. Probab. 24(3): 1199-1225 (June 2014). 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 $\mathbf{M}_{\tau}=\frac{1}{2T}\sum_{j=1}^{T}(\mathbf{e} _{j}\mathbf{e} _{j+\tau}^{*}+\mathbf{e} _{j+\tau}\mathbf{e} _{j}^{*})$, where $\mathbf{e} _{j}$ is an $N$ dimensional vectors of independent standard complex components with properties stated in Theorem 1.1, and $\tau$ is the lag. $\mathbf{M}_{0}$ 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 $\mathbf{M}_{\tau}$ where $\tau\ge1$. It should be noted that the LSD of the $\mathbf{M}_{\tau}$ 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.


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Baisuo Jin. Chen Wang. Z. D. Bai. K. Krishnan Nair. Matthew Harding. "Limiting spectral distribution of a symmetrized auto-cross covariance matrix." Ann. Appl. Probab. 24 (3) 1199 - 1225, June 2014.


Published: June 2014
First available in Project Euclid: 23 April 2014

zbMATH: 1296.60006
MathSciNet: MR3199984
Digital Object Identifier: 10.1214/13-AAP945

Primary: 15A52, 60F15, 62H25
Secondary: 60F05, 60F17

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.24 • No. 3 • June 2014
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