Limiting spectral distribution of a symmetrized auto-cross covariance matrix

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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 Marcenko–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.