a sequence of $m$-dimensional “latent factor” ($m\ll p$) uncorrelated with the error process $\{\varepsilon_t\}$ and $\mu \in \mathbb{R}^p$ is the general mean. Therefore, the lag-$s$ auto-covariance matrix of the time series $x_t$ can be considered as a finite rank (rank $m$) perturbation of the lag-$s$ auto-covariance matrix of $\varepsilon_t$. Therefore, the first step is to study the base component, which is the lag-$s$ auto-covariance matrix of the error term. Besides, we are considering the random matrix framework, where the dimension $p$ and the sample size $T$ both tend to infinity with their ratio converging to a constant: $\lim p/T \rightarrow y>0$.

One of the main problems in random matrix theory is to investigate the convergence of the sequence of [*empirical spectral distribution*]{} $\{F^{A_n}\}$ for a given sequence of symmetric or Hermitian random matrices $\{A_n\}$, where $$\begin{aligned}
F^{A_n}(x):=\frac 1p \sum_{j=1}^p\delta_{l_j}~,\end{aligned}$$ $l_j$ are the eigenvalues of $A_n$. The limit distribution $F$, which is usually nonrandom, is called the [*limiting spectral distribution*]{} (LSD) of the sequence $\{A_n\}$. The study of spectral analysis of large dimensional random matrices dates back to the Wigner’s famous semicircular law ([@wigner]) for Wigner matrix, which is further extended in various aspects: Marčenko-Pastur (M-P) law ([@MP]) for large dimensional 