sample covariance matrix; and circular law for complex random matrix ([@girko]). Another aspect is the bound on extreme eigenvalues. The literature dates back to [@geman], who proved the almost sure convergence of the largest eigenvalue of a sample covariance matrix under however some moment restrictions, which is later improved by [@yin]. For Wigner matrix, [@baiyin] found the sufficient and necessary condition for the almost sure convergence of its largest eigenvalue. [@vu] presented an upper bound for the spectral norm of symmetric random matrices with independent entries and [@peche] derived the lower bound. [@Vershynin] studied the sharp upper bound of the spectral norm of products of random and deterministic matrices, which behave similarly to random matrices with independent entries, etc.

Notice that lag-$0$ auto-covariance matrix of $\varepsilon_t$ reduces to the standard sample covariance matrix $\frac 1T \sum_{t=1}^{T}\varepsilon_t\varepsilon^T_{t}$ and its property in large-dimension has been well developed in the literature. In contrast, very few is known for the lag-$s$ auto-covariance matrix $X_T$. Recent related work include [@liu], [@li] and [@jin] for the LSD of the symmetrized auto-covariance matrix and [@wang] for its exact separation, which also ensures the convergence of its largest eigenvalue.

Since $X_T$ is not symmetric, its singular values are 