The auto-cross covariance matrix is defined as [mathbf{M}_n=frac{1} {2T}sum_{j=1}^Tbigl(mathbf{e}_jmathbf{e}_{j+tau}^*+mathbf{e}_{j+ tau}mathbf{e}_j^*bigr),] where $mathbf{e}_j$s are $n$-dimensional vectors of independent standard complex components
with a common mean 0, variance $sigma^2$, and uniformly bounded $2+eta$th moments and $tau$ is the lag. Jin et al. [Ann. Appl. Probab. 24 (2014) 1199-1225] has proved that the LSD of $mathbf{M}_n$ exists uniquely and nonrandomly, and independent of $tau$ for all $tauge 1$. And in addition they gave an analytic expression of the LSD. As a continuation of Jin et al. [Ann. Appl. Probab. 24 (2014) 1199-1225], this paper proved that under the condition of uniformly bounded fourth moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of $mathbf{M}_n$ for all large $n$. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of $mathbf{M}_n$ are also obtained.
