Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix

The auto-cross covariance matrix is defined as \[\mathbf{M}_n=\frac{1} {2T}\sum_{j=1}^T\bigl(\mathbf{e}_j\mathbf{e}_{j+\tau}^*+\mathbf{e}_{j+ \tau}\mathbf{e}_j^*\bigr),\] 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 $\tau\ge 1$. 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.