Strong Limit of the Extreme Eigenvalues of a Symmetrized Auto-Cross Covariance Matrix

The auto-cross covariance matrix is defined as $${\bf M}_n=\frac{1}{2T}\sum_{j=1}^{T} ({\bf e}_{j}{\bf e}_{j+\tau}^{*}+{\bf e}_{j+\tau}{\bf e}_{j}^{*}),$$ where ${\bf e}_{j}$'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. (2013) has proved that the LSD of ${\bf M}_n$ exists uniquely and non-randomly, 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. (2013), this paper proved that under the condition of uniformly bounded $4$th moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of ${\bf M}_{n}$ for all large $n$. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of ${\bf M}_n$ are also obtained.