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

Abstract

The auto-cross covariance matrix is defined as

\[\mathbf{M}_{n}=\frac{1}{2T}\sum_{j=1}^{T}(\mathbf{e}_{j}\mathbf{e}_{j+\tau}^{*}+\mathbf{e}_{j+\tau}\mathbf{e}_{j}^{*}),\] 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 $\tau\ge1$. 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.