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

Abstract

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

Published at http://dx.doi.org/10.1214/14-AAP1092 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Central limit theorems for eigenvalues in a spiked population model
• Identifying the number of factors from singular values of a large sample auto-covariance matrix

Cited by in corpus (1)

• Moment approach for singular values distribution of a large auto-covariance matrix