Open Access
December 2015 Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix
Chen Wang, Baisuo Jin, Z. D. Bai, K. Krishnan Nair, Matthew Harding
Ann. Appl. Probab. 25(6): 3624-3683 (December 2015). DOI: 10.1214/14-AAP1092


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.


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Chen Wang. Baisuo Jin. Z. D. Bai. K. Krishnan Nair. Matthew Harding. "Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix." Ann. Appl. Probab. 25 (6) 3624 - 3683, December 2015.


Received: 1 September 2014; Revised: 1 December 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1328.60088
MathSciNet: MR3404646
Digital Object Identifier: 10.1214/14-AAP1092

Primary: 15A52 , 60F15 , 62H25
Secondary: 60F05 , 60F17

Keywords: Auto-cross covariance , dynamic factor analysis , Limiting spectral distribution , Marčenko–Pastur law , order detection , Random matrix theory , Stieltjes transform , strong limit of extreme eigenvalues

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.25 • No. 6 • December 2015
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