The Annals of Applied Probability

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

Chen Wang, Baisuo Jin, Z. D. Bai, K. Krishnan Nair, and Matthew Harding

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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.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3624-3683.

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1443703784

Digital Object Identifier
doi:10.1214/14-AAP1092

Mathematical Reviews number (MathSciNet)
MR3404646

Zentralblatt MATH identifier
1328.60088

Subjects
Primary: 60F15: Strong theorems 15A52 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles

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

Citation

Wang, Chen; Jin, Baisuo; Bai, Z. D.; Nair, K. Krishnan; Harding, Matthew. Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix. Ann. Appl. Probab. 25 (2015), no. 6, 3624--3683. doi:10.1214/14-AAP1092. https://projecteuclid.org/euclid.aoap/1443703784


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