[Paper Review] Extreme Eigenvalues of Large Dimensional Quaternion Sample Covariance Matrix

Huiqin Li, Zhidong Bai|arXiv (Cornell University)|Dec 17, 2013

Random Matrices and Applications4 references3 citations

TL;DR

This paper establishes the almost sure limits of the extreme eigenvalues of large-dimensional quaternion sample covariance matrices. Under i.i.d. quaternion entries with mean zero, unit variance, and finite fourth moments, the largest eigenvalue almost surely converges to $(1 + ilde{y})^2$ and the smallest to $(1 - ilde{y})^2$, where $ ilde{y} = ilde{p}/n$, extending classical Marčenko-Pastur laws to the quaternion setting with sharp moment conditions.

ABSTRACT

In this paper, we shall investigate the almost sure limits of the largest and smallest eigenvalues of a quaternion sample covariance matrix. Suppose that $\mathbf X_n$ is a $p imes n$ matrix whose elements are independent quaternion variables with mean zero, variance 1 and uniformly bounded fourth moments. Denote $\mathbf S_n=\frac{1}{n}\mathbf X_n\mathbf X_n^*$. In this paper, we shall show that $s_{\max}\left(\mathbf S_n ight)=s_{p}\left(\mathbf S_n ight) o\left(1+\sqrt y ight)^2, a.s.$ and $s_{\min}\left(\mathbf S_n ight) o\left(1-\sqrt y ight)^2,a.s.$ as $n o\infty$, where $y=\lim p/n$, $s_1\left(\mathbf S_n ight)\le\cdots\le s_{p}\left(\mathbf S_n ight)$ are the eigenvalues of $\mathbf{S}_n$, $s_{\min}\left(\mathbf S_n ight)=s_{p-n+1}\left(\mathbf S_n ight)$ when $p>n$ and $s_{\min}\left(\mathbf S_n ight)=s_1\left(\mathbf S_n ight)$ when $p\le n$. We also prove that the set of conditions are necessary for $s_{\max}\left(\mathbf S_n ight) o\left(1+\sqrt y ight)^2, a.s.$ when the entries of $\mathbf {X}_n$ are i. i. d.

Motivation & Objective

• To determine the almost sure limits of the largest and smallest eigenvalues of large-dimensional quaternion sample covariance matrices.
• To establish the necessary and sufficient moment conditions for the almost sure convergence of the extreme eigenvalues.
• To extend classical random matrix theory results—previously known for real and complex cases—to the quaternion algebra setting.
• To prove that finite fourth moments are necessary and sufficient for the almost sure convergence of the largest eigenvalue in the quaternion case.
• To analyze the limiting spectral behavior under non-i.i.d. but mean-zero, finite-variance, and uniformly bounded fourth-moment conditions on quaternion entries.

Proposed method

• Define the quaternion sample covariance matrix $\mathbf{S}_n = \frac{1}{n}\mathbf{X}_n\mathbf{X}_n^*$, where $\mathbf{X}_n$ has i.i.d. quaternion entries with mean zero, variance 1, and uniformly bounded fourth moments.
• Use the spectral norm and singular value decomposition to analyze the deviation of $\mathbf{S}_n$ from its expected value $\sigma^2(1 + y)\mathbf{I}_p^Q$.
• Apply a matrix perturbation argument to bound $\|\mathbf{S}_n - \sigma^2(1 + y)\mathbf{I}_p^Q\|_2$ using the maximum of normalized quadratic forms of quaternion entries.
• Leverage Lemma 7.1 (a strong law for double arrays of i.i.d. random variables) to control the convergence of normalized sums of squared quaternion norms.
• Use the min-max principle and eigenvalue interlacing to derive almost sure bounds on extreme eigenvalues via comparison with the expected matrix.
• Prove necessity of moment conditions by contradiction: if $\mathbb{E}\|z_{11}\|^4 = \infty$, the largest eigenvalue diverges almost surely.

Experimental results

Research questions

• RQ1 What are the almost sure limits of the largest and smallest eigenvalues of a large-dimensional quaternion sample covariance matrix?
• RQ2 Are finite fourth moments necessary for the almost sure convergence of the extreme eigenvalues in the quaternion case?
• RQ3 Does the Marčenko-Pastur law for extreme eigenvalues extend to the quaternion setting under the same moment conditions as in the real and complex cases?
• RQ4 How do the extreme eigenvalues behave when the number of samples $n$ and dimension $p$ grow with $y = \lim p/n$?
• RQ5 What conditions on the quaternion entries are necessary and sufficient for the almost sure convergence of the extreme eigenvalues to the Marčenko-Pastur bounds?

Key findings

• The largest eigenvalue $s_p(\mathbf{S}_n)$ almost surely converges to $(1 + \sqrt{y})^2$ as $n \to \infty$, where $y = \lim p/n$.
• The smallest eigenvalue $s_{\min}(\mathbf{S}_n)$ almost surely converges to $(1 - \sqrt{y})^2$ under the same asymptotic regime.
• Finite fourth moments $\mathbb{E}\|z_{11}\|^4 < \infty$ are necessary and sufficient for the almost sure convergence of the largest eigenvalue.
• Non-zero mean of the quaternion entries ($\mathbb{E}[z_{11}] \neq 0$) leads to almost sure divergence of the spectral norm, invalidating convergence.
• The convergence results hold under the assumption that the quaternion entries are i.i.d. with mean zero, unit variance, and uniformly bounded fourth moments.
• The proof establishes that the limiting spectral distribution of the quaternion sample covariance matrix follows the Marčenko-Pastur law, with extreme eigenvalues confined to the almost sure limits $(1 \pm \sqrt{y})^2$.