Convergence of empirical spectral distributions of large dimensional quaternion sample covariance matrices

Abstract

In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices. Motivated by Bai and Silverstein (Spectral analysis of large dimensional random matrices, Springer, New York, 2010) and Marčenko and Pastur (Matematicheskii Sb, 114:507–536, 1967), we can extend the results of the real or complex sample covariance matrix to the quaternion case. Suppose $\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}$ is a quaternion random matrix. For each $n$, the entries $\{x_{ij}^{(n)}\}$ are independent random quaternion variables with a common mean $\mu$ and variance $\sigma ^2>0$. It is shown that the empirical spectral distribution of the quaternion sample covariance matrix $\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*$ converges to the Marčenko–Pastur law as $p\rightarrow \infty$, $n\rightarrow \infty$ and $p/n\rightarrow y\in (0,+\infty )$.

Appendix

In this section, some results are listed which are used in the proof of the main theorem.

Lemma 15

Suppose for any $\eta >0$ $\sum _{n=1}^{\infty }f\left( \eta ,n\right) <\infty$, then we can select a slowly decreasing sequence of constants $\eta _n\rightarrow 0$ such that

$$\begin{aligned} \sum _{n=1}^{\infty }f\left( \eta _n,n\right) <\infty \end{aligned}$$

where $f$ is a nonnegative function.

Similarly, if $f(\eta ,n)\rightarrow 0$ for any fixed $\eta >0$, then there exists a decreasing sequence $\eta _n\rightarrow 0$ such that $f(\eta _n,n)\rightarrow 0$.

Proof

Letting $\eta =\frac{1}{m}$, one has $\sum _{n=1}^{\infty }f\left( \frac{1}{m},n\right) <\infty .$ Moreover, there exists a increasing sequence $N_m$ such that $\sum _{n=N_m}^{\infty }f\left( \frac{1}{m},n\right) \le \frac{1}{2^m}$. Define a sequence $\eta _n=\frac{1}{m}$ when $N_m\le n<N_{m+1}$. We get

$$\begin{aligned} \sum _{n=1}^{\infty }f\left( \eta _n,n\right)&=\sum _{m=1}^{\infty }\sum _{n=N_m}^{N_{m+1}-1}f\left( \frac{1}{m},n\right) \le \sum _{m=1}^{\infty }\sum _{n=N_m}^{\infty }f\left( \frac{1}{m},n\right) \\&\le \sum _{n=1}^{N_1-1} f(1,n) +\sum _{m=1}^{\infty }\frac{1}{2^m}< \infty . \end{aligned}$$

This completes the proof of this lemma.$\square$

Lemma 16

(Corollary A.42 of Bai and Silverstein 2010) Let ${\mathbf A}$ and ${\mathbf B}$ be two $p \times n$ matrices and denote the ESD of ${\mathbf S} = {\mathbf A}{{\mathbf A}^ * }$ and $\widetilde{\mathbf S} = {\mathbf B}{{\mathbf B}^ * }$ by ${F^{\mathbf S}}$ and ${F^{\widetilde{\mathbf S}}}$, respectively. Then,

$$\begin{aligned} {L^4}({F^{\mathbf S}},{F^{\widetilde{\mathbf S}}}) \le \frac{2}{{{p^2}}}(\mathrm{tr}({\mathbf A}{{\mathbf A}^ * } + {\mathbf B}{{\mathbf B}^ * }))(\mathrm{tr}[({\mathbf A} - {\mathbf B}){({\mathbf A} - {\mathbf B})^ * }]), \end{aligned}$$

where $L(\cdot ,\cdot )$ denotes the Lévy distance, that is,

$$\begin{aligned} L({F^{\mathbf S}},{F^{\widetilde{\mathbf S}}})=\inf \{\varepsilon :{F^{\mathbf S}\left( x-\varepsilon ,y-\varepsilon \right) -\varepsilon }\le {F^{\widetilde{\mathbf S}}}\le {F^{\mathbf S}\left( x+\varepsilon ,y+\varepsilon \right) +\varepsilon }\}. \end{aligned}$$

Lemma 17

(Theorem A.44 of Bai and Silverstein 2010) Let ${\mathbf A}$ and ${\mathbf B}$ be $p \times n$ complex matrices. Then,

$$\begin{aligned} \Vert {{F^{{\mathbf A}{{\mathbf A}^ * }}} - {F^{{\mathbf B}{{\mathbf B}^ * }}}} \Vert _{KS} \le \frac{1}{p}\mathrm{rank}\left( {\mathbf A} - {\mathbf B}\right) . \end{aligned}$$

Lemma 18

(Bernstein’s inequality) If $\varvec{\tau }_1,\ldots ,\varvec{\tau }_n$ are independent random variables with means zero and uniformly bounded by $b$, then, for any $\varepsilon > 0$,

$$\begin{aligned} \mathrm{P}\left( \left| \sum _{j=1}^{n}\varvec{\tau }_j\right| \ge \varepsilon \right) \le 2{\exp }\left( -\varepsilon ^2/\left[ 2(B_n^2+b\varepsilon )\right] \right) \end{aligned}$$

where $B_n^2={ E}(\varvec{\tau }_1+\cdots +\varvec{\tau }_n)^2.$

Lemma 19

(see $\left( A.1.12\right)$ of Bai and Silverstein 2010) Let $z = u + iv, v > 0,$ and ${\mathbf A}$ be an $n \times n$ Hermitian matrix. Denote by ${{\mathbf A}_k}$ the kth major sub-matrix of ${\mathbf A}$ of order $(n-1)$, to be the matrix resulting from deleting the $k$th row and column from ${\mathbf A}$. Then,

$$\begin{aligned} | {\mathrm{tr}{{({\mathbf A} - z{\mathbf I}_n)}^{ - 1}} -{ \mathrm tr}{{({{\mathbf A}_k} - z{{\mathbf I}_{n - 1}})}^{ - 1}}} | \le \frac{1}{\upsilon }. \end{aligned}$$

Lemma 20

Suppose that the matrix $\varvec{\Sigma }$ has the partition as given by $\begin{pmatrix}\varvec{\Sigma }_{11}&{}\quad \varvec{\Sigma }_{12}\\ \varvec{\Sigma }_{21}&{}\quad \varvec{\Sigma }_{22}\end{pmatrix}$. If $\varvec{\Sigma }$ and $\varvec{\Sigma }_{11}$ are nonsingular, then the inverse of $\varvec{\Sigma }$ has the form

$$\begin{aligned} \varvec{\Sigma }^{-1}=\left( \begin{array}{l@{\quad }l} \varvec{\Sigma }_{11}^{-1}+\varvec{\Sigma }_{11}^{-1}\varvec{\Sigma }_{12}\varvec{\Sigma }_{22.1}^{-1}\varvec{\Sigma }_{21}\varvec{\Sigma }_{11}^{-1}&{}\quad -\varvec{\Sigma }_{11}^{-1}\varvec{\Sigma }_{12}\varvec{\Sigma }_{22.1}^{-1}\\ -\varvec{\Sigma }_{22.1}^{-1}\varvec{\Sigma }_{21}\varvec{\Sigma }_{11}^{-1}&{}\quad \varvec{\Sigma }_{22.1}^{-1} \end{array}\right) \end{aligned}$$

where $\varvec{\Sigma }_{22.1}=\varvec{\Sigma }_{22}-\varvec{\Sigma }_{21}\varvec{\Sigma }_{11}^{-1}\varvec{\Sigma }_{12}$.

Lemma 21

(Burkholder’s inequality) Let $\{ {{\mathbf X}_k}\}$ be a complex martingale difference sequence with respect to the increasing $\sigma$-field. Then, for $p > 1,$

$$\begin{aligned} { E}{\left| {\sum _k {{{\mathbf X}_k}} } \right| ^p} \le {K_p}{ E}{\left( {\sum _k{| {{{\mathbf X}_k}} |} ^2}\right) ^{p/2}}. \end{aligned}$$

Lemma 22

(Rosenthal’s inequality) Let ${\mathbf X}_i$ be independent with zero means, then we have, for some constant $C_k$,

$$\begin{aligned} { E}\left| \sum _i {\mathbf X}_i\right| ^{2k} \le C_k\left( \sum _i{ E}\left| {\mathbf X}_i\right| ^{2k}+\left( \sum _i { E}\left| {\mathbf X}_i\right| ^2\right) ^k\right) . \end{aligned}$$

Lemma 23

(Lemma 2.14 of Bai and Silverstein 2010) Let $f_1,f_2,\ldots$ be analytic in $D$, a connected open set of $\mathbb C$, satisfying $\left| f_n\left( z\right) \right| \le M$ for every $n$ and $z$ in $D$, and $f_n\left( z\right)$ converges as $n \rightarrow \infty$ for each $z$ in a subset of $D$ having a limit point in $D$. Then, there exists a function $f$ analytic in $D$ for which $f_n\left( z\right) \rightarrow f\left( z\right)$ and $f_n^{\prime } \rightarrow f^{\prime }\left( z\right)$ for all $z\in D$. Moreover, on any set bounded by a contour interior to $D$, the convergence is uniform and $\{f_n^{\prime }\left( z\right) \}$ is uniformly bounded.

Lemma 24

(Theorem B.9 of Bai and Silverstein 2010) Assume that $\left\{ G_n\right\}$ is a sequence of functions of bounded variation and $G_n\left( -\infty \right) =0$ for all $n$. Then,

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathbf m}_{G_n}\left( z\right) ={\mathbf m}\left( z\right) \quad \forall z\in D \end{aligned}$$

where $D\equiv \left\{ z\in \mathbb {C}:\mathfrak {I}z>0\right\}$ if and only if there is a function of bounded variation $G$ with $G\left( -\infty \right) =0$ and Stieltjes transform ${\mathbf m}\left( z\right)$ and such that $G_n\rightarrow G$ vaguely.