Title: LSD of the Commutator of two data Matrices Show Chinese title

Title: 两个数据矩阵的交换子的LSD

Abstract: We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma_k^{1/2}Z_k$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and $\Sigma_k$ are $p\times p$ positive semi-definite matrices that commute and are independent of the $Z_k$'s for $k=1,2$. We assume that $Z_k$'s have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix $S_n^{-}$ will be referred to as a random commutator matrix associated with the samples $X_1$ and $X_2$. We show that, when the dimension $p$ and sample size $n$ increase simultaneously, so that $p/n \to c \in (0,\infty)$, there exists a limiting spectral distribution (LSD) for $S_n^{-}$, supported on the imaginary axis, under the assumptions that the joint spectral distribution of $\Sigma_1, \Sigma_2$ converges weakly and the entries of $Z_k$'s have moments of sufficiently high order. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a system of Mar\v{c}enko-Pastur-type functional equations. Moreover, we show that the companion matrix $S_n^{+} = n^{-1}(X_1X_2^* + X_2X_1^*)$, under identical assumptions, has an LSD supported on the real line, which can be similarly characterized. Show Chinese abstract

Abstract: 我们研究了一类形式为 $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ 的随机矩阵的谱性质，其中 $X_k = \Sigma_k^{1/2}Z_k$, $Z_k$ 是独立的 $p\times n$ 复值随机矩阵，而 $\Sigma_k$ 是 $p\times p$ 阶正半定矩阵且相互交换，并且对于 $k=1,2$，它们与 $Z_k$ 独立。 我们假设 $Z_k$的元素相互独立，且均值为零，方差为一。 斜对称/斜埃尔米特矩阵 $S_n^{-}$将被称为与样本 $X_1$和 $X_2$相关的随机换位子矩阵。 我们证明了当维度$p$和样本容量$n$同时趋于无穷大，并且满足$p/n \to c \in (0,\infty)$的条件下，假设$\Sigma_1, \Sigma_2$的联合谱分布弱收敛，且$Z_k$的元素具有足够高阶的矩，则矩阵$S_n^{-}$存在一个支持在虚轴上的非随机极限谱分布（LSD）。此 LSD 可通过其 Stieltjes 变换来描述，该变换满足一个类似于 Marčenko-Pastur 方程组的功能方程系统。 此外，我们在相同的假设下证明了伴随矩阵$S_n^{+} = n^{-1}(X_1X_2^* + X_2X_1^*)$存在一个支持在实轴上的 LSD，也可以用类似的方式刻画。