arXiv 2103.119462021-03-22math.PR

样本互协方差阵的联合收敛性

Joint convergence of sample cross-covariance matrices

Monika Bhattacharjee, Arup Bose, Apratim Dey

摘要：

假设$X$和$Y$是$p\times n$矩阵，每个矩阵具有均值$0$和方差$1$，并且其中任何阶的所有矩都一致有界为$p,n \to \infty$。此外，条目$(X_{ij}, Y_{ij})$在$i,j$上是独立的，具有共同的相关性$ρ$。设$C=n^{-1}XY^*$为样本互协方差矩阵。我们证明了，如果$n, p\to \infty, p/n\to y\neq 0$，那么$C$收敛在代数意义下和极限力矩只依赖于$ρ$。具有相同$p$但不同$n$（例如$\{n_l\}$）、不同相关性$\{ρ_l\}$和不同非零$y$（例如$\{y_l\}$）的这种矩阵的独立副本也联合收敛并且是渐近自由的。当$y=0$时，矩阵$\sqrt{np^{-1}}(C-ρI_p)$收敛于具有参数$ρ^2$的椭圆变量。特别地，当$ρ=0$时该椭圆变量是圆形的，当$ρ=1$时该椭圆变量是半圆形的。如果我们取独立的$C_l$，则矩阵$\{\sqrt{n_lp^{-1}}(C_l-ρ_l I_p)\}$联合收敛并且也是渐近自由的。因此，任何对称矩阵多项式的极限谱分布都是存在的，并且具有紧支集。

英文摘要：

Suppose $X$ and $Y$ are $p\times n$ matrices each with mean $0$, variance $1$ and where all moments of any order are uniformly bounded as $p,n \to \infty$. Moreover, the entries $(X_{ij}, Y_{ij})$ are independent across $i,j$ with a common correlation $ρ$. Let $C=n^{-1}XY^*$ be the sample cross-covariance matrix. We show that if $n, p\to \infty, p/n\to y\neq 0$, then $C$ converges in the algebraic sense and the limit moments depend only on $ρ$. Independent copies of such matrices with same $p$ but different $n$, say $\{n_l\}$, different correlations $\{ρ_l\}$, and different non-zero $y$'s, say $\{y_l\}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $\sqrt{np^{-1}}(C-ρI_p)$ converges to an elliptic variable with parameter $ρ^2$. In particular, this elliptic variable is circular when $ρ=0$ and is semi-circular when $ρ=1$. If we take independent $C_l$, then the matrices $\{\sqrt{n_lp^{-1}}(C_l-ρ_l I_p)\}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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