Limiting distribution of the sample canonical correlation coefficients of high-dimensional random vectors

Abstract

In this paper, we prove a CLT for the sample canonical correlation coefficients between two high-dimensional random vectors with finite rank correlations. More precisely, consider two random vectors $\tilde{\mathbf{x}}=\mathbf{x}+A\mathbf{z}$ and $\tilde{\mathbf{y}}=\mathbf{y}+B\mathbf{z}$, where $\mathbf{x}\in {\mathbb{R}}^p$, $\mathbf{y}\in {\mathbb{R}}^q$ and $\mathbf{z}\in {\mathbb{R}}^r$ are independent random vectors with i.i.d. entries of mean zero and variance one, and $A\in {\mathbb{R}}^{p\times r}$ and $B\in {\mathbb{R}}^{q\times r}$ are two arbitrary deterministic matrices. Given n samples of $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{y}}$, we stack them into two matrices $\mathcal{X}=X+AZ$ and $\mathcal{Y}=Y+BZ$, where $X\in {\mathbb{R}}^{p\times n}$, $Y\in {\mathbb{R}}^{q\times n}$ and $Z\in {\mathbb{R}}^{r\times n}$ are random matrices with i.i.d. entries of mean zero and variance one. Let ${\tilde{\mathit{\lambda }}}_1\ge {\tilde{\mathit{\lambda }}}_2\ge \cdots \ge {\tilde{\mathit{\lambda }}}_r$ be the largest r eigenvalues of the sample canonical correlation (SCC) matrix ${\mathcal{C}}_{\mathcal{X}\mathcal{Y}}={(\mathcal{X}{\mathcal{X}}^{\top })}^{-1∕2}\mathcal{X}{\mathcal{Y}}^{\top }{(\mathcal{Y}{\mathcal{Y}}^{\top })}^{-1}\mathcal{Y}{\mathcal{X}}^{\top }{(\mathcal{X}{\mathcal{X}}^{\top })}^{-1∕2}$, and let $t_1\ge t_2\ge \cdots \ge t_r$ be the squares of the population canonical correlation coefficients between $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{y}}$. Under certain moment assumptions, we show that there exists a threshold $t_c\in (0,1)$ such that if $t_i>t_c$, then $\sqrt{n}({\tilde{\mathit{\lambda }}}_i-{\mathit{\theta }}_i)$ converges weakly to a centered normal distribution, where ${\mathit{\theta }}_i$ is a fixed outlier location determined by $t_i$. Our proof uses a self-adjoint linearization of the SCC matrix and a sharp local law on the inverse of the linearized matrix.