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

Consider two high-dimensional random vectors $\widetilde{\mathbf x}\in\mathbb R^p$and$\widetilde{\mathbf y}\in\mathbb R^q$ with finite rank correlations. More precisely, suppose that $\tilde{\mathbf{x}}=\mathbf{x}+A\mathbf{z}$ and $\tilde{\mathbf{y}}=\mathbf{y}+B\mathbf{z}$, for independent random vectors $\mathbf{x}\in {\mathbb{R}}^p$, $\mathbf{y}\in {\mathbb{R}}^q$ and $\mathbf z\in\mathbb R^r$withiidentriesofmean0andvariance1,andtwodeterministicmatrices$A\in\mathbb R^{p\times r}$and$B\in\mathbb R^{q\times r}$\mathrm{.}With$n$ iid observations of $(\tilde{\mathbf{x}},\tilde{\mathbf{y}})$, we study the sample canonical correlations between them. In this paper, we focus on the high-dimensional setting with a rank-$r$ correlation. Let $t_1\ge \cdots \ge t_r$ be the squares of the population canonical correlation coefficients (CCC) between $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{y}}$, and ${\tilde{\lambda }}_1\ge \cdots \ge {\tilde{\lambda }}_r$ be the squares of the largest $r$ sample CCC. Under certain moment assumptions on the entries of $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$, we show that there exists a threshold $t_c\in (0,1)$ such that if $t_i>t_c$, then $\sqrt{n}({\tilde{\lambda }}_i-{\theta }_i)$ converges in law to a centered normal distribution, where ${\theta }_i>{\lambda }_{+}$ is a fixed outlier location determined by $t_i$. Our results extend the ones in [4] for Gaussian vectors. Moreover, we find that the variance of the limiting distribution of $\sqrt{n}({\tilde{\lambda }}_i-{\theta }_i)$ also depends on the fourth cumulants of the entries of $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$, a phenomenon that cannot be observed in the Gaussian case.