  We 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. Suppose that $\widetilde{\mathbf x}=\mathbf x+A\mathbf z$ and
$\widetilde{\mathbf y}=\mathbf y+B\mathbf z$, for independent random vectors
$\mathbf x\in\mathbb R^p$, $\mathbf y\in\mathbb R^q$ and $\mathbf z\in\mathbb
R^r$ with iid entries of mean 0 and variance 1, and two deterministic matrices
$A\in\mathbb R^{p\times r}$ and $B\in\mathbb R^{q\times r}$ . With $n$ iid
observations of $(\widetilde{\mathbf x},\widetilde{\mathbf y})$, 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 $\widetilde{\mathbf x}$ and $\widetilde{\mathbf y}$, and
$\widetilde\lambda_1\ge\cdots\ge\widetilde\lambda_r$ 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 the following dichotomy happens. If $t_i<t_c$,
$n^{2/3}(\widetilde\lambda_i-\lambda_+)$ converges in law to the Tracy-Widom
distribution, where $\lambda_+$ is the right-edge of the bulk spectrum of the
sample canonical correlation matrix; if $t_i>t_c$,
$\sqrt{n}(\widetilde\lambda_i-\theta_i)$ 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 [5] for Gaussian vectors. Moreover, we
find that the variance of the limiting distribution of
$\sqrt{n}(\widetilde\lambda_i-\theta_i)$ 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.
