  Consider a normal vector $\mathbf{z}=(\mathbf{x}',\mathbf{y}')'$, consisting
of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$
respectively. With $n$ independent observations of $\mathbf{z}$ at hand, we
study the correlation between $\mathbf{x}$ and $\mathbf{y}$, from the
perspective of the Canonical Correlation Analysis, under the high-dimensional
setting: both $p$ and $q$ are proportional to the sample size $n$. In this
paper, we focus on the case that $\Sigma_{\mathbf{x}\mathbf{y}}$ is of finite
rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose
squares are denoted by $r_1\geq\cdots\geq r_k>0$. Under the additional
assumptions $(p+q)/n\to y\in (0,1)$ and $p/q\not\to 1$, we study the sample
counterparts of $r_i,i=1,\ldots,k$, i.e. the largest k eigenvalues of the
sample canonical correlation matrix
$S_{\mathbf{x}\mathbf{x}}^{-1}S_{\mathbf{x}\mathbf{y}}S_{\mathbf{y}\mathbf{y}}^{-1}S_{\mathbf{y}\mathbf{x}}$,
namely $\lambda_1\geq\cdots\geq \lambda_k$. We show that there exists a
threshold $r_c\in(0,1)$, such that for each $i\in\{1,\ldots,k\}$, when $r_i\leq
r_c$, $\lambda_i$ converges almost surely to the right edge of the limiting
spectral distribution of the sample canonical correlation matrix, denoted by
$d_r$. When $r_i>r_c$, $\lambda_i$ possesses an almost sure limit in $(d_r,1]$,
from which we can recover $r_i$ in turn, thus provide an estimate of the latter
in the high-dimensional scenario.
