The convergence of the empirical distribution of canonical correlation coefficients

Abstract

Suppose that $\{X_{jk}, j=1,\cdots,p_1; k=1,\cdots,n\}$ are independent and identically distributed (i.i.d) real random variables with $EX_{11}=0$ and $EX_{11}^{2}=1$, and that $\{Y_{jk}, j=1,\cdots,p_2; k=1,\cdots,n\}$ are i.i.d real random variables with $EY_{11}=0$ and $EY_{11}^{2}=1$, and that $\{X_{jk}, j=1,\cdots,p_1; k=1,\cdots,n\}$ are independent of $\{Y_{jk}, j=1,\cdots,p_2; k=1,\cdots,n\}$. This paper investigates the canonical correlation coefficients $r_1 \geq r_2 \geq \cdots \geq r_{p_1}$, whose squares $\lambda_1=r_1^2, \lambda_2=r_2^2, \cdots, \lambda_{p_1}=r_{p_1}^2$ are the eigenvalues of the matrix
\begin{equation*}
S_{xy}=A_x^{-1}A_{xy}A_y^{-1}A_{xy}^{T},
\end{equation*}
where
\begin{equation*}
A_x=\frac{1}{n}\sum^{n}_{k=1}x_kx_k^{T},\
A_y=\frac{1}{n}\sum^{n}_{k=1}y_ky_k^{T},\
A_{xy}=\frac{1}{n}\sum^{n}_{k=1}x_ky_k^{T},
\end{equation*}
and
\begin{equation*}
x_k=(X_{1k},\cdots,X_{p_1k})^{T},\
y_k=(Y_{1k},\cdots,Y_{p_2k})^{T},\ k=1,\cdots,n.
\end{equation*}

When $p_1\rightarrow \infty$, $p_2\rightarrow \infty$ and $n\rightarrow \infty$ with $\frac{p_1}{n}\rightarrow c_1$, $\frac{p_2}{n}\rightarrow c_2$, $c_1, c_2\in (0,1)$, it is proved that the empirical distribution of $r_1, r_2, \cdots, r_{p_1}$ converges, with probability one, to a fixed distribution under the finite second moment condition.