Sample canonical correlation coefficients of high-dimensional random vectors: local law and Tracy-Widom limit

Abstract

Consider two random vectors $\mathbf C_1^{1/2}\mathbf x \in \mathbb R^p$ and $\mathbf C_2^{1/2}\mathbf y\in \mathbb R^q$, where the entries of $\mathbf x$ and $\mathbf y$ are i.i.d. random variables with mean zero and variance one, and $\mathbf C_1$ and $\mathbf C_2$ are $p \times p$ and $q\times q$ deterministic population covariance matrices. With $n$ independent samples of $(\mathbf C_1^{1/2}\mathbf x,\mathbf C_2^{1/2}\mathbf y)$, we study the sample correlation between these two vectors using canonical correlation analysis. We denote by $S_{xx}$ and $S_{yy}$ the sample covariance matrices for $\mathbf C_1^{1/2}\mathbf x$ and $\mathbf C_2^{1/2}\mathbf y$, respectively, and $S_{xy}$ the sample cross-covariance matrix. Then the sample canonical correlation coefficients are the square roots of the eigenvalues of the sample canonical correlation matrix $\cal C_{XY}:=S_{xx}^{-1}S_{xy}S_{yy}^{-1}S_{yx}$. Under the high-dimensional setting with ${p}/{n}\to c_1 \in (0, 1)$ and ${q}/{n}\to c_2 \in (0, 1-c_1)$ as $n\to \infty$, we prove that the largest eigenvalue of $\mathcal C_{XY}$ converges to the Tracy-Widom distribution as long as we have $\lim_{s \rightarrow \infty}s^4 [\mathbb{P}(\vert x_{ij} \vert \geq s)+ \mathbb{P}(\vert y_{ij} \vert \geq s)]=0$. This extends the result in [16], which established the Tracy-Widom limit of the largest eigenvalue of $\mathcal C_{XY}$ under the assumption that all moments are finite. Our proof is based on a linearization method, which reduces the problem to the study of a $(p+q+2n)\times (p+q+2n)$ random matrix $H$. In particular, we shall prove an optimal local law on its inverse $G:=H^{-1}$, i.e the resolvent. This local law is the main tool for both the proof of the Tracy-Widom law in this paper, and the study in [22,23] on the canonical correlation coefficients of high-dimensional random vectors with finite rank correlations.