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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 samplecorrelation between these two vectors using canonical correlation analysis. Wedenote by $S_{xx}$ and $S_{yy}$ the sample covariance matrices for $\mathbfC_1^{1/2}\mathbf x$ and $\mathbf C_2^{1/2}\mathbf y$, respectively, and$S_{xy}$ the sample cross-covariance matrix. Then the sample canonicalcorrelation coefficients are the square roots of the eigenvalues of the samplecanonical 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 largesteigenvalue of $\mathcal C_{XY}$ converges to the Tracy-Widom distribution aslong 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 proofis 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 anoptimal local law on its inverse $G:=H^{-1}$, i.e the resolvent. This local lawis the main tool for both the proof of the Tracy-Widom law in this paper, andthe study in [22,23] on the canonical correlation coefficients ofhigh-dimensional random vectors with finite rank correlations.

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