Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Consider two random vectors (similar to)x = Az + C-1/2 (1) x is an element of R-p and (similar to)y = Bz + C-1/2 (2) y is an element of R-q, where x is an element of R-p, y is an element of R-q and z is an element of R-r are independent random vectors with i.i.d. entries of zero mean and unit variance, C-1 and C-2 are p x p and q x q deterministic population covariance matrices, and A and B are p x r and q x r deterministic factor loading matrices. With n independent observations of (similar to)x and (similar to)y, we study the sample canonical correlations between them. Under the sharp fourth moment condition on the entries of x, y and z, we prove the BBP transition for the sample canonical correlation coefficients (CCCs). More precisely, if a population CCC is below a threshold, then the corresponding sample CCC converges to the right edge of the bulk eigenvalue spectrum of the sample canonical correlation matrix and satisfies the famous Tracy-Widom law; if a population CCC is above the threshold, then the corresponding sample CCC converges to an outlier that is detached from the bulk eigenvalue spectrum. We prove our results in full generality, in the sense that they also hold for near-degenerate population CCCs and population CCCs that are close to the threshold.