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

Consider two random vectors x˜=Az+C11∕2x∈Rp and y˜=Bz+C21∕2y∈Rq, where x∈Rp, y∈Rq z∈Rr are independent with i.i.d. entries of zero mean unit variance, C1 C2 p×p q×q deterministic population covariance matrices, A B p×r q×r factor loading matrices. With n observations x˜ y˜, we study the sample canonical correlations between them. Under sharp fourth moment condition on x, y z, prove BBP transition for correlation coefficients (CCCs). More precisely, if a CCC is below threshold, then corresponding converges to right edge bulk eigenvalue spectrum matrix satisfies famous Tracy-Widom law; above an outlier that detached from spectrum. We our results in full generality, sense they also hold near-degenerate CCCs close threshold.