Limiting distribution of the sample canonical correlation coefficients of high-dimensional random vectors

In this paper, we prove a CLT for the sample canonical correlation coefficients between two high-dimensional random vectors with finite rank correlations. More precisely, consider two random vectors (x ) over tilde = x + Az and (y) over tilde = y Bz, 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 mean zero and variance one, and A is an element of R-P(xr) and B is an element of R-q(xr) are two arbitrary deterministic matrices. Given n samples of (x ) over tilde - and (y) over tilde, we stack them into two matrices X = X + AZ and = Y + BZ, where X is an element of R-P(xn), Y is an element of R-q(xn) and Z is an element of R-rxn are random matrices with i.i.d. entries of mean zero and variance one. Let (lambda) over tilde (1) >= (lambda) over tilde (2) >= ... >= (lambda) over tilde (r) be the largest r eigenvalues of the sample canonical correlation (SCC) matrix C-xy = (XXT)(-1/2XYT) (YYT)(-1YXT) (XXT)(-1/2), and let t(1) >= t(2) >= ... >= t(r) be the squares of the population canonical correlation coefficients between (x ) over tilde and (y) over tilde. Under certain moment assumptions, we show that there exists a threshold t(c) is an element of (0, 1) such that if t(i) > t(c), then root n((lambda) over tilde (i) - theta(i)) converges weakly to a centered normal distribution, where theta(i), is a fixed outlier location determined by t(i). Our proof uses a self-adjoint linearization of the SCC matrix and a sharp local law on the inverse of the linearized matrix.