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 x˜=x+Az and y˜=y+Bz, where x∈Rp, y∈Rq z∈Rr are independent i.i.d. entries of mean zero variance one, A∈Rp×r B∈Rq×r arbitrary deterministic matrices. Given n samples x˜ y˜, stack them into matrices X=X+AZ Y=Y+BZ, X∈Rp×n, Y∈Rq×n Z∈Rr×n one. Let λ˜1≥λ˜2≥⋯≥λ˜r be largest r eigenvalues (SCC) matrix CXY=(XX⊤)−1∕2XY⊤(YY⊤)−1YX⊤(XX⊤)−1∕2, let t1≥t2≥⋯≥tr squares population y˜. Under certain moment assumptions, show that there exists threshold tc∈(0,1) such if ti>tc, then n(λ˜ i−θi) converges weakly to centered normal distribution, θi is fixed outlier location determined by ti. Our proof uses self-adjoint linearization SCC sharp local law on inverse linearized matrix.