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

Abstract

We consider two high-dimensional random vectors rx P R p and r y P R q with finite rank correlations. More precisely, suppose that rx " x `Az and r y " y `Bz, for independent random vectors x P R p , y P R q and z P R r with i.i.d. entries of mean zero and variance one, and two deterministic factor loading matrices A P R pˆr and B P R qˆr . With n i.i.d. observations of pr x, r yq, we study the sample canonical correlations between rx and r y. In this paper, we focus on the high-dimensional setting with a rank-r correlation. Let t1 ě t2 ě ¨¨¨ě tr be the squares of the population canonical correlation coefficients between rx and r y, and r λ1 ě r λ2 ě ¨¨¨ě r λr be the squares of the largest r sample canonical correlation coefficients. Under certain moment assumptions on the entries of x, y and z, we show that there exists a threshold tc P p0, 1q such that the following dichotomy happens. If ti ă tc, n 2{3 p r λi ´λ`q converges in law to the Tracy-Widom distribution, where λ`is the right-edge of the bulk spectrum of the sample canonical correlation matrix; if ti ą tc, ? np r λi ´θiq converges in law to a centered normal distribution, where θi ą λ`is a fixed outlier location determined by ti. Our results extend the ones in [5] for Gaussian vectors rx and r y. Moreover, we find that the variance of the limiting distribution of ? np r λi ´θiq also depends on the fourth cumulants of the entries of x, y and z, a phenomenon that cannot be observed in the Gaussian case.

Spiked eigenvalues of noncentral Fisher matrix with applications

“…Under the sharp moment condition, [36,28] prove the largest eigenvalue of sample canonical correlation matrix converges to T-W law to test the independence of random vectors with two different structures. Moreover, [37] shows the limiting distribution of the spiked eigenvalues depends on the fourth cumulants of the population distribution. Note that [3,36,28,37] only focused on the finite rank case, where the number of the positive population canonical correlation coefficients of two groups of high-dimensional Gaussian vectors are finite.…”

“…Moreover, [37] shows the limiting distribution of the spiked eigenvalues depends on the fourth cumulants of the population distribution. Note that [3,36,28,37] only focused on the finite rank case, where the number of the positive population canonical correlation coefficients of two groups of high-dimensional Gaussian vectors are finite. However, we popularize finite rank case to infinite rank case, in other words, we get the limits and fluctuations of the sample canonical correlation coefficients under the infinite rank case.…”