Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case

Abstract

Consider a Gaussian vector z = (x, y), consisting of two sub-vectors x and y with dimensions p and q, respectively. With n independent observations of z, we study the correlation between x and y, from the perspective of the canonical correlation analysis. We investigate the high-dimensional case: both p and q are proportional to the sample size n. Denote by uv the population cross-covariance matrix of random vectors u and v, and denote by Suv the sample counterpart. The canonical correlation coefficients between x and y are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix xx−1xyyy−1yx. In this paper, we focus on the case that xy is of finite rank k, that is, there are k nonzero canonical correlation coefficients, whose squares are denoted by r1 ≥ · · · ≥ rk > 0. We study the sample counterparts of ri, i = 1, . . ., k, that is, the largest k eigenvalues of the sample canonical correlation matrix Sxx−1SxySyy−1Syx, denoted by λ1 ≥ · · · ≥ λk. We show that there exists a threshold rc ∈ (0, 1), such that for each i ∈ {1, . . ., k}, when ri ≤ rc, λi converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by d+. When ri > rc, λi possesses an almost sure limit in (d+, 1], from which we can recover ri’s in turn, thus provide an estimate of the latter in the high-dimensional scenario. We also obtain the limiting distribution of λi’s under appropriate normalization. Specifically, λi possesses Gaussian type fluctuation if ri > rc, and follows Tracy–Widom distribution if ri < rc. Some applications of our results are also discussed.