CANONICAL CORRELATION COEFFICIENTS OF HIGH-DIMENSIONAL GAUSSIAN VECTORS: FINITE RANK CASE

• 成果类型: Article
• 署名作者: Bao, Zhigang; Hu, Jiang; Pan, Guangming; Zhou, Wang
• 署名单位: Hong Kong University of Science & Technology; Northeast Normal University - China; Nanyang Technological University; National University of Singapore
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1704
• 发表日期: 2019
• 页码: 612-640
• 关键词: CENTRAL LIMIT-THEOREMS; LARGEST EIGENVALUE; MULTIVARIATE-ANALYSIS; multiple correlation; matrices; INDEPENDENCE; deformation; CONVERGENCE; Consistency; components

摘要

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 Sigma(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 Sigma(-1)(xx) Sigma(xy) Sigma(-1)(yy) Sigma(yx). In this paper, we focus on the case that Sigma(xy) is of finite rank k, that is, there are k nonzero canonical correlation coefficients, whose squares are denoted by r(1) >= ... >= r(k) > 0. We study the sample counterparts of r(i), i = 1, ... , k, that is, the largest k eigen-values of the sample canonical correlation matrix S-xx(-1) S-xy S-yy(-1) S-yx, denoted by lambda(1) >= ... >= lambda(k). We show that there exists a threshold r(C) epsilon (0, 1), such that for each i epsilon {1, ... , k}, when r(i) <= r(C), lambda(i) converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by d(+). When r(i) > r(C), lambda(i) possesses an almost sure limit in (d(+), 1], from which we can recover r(i)'s in turn, thus provide an estimate of the latter in the high-dimensional scenario. We also obtain the limiting distribution of lambda(i)'s under appropriate normalization. Specifically, lambda(i) possesses Gaussian type fluctuation if r(i) > r(C), and follows Tracy-Widom distribution if r(i) < r(C). Some applications of our results are also discussed.