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

报告人：周望（新加坡国立大学）

时间: 2019年6月1日(周六)上午10:00-11:00

地点：管理科研楼1418 

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 Σuv 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 yare known as the square roots of the nonzero eigenvalues of the canonical correlation matrix Σ−1xxΣxyΣ−1yyΣyx. In this paper, we focus on the case that Σxy is of finite rank k, that is, there are knonzero 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 S−1xxSxyS−1yySyx, 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.

Joint work with Zhigang Bao, Jiang Hu and Guangming Pan 

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