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

By Z. Bao, Jiang Hu, Guangming Pan, Wang Zhou. Annals of Statistics, 2017

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Consider a Gaussian vector $\mathbf{z}=(\mathbf{x}',\mathbf{y}')'$, consisting of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$ respectively, where both $p$ and $q$ are proportional to the sample size $n$. Denote by $\Sigma_{\mathbf{u}\mathbf{v}}$ the population cross-covariance matrix of random vectors $\mathbf{u}$ and $\mathbf{v}$, and denote by $S_{\mathbf{u}\mathbf{v}}$ the sample counterpart. The canonical correlation coefficients between $\mathbf{x}$ and $\mathbf{y}$ are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix $\Sigma_{\mathbf{x}\mathbf{x}}^{-1}\Sigma_{\mathbf{x}\mathbf{y}}\Sigma_{\mathbf{y}\mathbf{y}}^{-1}\Sigma_{\mathbf{y}\mathbf{x}}$. In this paper, we focus on the case that $\Sigma_{\mathbf{x}\mathbf{y}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1\geq\cdots\geq r_k>0$. We study the sample counterparts of $r_i,i=1,\ldots,k$, i.e. the largest $k$ eigenvalues of the sample canonical correlation matrix $\S_{\mathbf{x}\mathbf{x}}^{-1}\S_{\mathbf{x}\mathbf{y}}\S_{\mathbf{y}\mathbf{y}}^{-1}\S_{\mathbf{y}\mathbf{x}}$, denoted by $\lambda_1\geq\cdots\geq \lambda_k$. We show that there exists a threshold $r_c\in(0,1)$, such that for each $i\in\{1,\ldots,k\}$, when $r_i\leq r_c$, $\lambda_i$ 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 $(d_{+},1]$. 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

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/da2b060f16e0ac5d82990925fff3ed37354dc3cc

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In Discrete Mathematics and Combinatorics

  1. Xiucai Ding
  2. Z. Bao
  3. A. Onatski
  4. Iain M. Johnstone
  5. Ke Wang
  6. Jun Yin
  7. Fan Yang
  8. Guangming Pan

In Mathematical Physics

  1. Z. Bao
  2. Xiucai Ding
  3. A. Onatski
  4. Iain M. Johnstone
  5. Ke Wang
  6. Jun Yin
  7. Guangming Pan
  8. Wang Zhou

In Statistics and Probability

  1. Xiucai Ding
  2. Z. Bao
  3. Ke Wang
  4. A. Onatski
  5. Iain M. Johnstone
  6. Guangming Pan
  7. Jun Yin
  8. Tommaso Cai

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