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Sample canonical correlation coefficients of high-dimensional random vectors: Local law and Tracy–Widom limit
Random Matrices: Theory and Applications ( IF 0.9 ) Pub Date : 2021-05-31 , DOI: 10.1142/s2010326322500071 Fan Yang 1
Random Matrices: Theory and Applications ( IF 0.9 ) Pub Date : 2021-05-31 , DOI: 10.1142/s2010326322500071 Fan Yang 1
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Consider two random vectors C 1 1 / 2 x ∈ ℝ p and C 2 1 / 2 y ∈ ℝ q , where the entries of x and y are i.i.d. random variables with mean zero and variance one, and C 1 and C 2 are respectively, p × p and q × q deterministic population covariance matrices. With n independent samples of ( C 1 1 / 2 x , C 2 1 / 2 y ) , we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional setting with p / n → c 1 ∈ ( 0 , 1 ) and q / n → c 2 ∈ ( 0 , 1 − c 1 ) as n → ∞ , we prove that the largest sample canonical correlation coefficient converges to the Tracy–Widom distribution as long as we have lim s → ∞ s 4 ℙ ( | x i j | ≥ s ) = 0 and lim s → ∞ s 4 ℙ ( | y i j | ≥ s ) = 0 , which we believe to be a sharp moment condition. This extends the result in [19], which established the Tracy–Widom limit under the assumption that all moments exist for the entries of x and y . Our proof is based on a new linearization method, which reduces the problem to the study of a ( p + q + 2 n ) × ( p + q + 2 n ) random matrix H . In particular, we shall prove an optimal local law on its inverse G : = H − 1 , called resolvent. This local law is the main tool for both the proof of the Tracy–Widom law in this paper, and the study in [26, 27] on the canonical correlation coefficients of high-dimensional random vectors with finite rank correlations.
中文翻译:
高维随机向量的样本典型相关系数:局部定律和 Tracy-Widom 极限
考虑两个随机向量C 1 1 / 2 X ∈ ℝ p 和C 2 1 / 2 是的 ∈ ℝ q ,其中的条目X 和是的 是具有均值零和方差一的独立同分布随机变量,并且C 1 和C 2 分别是,p × p 和q × q 确定性总体协方差矩阵。和n 的独立样本( C 1 1 / 2 X , C 2 1 / 2 是的 ) ,我们使用典型相关分析研究这两个向量之间的样本相关性。在高维设置下p / n → C 1 ∈ ( 0 , 1 ) 和q / n → C 2 ∈ ( 0 , 1 - C 1 ) 作为n → ∞ ,我们证明了最大样本典型相关系数收敛到 Tracy-Widom 分布,只要我们有林 s → ∞ s 4 ℙ ( | X 一世 j | ≥ s ) = 0 和林 s → ∞ s 4 ℙ ( | 是的 一世 j | ≥ s ) = 0 ,我们认为这是一个尖锐的时刻条件。这扩展了 [19] 中的结果,它建立了 Tracy-Widom 极限,假设所有矩都存在于X 和是的 . 我们的证明基于一种新的线性化方法,将问题简化为研究( p + q + 2 n ) × ( p + q + 2 n ) 随机矩阵H . 特别是,我们将证明其逆的最优局部律G : = H - 1 ,称为分解剂。该局部定律是本文证明 Tracy-Widom 定律的主要工具,也是 [26, 27] 中研究具有有限秩相关的高维随机向量的典型相关系数的主要工具。
更新日期:2021-05-31
中文翻译:
高维随机向量的样本典型相关系数:局部定律和 Tracy-Widom 极限
考虑两个随机向量