Consider two random vectors C11/2x ∈ ℝp and C21/2y ∈ ℝq, where the entries of x and y are i.i.d. random variables with mean zero and variance one, and C1 and C2 are respectively, p × p and q × q deterministic population covariance matrices. With n independent samples of (C11/2x,C 21/2y), we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional setting with p/n → c1 ∈ (0, 1) and q/n → c2 ∈ (0, 1 − c1) as n →∞, we prove that the largest sample canonical correlation coefficient converges to the Tracy–Widom distribution as long as we have lims→∞s4ℙ(|x ij|≥ s) = 0 and lims→∞s4ℙ(|y ij|≥ 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 + 2n) × (p + q + 2n) 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：

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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C11/2X ∈ ℝp和C21/2是的 ∈ ℝq，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C1和C2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(C11/2X,C 21/2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 → C1 ∈ (0, 1)和q/n → C2 ∈ (0, 1 - C1)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→∞s4ℙ(|X 一世j|≥ s) = 0和林s→∞s4ℙ(|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 + 2n) × (p + q + 2n)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。