论文标题
高维随机向量的样品规范相关系数：地方法和Tracy-Widom极限
Sample canonical correlation coefficients of high-dimensional random vectors: local law and Tracy-Widom limit
论文作者
论文摘要
考虑两个随机矢量$ \ MATHBF C_1^{1/2} \ MATHBF X \ in \ MATHBB r^p $和$ \ MATHBF C_2^{1/2} \ MATHBF y \ in \ MATHBB R^Q $平均零和差异的随机变量，以及$ \ Mathbf C_1 $和$ \ Mathbf C_2 $是$ P \ times P $和$ Q \ times Q $确定性种群协方差矩阵。使用$（\ Mathbf C_1^{1/2} \ Mathbf X，\ Mathbf C_2^{1/2} \ Mathbf Y）$的$ N $独立样本，我们使用构成相关性分析研究了这两个矢量之间的样本相关。我们用$ s_ {xx} $和$ s_ {yy} $表示$ \ MathBf C_1^{1/2} \ Mathbf X $和$ \ MathBf C_2^{1/2} {1/2} {1/2} \ Mathbf y $的样本协方差矩阵，以及$ s_}然后，样本规范相关系数是样品规范相关矩阵$ \ cal c_ {xy}：= s_ {xx}^{ - 1} s_ {xy} s_ {xy} s_ {yy} {yy}^{ - 1} s_ {yx s_ {yx} $。在具有$ {p}/{n} \至c_1 \ in（0，1）$和$ {q}/{n}/{n} \ to c_2 \ in（0，1-c_1）$ as $ n \ to \ in（$ n \ to \ in）中的高维设置下只要我们有$ \ lim_ {s \ rightArrow \ infty} s^4 [\ Mathbb {p}（\ vert x_ {ij} \ vert \ geq s）+ \ geq s）+ \ mathbb {p}（\ vert y__ {ij}}} \ vert \ vert \ geq s）= 0 $。这扩展了[16]的结果，该结果确立了$ \ Mathcal C_ {Xy} $的最大特征值的Tracy-Widom限制，假设所有矩都是有限的。我们的证明是基于线性化方法，该方法将问题降低为$（p+q+2n）\ times（p+q+2n）$随机矩阵$ h $。特别是，我们将证明其反向$ g：= h^{ - 1} $，即分解。本文法律是本文中Tracy-Widom定律证明的主要工具，也是[22,23]中对具有有限秩相关的高维随机向量的规范相关系数的研究。
Consider two random vectors $\mathbf C_1^{1/2}\mathbf x \in \mathbb R^p$ and $\mathbf C_2^{1/2}\mathbf y\in \mathbb R^q$, where the entries of $\mathbf x$ and $\mathbf y$ are i.i.d. random variables with mean zero and variance one, and $\mathbf C_1$ and $\mathbf C_2$ are $p \times p$ and $q\times q$ deterministic population covariance matrices. With $n$ independent samples of $(\mathbf C_1^{1/2}\mathbf x,\mathbf C_2^{1/2}\mathbf y)$, we study the sample correlation between these two vectors using canonical correlation analysis. We denote by $S_{xx}$ and $S_{yy}$ the sample covariance matrices for $\mathbf C_1^{1/2}\mathbf x$ and $\mathbf C_2^{1/2}\mathbf y$, respectively, and $S_{xy}$ the sample cross-covariance matrix. Then the sample canonical correlation coefficients are the square roots of the eigenvalues of the sample canonical correlation matrix $\cal C_{XY}:=S_{xx}^{-1}S_{xy}S_{yy}^{-1}S_{yx}$. Under the high-dimensional setting with ${p}/{n}\to c_1 \in (0, 1)$ and ${q}/{n}\to c_2 \in (0, 1-c_1)$ as $n\to \infty$, we prove that the largest eigenvalue of $\mathcal C_{XY}$ converges to the Tracy-Widom distribution as long as we have $\lim_{s \rightarrow \infty}s^4 [\mathbb{P}(\vert x_{ij} \vert \geq s)+ \mathbb{P}(\vert y_{ij} \vert \geq s)]=0$. This extends the result in [16], which established the Tracy-Widom limit of the largest eigenvalue of $\mathcal C_{XY}$ under the assumption that all moments are finite. Our proof is based on a linearization method, which reduces the problem to the study of a $(p+q+2n)\times (p+q+2n)$ random matrix $H$. In particular, we shall prove an optimal local law on its inverse $G:=H^{-1}$, i.e the resolvent. This local law is the main tool for both the proof of the Tracy-Widom law in this paper, and the study in [22,23] on the canonical correlation coefficients of high-dimensional random vectors with finite rank correlations.
