Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
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样本相关矩阵极值特征值的 Tracy-Widom 定律
DOI：
10.1214/ejp.v17-1962
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发表时间：
2012-10-04
影响因子：
1.4
通讯作者：
Zhou, Wang
中科院分区：
文献类型：
--
作者：
Bao, Zhigang;Pan, Guangming;Zhou, Wang
设样本相关矩阵为\(W = YYT\)，其中\(Y=(y_{(ij)})_{p\times n}\)，且\(y_{(ij)}=\frac{w_{e}x_{(ij)}}{\sqrt{\sum_{j = 1}^{n}x_{(ij)}^{2}}}\)。我们假设\(\{x_{(ij)}:i\)（最后这里不完整，可能影响准确理解）
Let the sample correlation matrix be $W=YY^T$ , where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0<p<n$ and $p/n\rightarrow y$ with some $y\in(0,1)$ as $p,n\rightarrow\infty$. In this paper, we provide the Tracy-Widom law ($TW_1$) for both the largest and smallest eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard normal, we can derive the $TW_1$ for both the largest and smallest eigenvalues of the matrix $\mathcal{R}=RR^T$, where $R=(r_{ij})_{p,n}$ with $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.