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Tracy-Widom law for the extreme eigenvalues of sample correlation matrices. (English) Zbl 1254.15036
Summary: 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 $$r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$$, $$\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$$.

##### MSC:
 15B52 Random matrices (algebraic aspects) 60B20 Random matrices (probabilistic aspects) 62H20 Measures of association (correlation, canonical correlation, etc.) 15A18 Eigenvalues, singular values, and eigenvectors 62J10 Analysis of variance and covariance (ANOVA)
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