The logarithmic law of sample correlation matrices

Abstract

Let $\mathbf{R}$ be the sample correlation matrix constructed from $\mathbf{X}\in \mathbb{R}^{p\times n}$, whose entries are independent and identically distributed random variables with mean zero and tail probability condition $\lim_{x\rightarrow \infty}x^3\mathbb{P}(|\xi|>x)=0$. We derive the universal logarithmic law for $\log \det \mathbf{R}$, \begin{equation*} \frac{\log \det \mathbf{R}-(p-n+1/2)\log (1-\frac{p-1}{n})+p-\frac{p}{n}}{\sqrt{-2\log (1-\frac{p-1}{n})-2\frac{p}{n}}}\stackrel{d}{\rightarrow} {N}(0,1), \end{equation*} if $p\le n$ as $p,n\rightarrow \infty$. Moreover, under the near-singularity case $0\le n-p\le n^{1-w}$ for any $w\in (0,1)$, it is shown that the tail probability condition can be weakened to $\lim_{x\rightarrow \infty}x^3(\log x)^{-1/4+\mathfrak{c}}\mathbb{P}(|\xi|>x)<\infty$ for any constant $0<\mathfrak{c}<1/4$.