The logarithmic law of sample correlation matrices
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908902666272768 |
|---|---|
| author | Li, Yanpeng Liu, Zhi Xie, Jiahui Zhou, Wang |
| author_facet | Li, Yanpeng Liu, Zhi Xie, Jiahui Zhou, Wang |
| contents | 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 $\lim_{x\rightarrow \infty}x^3\mathbb{P}(|ξ|>x)=0$. 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 $\lim_{x\rightarrow \infty}x^3(\log x)^{-1/4+\mathfrak{c}}\mathbb{P}(|ξ|>x)<\infty$ for any constant $0<\mathfrak{c}<1/4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19800 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The logarithmic law of sample correlation matrices Li, Yanpeng Liu, Zhi Xie, Jiahui Zhou, Wang Probability 60B20, 60F05 (Primary) 62H20 (Secondary) 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 $\lim_{x\rightarrow \infty}x^3\mathbb{P}(|ξ|>x)=0$. 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 $\lim_{x\rightarrow \infty}x^3(\log x)^{-1/4+\mathfrak{c}}\mathbb{P}(|ξ|>x)<\infty$ for any constant $0<\mathfrak{c}<1/4$. |
| title | The logarithmic law of sample correlation matrices |
| topic | Probability 60B20, 60F05 (Primary) 62H20 (Secondary) |
| url | https://arxiv.org/abs/2603.19800 |