February 2018 The logarithmic law of sample covariance matrices near singularity
Xuejun Wang, Xiao Han, Guangming Pan
Bernoulli 24(1): 80-114 (February 2018). DOI: 10.3150/16-BEJ867

## Abstract

Let $B=(b_{jk})_{p\times n}=(Y_{1},Y_{2},\ldots,Y_{n})$ be a collection of independent real random variables with mean zero and variance one. Suppose that $\Sigma$ is a $p$ by $p$ population covariance matrix. Let $X_{k}=\Sigma^{1/2}Y_{k}$ for $k=1,2,\ldots,n$ and $\hat{\Sigma}_{1}=\frac{1}{n}\sum_{k=1}^{n}X_{k}X_{k}^{T}$. Under the moment condition $\mathop{\mathrm{sup}}_{p,n}\max_{1\leq j\leq p,1\leq k\leq n}\mathbb{E}b_{jk}^{4}<\infty$, we prove that the log determinant of the sample covariance matrix $\hat{\Sigma}_{1}$ satisfies

$\frac{\log\operatorname{det}\hat{\Sigma}_{1}-\sum_{k=1}^{p}\log(1-\frac{k}{n})-\log\det\Sigma}{\sqrt{-2\log(1-\frac{p}{n})}}\xrightarrow[\qquad]{d}N(0,1),$ when $p/n\rightarrow1$ and $p<n$. For $p=n$, we prove that

$\frac{\log\det\hat{\Sigma}_{1}+n\log n-\log(n-1)!-\log\det\Sigma}{\sqrt{2\log n}}\xrightarrow[\qquad]{d}N(0,1).$

## Citation

Xuejun Wang. Xiao Han. Guangming Pan. "The logarithmic law of sample covariance matrices near singularity." Bernoulli 24 (1) 80 - 114, February 2018. https://doi.org/10.3150/16-BEJ867

## Information

Received: 1 September 2015; Revised: 1 February 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 06778322
MathSciNet: MR3706751
Digital Object Identifier: 10.3150/16-BEJ867

Keywords: central limit theorem , Covariance matrix , determinant , logarithmic law