The logarithmic law of sample covariance matrices near singularity

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

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