The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants

Abstract

Let $A=(A_n{)}_{n\ge 2}$ be a triangular array of random matrices, where $A_n=(a_{ij}{)}_{1\le i,j\le n}$ is an $n\times n$ random matrix with independent real entries satisfying $\mathbb{E}{a}_{ij}=0$ and $\mathbb{E}{a}_{ij}^2=1$, and put ${\mathcal{L}}_n=\log |\det A_n|$ and \[ W_n^{\mathrm d}(A_n):=\frac{\mathcal L_n - \frac12\log(n-1)!}{\sqrt{\frac12\log n}},\quad W_n^{\mathrm e}(A_n):= \frac{\mathcal L_n-\mathbb E \mathcal L_n}{\sqrt{\frac12\log n}}. \] We prove that $W_n^{\mathrm{d}}(A_n)\Rightarrow \mathcal{N}(0,1)$, whenever the family ${\left\{\frac{|a_{ij}{|}^4}{\sqrt{\log (e+|a_{ij}|)}}\right\}}_{n\ge 2;1\le i,j\le n}$ is uniformly integrable. If, in addition, the entries have uniformly bounded densities, then $W_n^{\mathrm{e}}(A_n)\Rightarrow \mathcal{N}(0,1)$ whenever the family ${\left\{\frac{|a_{ij}{|}^4}{\log (e+|a_{ij}|)}\right\}}_{n\ge 2;1\le i,j\le n}$ is uniformly integrable. These two conditions are optimal at the level of universal moment assumptions. We further establish the corresponding Berry--Esseen bounds, and show that for $0<\delta \le \frac{1}{2}$, if $\underset{n}{\sup }\underset{1\le i,j\le n}{\max }\mathbb{E}\frac{|a_{ij}{|}^4}{\{\log (e+|a_{ij}|){\}}^{1/2-\delta }}<\infty$, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm d}(A_n),\mathcal N(0,1))\le C(\log n)^{-δ}. \end{align*} For $0<\gamma \le 1$, if $\underset{n}{\sup }\underset{1\le i,j\le n}{\max }\mathbb{E}\frac{|a_{ij}{|}^4}{\{\log (e+|a_{ij}|){\}}^{1-\gamma }}<\infty$ and the entries have uniformly bounded densities, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm e}(A_n),\mathcal N(0,1))\le C(\log n)^{-γ}. \end{align*} When $\delta =1/2$ and $\gamma =1$, the bounds $(\log n{)}^{-1/2}$ and $(\log n{)}^{-1}$ are optimal, respectively. Our results improve the earlier Central Limit Theorem by \cite{BaoPanZhou2015} and the Berry--Esseen bound by \cite{NguyenVu2014}.