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

Abstract

Let $A=(A_n)_{n\ge2}$ 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 $\mathbb Ea_{ij}^2=1$, 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 $\left\{\frac{|a_{ij}|^{4}}{\sqrt{\log(e+|a_{ij}|)}} \right\}_{n\geq 2;1\leq i,j\leq n}$ 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\geq 2;1\leq i,j\leq 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 $0<\delta\le\tfrac12$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1/2-\delta}}<\infty$, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm d}(A_n),\mathcal N(0,1))\le C(\log n)^{-\delta}. \end{align*} For $0<\gamma\le1$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1-\gamma}}<\infty$ 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)^{-\gamma}. \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}.