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

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{\ma