From Berry-Esseen to super-exponential

For any integer $m<n$, where $m$ can depend on $n$, we study the rate of convergence of $\frac{1}{\sqrt{m}}\mathrm{Tr} \mathbf{U}^m$ to its limiting Gaussian as $n\to\infty$ for orthogonal, unitary and symplectic Haar distributed random matrices $\mathbf{U}$ of size $n$. In the unitary case, we prove that the total variation distance is less than $\Gamma(\lfloor n/m \rfloor+2)^{-1} m^{- \lfloor n/m\rfloor} \lfloor n/m \rfloor^{1/4}\sqrt{\log n}$ times a constant. This result interpolates between the super-exponential bound obtained for fixed $m$ and the $1/n$ bound coming from the Berry-Esseen theorem applicable when $m\ge n$ by a result of Rains. We obtain analogous results for the orthogonal and symplectic groups. In these cases, our total variation upper bound takes the form $\Gamma(2\lfloor n/m\rfloor+1)^{-1/2}m^{-\lfloor n/m\rfloor +1}(\log n)^{1/4}$ times a constant and the result holds provided $n \geq 2m$. For $m=1$, we obtain complementary lower bounds and precise asymptotics for the $L^2$-distances as $n\to\infty$, which show how sharp our results are.