From Berry-Esseen to super-exponential

Abstract

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 $n\ge 2m$. 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.

44 pages