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 $\frac{1}{\sqrt{m}}Tr{\mathbf{U}}^m$ to its limiting Gaussian as $n\to \mathrm{\infty }$ for orthogonal, unitary and symplectic Haar distributed random matrices U of size n. In the unitary case, we prove that the total variation distance is less than $\mathrm{\Gamma }{(\lfloor n∕m\rfloor +2)}^{-1}{m}^{-\lfloor n∕m\rfloor }{\lfloor n∕m\rfloor }^{1∕4}\sqrt{logn}$ 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 $\mathrm{\Gamma }{(2\lfloor n∕m\rfloor +1)}^{-1∕2}{m}^{-\lfloor n∕m\rfloor +1}{(logn)}^{1∕4}$ times a constant and the result holds provided $n>2m$. For $m=1$, we obtain complementary lower bounds and precise asymptotics for the $L^2$-distances as $n\to \mathrm{\infty }$, which show how sharp our results are.