Multivariate normal approximation for traces of random unitary matrices

Abstract

In this article we obtain a superexponential rate of convergence in total variation between the traces of the first m powers of a $\mathit{n}\times \mathit{n}$ random unitary matrices and a $2\mathit{m}$-dimensional Gaussian random variable. This generalizes previous results in the scalar case to the multivariate setting, and we also give the precise dependence on the dimensions m and n in the estimates with explicit constants. We are especially interested in the regime where m grows with n and our main result basically states that if $\mathit{m}\ll \sqrt{\mathit{n}}$, then the rate of convergence in the Gaussian approximation is $\mathrm{\Gamma }{(\frac{\mathit{n}}{\mathit{m}}+1)}^{-1}$ times a correction. We also show that the Gaussian approximation remains valid for all $\mathit{m}\ll {\mathit{n}}^{2/3}$ without a fast rate of convergence.