Multivariate normal approximation for traces of orthogonal and symplectic matrices

We show that the distance in total variation between $($\mathrm{T}\mathrm{r}$\ U, \frac{1}{\sqrt{2}}$\mathrm{T}\mathrm{r}$\ U^2, \cdots, \frac{1}{\sqrt{m}}$\mathrm{T}\mathrm{r}$
U^m)$andarealGaussianvector,where$U$ is a Haar distributed orthogonal or symplectic matrix of size $2n$ or $2n+1$, is bounded by $\mathrm{\Gamma }(2\frac{n}{m}+1{)}^{-\frac{1}{2}}$ times a correction. The correction term is explicit and holds for all $n\ge m^4$, for $m$ sufficiently large. For $n\ge m^3$ we obtain the bound $(\frac{n}{m}{)}^{-c\sqrt{\frac{n}{m}}}$ with an explicit constant $c$. Our method of proof is based on an identity of Toeplitz+Hankel determinants due to Basor and Ehrhardt, see \cite{BE}, which is also used to compute the joint moments of the traces.