We show that the distance in total variation between $(\mathrm{ Tr}\ U, \frac{ 1}{ \sqrt{ 2}}\mathrm{ Tr}\ U^2, \cdots, \frac{ 1}{ \sqrt{ m}}\mathrm{ Tr}\ U^m)$ and a real Gaussian vector, where $U$ is a Haar distributed orthogonal or symplectic matrix of size $2n$ or $2n+1$, is bounded by $\Gamma(2\frac{ n}{ m}+1)^{ -\frac{ 1}{ 2}}$ times a correction. The correction term is explicit and holds for all $n\geq m^4$, for $m$ sufficiently large. For $n\geq m^3$ 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.
Categories: math.PR