  Let $M_n$ be a random element of the unitary, special orthogonal, or unitary
symplectic groups, distributed according to Haar measure. By a classical result
of Diaconis and Shahshahani, for large matrix size $n$, the vector $
(\on{Tr}(M_n), \on{Tr}(M_n^2),..., \on{Tr}(M_n^d))$ tends to a vector of
independent (real or complex) Gaussian random variables. Recently, Jason Fulman
has demonstrated that for a single power $j$ (which may grow with $n$), a speed
of convergence result may be obtained via Stein's method of exchangeable pairs.
In this note, we extend Fulman's result to the multivariate central limit
theorem for the full vector of traces of powers.
