  Let $M$ be a random matrix chosen according to Haar measure from the unitary
group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the
traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal
variables as $n \to \infty$, and Johansson proved that the rate of convergence
is superexponential in $n$.
  We prove a finite field analogue of these results. Fixing a prime power $q =
p^r$, we choose a matrix $M$ uniformly from the finite unitary group
$\mathrm{U}(n,q)\subseteq \mathrm{GL}(n,q^2)$ and show that the traces of $\{
M^i \}_{1 \le i \le k,\, p \nmid i}$ converge to independent uniform variables
in $\mathbb{F}_{q^2}$ as $n \to \infty$. Moreover we show the rate of
convergence is exponential in $n^2$. We also consider the closely related
problem of the rate at which characteristic polynomial of $M$ equidistributes
in `short intervals' of $\mathbb{F}_{q^2}[T]$. Analogous results are also
proved for the general linear, special linear, symplectic and orthogonal groups
over a finite field. In the two latter families we restrict to odd
characteristic.
  The proofs depend upon applying techniques from analytic number theory over
function fields to formulas due to Fulman and others for the probability that
the characteristic polynomial of a random matrix equals a given polynomial.
