  Stein's method is used to study the trace of a random element from a compact
Lie group or symmetric space. Central limit theorems are proved using very
little information: character values on a single element and the decomposition
of the square of the trace into irreducible components. This is illustrated for
Lie groups of classical type and Dyson's circular ensembles. The approach in
this paper will be useful for the study of higher dimensional characters, where
normal approximations need not hold.
