  We study random vectors of the form $(\operatorname
{Tr}(A^{(1)}V),...,\operatorname {Tr}(A^{(r)}V))$, where $V$ is a uniformly
distributed element of a matrix version of a classical compact symmetric space,
and the $A^{(\nu)}$ are deterministic parameter matrices. We show that for
increasing matrix sizes these random vectors converge to a joint Gaussian
limit, and compute its covariances. This generalizes previous work of Diaconis
et al. for Haar distributed matrices from the classical compact groups. The
proof uses integration formulas, due to Collins and \'{S}niady, for polynomial
functions on the classical compact groups.

Probability