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

Borel theorems for random matrices from the classical compact symmetric spaces