Symmetry tests for manifold-valued random variables

We start with a random variable defined on a complex metric structure. We are interested in obtaining simple tests to determine whether such random variable has a symmetry. More precisely, we consider a Riemannian manifold which admits a symmetry represented by a Killing vector field. In this setting, we take into account a random variable. Our problem is to analyze and test whether such random variable shares the symmetry related to the Killing vector field. In other words, if that random variable possesses such that symmetry. As particular cases, we consider the Euclidean space and the round sphere.