We refine recent existence and uniqueness results, for the barycenter of points at infinity of Hadamard manifolds, to measures on the sphere at infinity of symmetric spaces of non compact type and, more specifically, to measures concentrated on single orbits. The barycenter will be interpreted as the maximum likelihood estimate (MLE) of generalized Cauchy distributions on Furstenberg boundaries. As a spin-off, a new proof of the general Knight–Meyer characterization theorem will be given