  It is known that hyperbolic spaces have strict negative type, a condition on
the distances of any finite subset of points. We show that they have strong
negative type, a condition on every probability distribution of points (with
integrable distance to a fixed point). This implies that the function of
expected distances to points determines the probability measure uniquely. It
also implies that the distance covariance test for stochastic independence,
introduced by Sz\'ekely, Rizzo and Bakirov, is consistent against all
alternatives in hyperbolic spaces. We prove this by showing an analogue of the
Cram\'er-Wold device.
