  The general notion of a stochastic ordering is that one probability
distribution is smaller than a second one if the second attaches more
probability to higher values than the first. Motivated by recent work on
barycentric maps on spaces of probability measures on ordered Banach spaces, we
introduce and study a stochastic order on the space of probability measures
$\mathcal{P}(X)$, where $X$ is a metric space equipped with a closed partial
order, and derive several useful equivalent versions of the definition. We
establish the antisymmetry and closedness of the stochastic order (and hence
that it is a closed partial order) for the case of a partial order on a Banach
space induced by a closed normal cone with interior. We also consider
order-completeness of the stochastic order for a cone of a finite-dimensional
Banach space and derive a version of the arithmetic-geometric-harmonic mean
inequalities in the setting of the associated probability space on positive
matrices.
