  We show that a family of random variables is uniformly integrable if and only
if it is stochastically bounded in the increasing convex order by an integrable
random variable. This result is complemented by proving analogous statements
for the strong stochastic order and for power-integrable dominating random
variables. Especially, we show that whenever a family of random variables is
stochastically bounded by a p-integrable random variable for some p>1, there is
no distinction between the strong order and the increasing convex order. These
results also yield new characterizations of relative compactness in Wasserstein
and Prohorov metrics.
