## Stochastic order characterization of uniform integrability and tightness

*Statistics & Probability Letters*, 83(1):382–389, 2013.- doi:10.1016/j.spl.2012.09.023
- arXiv:1106.0607

### Abstract

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.

**Keywords**: stochastic order, uniformly integrable, tight, stochastically bounded,
bounded in probability, strong order, increasing convex order, integrated survival
function, Hardy-Littlewood maximal random variable

**AMS subject classification**: 60E15, 60B10, 60F25