Stochastic order characterization of uniform integrability and tightness


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