Stochastic order characterization of uniform integrability and tightness

TL;DR

This paper characterizes uniform integrability and tightness of random variable families using stochastic orders, providing new criteria and connections to Wasserstein and Prohorov metrics.

Contribution

It introduces stochastic order-based characterizations of uniform integrability and tightness, extending to strong orders and power integrable bounds, linking to metric compactness.

Findings

Uniform integrability characterized by stochastic boundedness in increasing convex order.

Equivalence of strong stochastic order and increasing convex order under p-integrable bounds.

New criteria for relative compactness in Wasserstein and Prohorov metrics.

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.