Convergences of Random Variables under Sublinear Expectations

TL;DR

This paper surveys convergence results for random variables under sublinear expectations, introduces new theorems, and provides a dominated convergence theorem, enhancing understanding of convergence types in this framework.

Contribution

It establishes new relationships among convergence types and proves a dominated convergence theorem under sublinear expectations, with the assumption of monotone continuity.

Findings

L^p convergence is stronger than convergence in capacity

Convergence in capacity is stronger than convergence in distribution

A dominated convergence theorem under sublinear expectations

Abstract

In this note, we will survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, we will prove that $L^p$ convergence is stronger than convergence in capacity, convergence in capacity is stronger than convergence in distribution, and give some equivalent characterizations of convergence in distribution. In addition, we give a dominated convergence theorem under sublinear expectations, which may have its own interest.