Forward-convex convergence in probability of sequences of nonnegative random variables

TL;DR

This paper establishes simple, measure-free conditions under which sequences of nonnegative random variables and their forward convex combinations converge in probability to the same limit, generalizing uniform integrability concepts.

Contribution

It introduces necessary and sufficient conditions for convergence of forward convex combinations in probability, extending the understanding of measure-free convergence criteria.

Findings

Conditions equivalent to measure-free uniform integrability.

Characterization of convergence for sequences of nonnegative random variables.

Applicability to general probability spaces.

Abstract

For a sequence of nonnegative random variables, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges in probability to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.