A Note on Uniform Nonintegrability of Random Variables

TL;DR

This paper introduces a weaker form of uniform nonintegrability for random variables, provides characterizations and conditions for it, and addresses an open problem related to the concept.

Contribution

It proposes a new, weaker definition of uniform nonintegrability, offers necessary and sufficient conditions, and resolves an open problem from prior research.

Findings

Introduces a weaker uniform nonintegrability (W-UNI) definition.

Provides necessary and sufficient conditions for W-UNI.

Offers an analogue of de La Vallée Poussin criterion for W-UNI.

Abstract

In a recent paper \cite{CHR16}, Chandra, Hu and Rosalsky introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. In this note, we introduce a weaker definition on uniform nonintegrability (W-UNI for short) of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W-UNI, one of which is a W-UNI analogue of the celebrated de La Vall\'{e}e Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised in \cite{CHR16}.