On Uniform Nonintegrability and Weak Uniform Nonintegrability of a Sequence of Random Variables with Respect to a Nonnegative Array

Abstract

Chandra, Hu and Rosalsky ^[1] introduced the notion of a sequence of random variables being uniformly nonintegrable and they established a de La Vallée Poussin type criterion for this notion. Inspired by the Chandra, Hu and Rosalsky ^[1] article, Hu and Peng ^[2] introduced the weaker notion of a sequence of random variables being weakly uniformly nonintegrable and they also established a de La Vallée Poussin type criterion for this notion using a modification of the Chandra, Hu and Rosalsky ^[1] argument. In this correspondence, we introduce the more general notion of uniform nonintegrability and weak uniform nonintegrability with respect to an array of nonnegative real numbers together with a de La Vallée Poussin type criterion for this notion. This criterion immediately yields as particular cases the criteria of Chandra, Hu and Rosalsky ^[1] and Hu and Peng ^[2] , and it has a substantially simpler and more straightforward proof.