On $B$-statistical uniform nonintegrability of a sequence of random variables

Abstract

In this paper, we introduce a notion of $B$-statistical uniform nonintegrability with respect to $A$ (B-UNI(c) w.r.t. $A$, in short), which is weaker than UNI(c) w.r.t. $A$ (see \cite{Chandra}). We establish the de La ${\rm Vall\acute{e}e}$ Poussin criterion for B-UNI(c) w.r.t. $A$, which extends Theorem 2.1 of Chandra et al. (2021) \cite{Chandra}. Moreover, we also give a necessary condition for the sequence of random variables $\{X_{k},k\in\mathds{N}\}$ to be B-UNI(c) w.r.t. $A$.