Dinh Quang Luu and Tran Quang Vinh

Abstract
Martingales in the limit and mils would be regarded as two important generalizations of martingales. It is known that every $L^1$-bounded such a sequence of random variables converges a.s. Recently, the first author of this note has noted that this convergence result still holds for another essentially larger class, that is, the class of quasi-martingales in the limit. The main aim of this note is to give a complete classification of the latter class into an increasing family of subclasses whose smallest element is just the class of mils.