Given a stochastic basic $(\mathcal A_n)$, a sequence $(X_n)$ of integrable random variables, adapted to $(\mathcal A_n)$ is said to be a quasi-martingale in the limit if for every $\varepsilon > 0$, there exists $p\in N$ such that for every $m\geq p$ there exists $p_m\geq m$ such that for all $n\geq p_m$ we have
$$P\left(\sup\limits_{p\leq q\leq m}|X_q(n)-X_q| > \varepsilon\right) < \varepsilon.$$
The main aim of this note is to prove that the class of all quasi-martingales in the limit would be classified into a nondecreasing directed family of subclasses whose smallest element is just the class of mils introduced by M. Talagrand (1985).