We know from Doob's (super-)martingale convergence theorem that $X_n \to X_\infty$ almost surely, if $(X_n)_{n \in \mathbb{N}}$ is a super-martingale $$ \mathbb{E}(X_{n+1}|F_n) \le X_n $$ adapted to some filtration $(F_n)_{n \in \mathbb{N}}$ and is bounded from below in a suitable sense (i.e. $\sup_k \mathbb{E}(\max(-X_k, 0)) < +\infty$) . In particular we consider the special case where the filtration is the sigma algebra generated by the process itself, i.e. we have $$ \mathbb{E}(X_{n+1}|\sigma(X_1,\dots,X_n)) \le X_n. $$
The question is, can we prove something in the case where we only have $$ \mathbb{E}(X_{n+1}|X_n) \le X_n. $$