Good morning,
imagine you have an adapted process $\{S(t):t\geq 1\}$ with $S(t)\geq 0$ and $E(S(t))<\infty$ for all $t$. The following holds (for every $t$):
$E[S(t+1)-S(t)|\mathcal{F}_t]\leq O\left(\frac{1}{t\ln(t)}\right)$.
So the process $\{S(t):t\geq 1\}$ is not a (non-negative) supermartingale, even if the term on the right-hand side of last equality tends to $0$ as $t\rightarrow \infty$. So the martingale property is satisfied "in the limit of large time". The point is that I need $(S(t))_t$ to be a supermartingale in order to apply the Martingale Conv. Thm. Do you know any trick to solve a problem of this type?
Thanks!