I have this problem. Suppose $(X_t)_{t\geq 1}$ is a sequence of random variables with values in $[0,\infty)$. I know that $E(X_t)\rightarrow 1$ as $t\rightarrow \infty$ and $(\exp(X_t))_{t\geq 1}$ is a sub-martingale. I would like to find whether $\sup_t E(\exp(X_t))<\infty$ in order to apply the martingale convergence theorem. Do you have any idea if this is possible/not feasible?
Thanks!