Suppose $\{X_n\}_{n = 1}^{\infty}$ is a discrete-time submartingale (a sequence of random variables, such that $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$), such, that $\exists C \in \mathbb{R} \forall n \in \mathbb{N} EX_n^2 < C \text{ and } P(X_n > 0) = 1$. Is it true, that $P(\exists \lim_{n \to \infty} X_n) = 1$?
I know, that as $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$, and, by Jensen inequality for conditional expectations $P(E[X_{n+1}^2|X_1, … X_n] \geq E^2[X_{n+1}|X_1, … X_n]) = 1$, we have that this question can be solved by proving one of the following statements: $$P(\exists \lim_{n \to \infty} E[X_{n+1}|X_1, … X_n]) = 1$$ $$P(\exists \lim_{n \to \infty} E[X_{n+1}^2|X_1, … X_n]) = 1$$ However, they do not seem to be any easier to prove.
Yesterday, I asked a similar question, but with weaker conditions («finite» instead of «bounded»): Do non-negative submartingales with finite second moment converge almost surely? Those conditions were too weak and so the answer was negative. However, the counterexample, provided in the answer, does not work in this case, as its second moments, despite being finite, form an unbounded sequence.
Any sub martingale $\{X_n\}$ such that $\sup_n E|X_n| <\infty$ converges almost surely. [See Theorem 9.4.4 in Chung's book]. If the submartingale is $L^{2}$ bounded then it is uniformly integrable, so it converges in the mean also. In fact, i t can also be shown that in the non-negative case it converges in $L^{2}$.