Exercise 5.3.1 in Durrett's "Probability Theory and Examples" states
Let $X_n$, $n\ge 0$, be a submartingale with $\sup X_n < \infty$. Let $\xi_n=X_n-X_{n-1}$, and suppose $E(\sup \xi_n^+)<\infty$. Show that $X_n$ converges almost surely.
Here are my thoughts:
It looks like we need to use theorem 5.2.8 (Martingale convergence theorem), which states that $X_n$ converges almost surely if $\sup EX_n^+<\infty$. I am trying to understand why we can just do the following (or at least I'm guessing we can't):
Suppose $X_n > 0$ (edit: on some set with positive measure), then $0< \sup X_n < \infty$, and therefore $EX_n^+ \le \sup X_n < \infty$. So we're done.
Otherwise, if $X_n \le 0$ a.e., then $\sup X_n^+ = 0$ a.e. Once again, we're done.
Am I missing something? Since I didn't use all the information, I probably am.
Thanks for the help.
Best Answer
You've stepped off a cliff in deducing that the expectation $E(X^+_n)$ is bounded above by the random variable $\sup_n X_n$.
Suggestion: Employ the argument used by Durrett in the proof of his Theorem 5.3.1. Fix a positive real $K$, define the stopping time $T=T_K$ to be the first time $n$ that $X_n$ is larger than $K$, and observe that the stopped process satisfies $$ X_{n\wedge T}\le K+\sup_m\xi_m^+, $$ so that $$ E(X_{n\wedge T})\le K+E(\sup_m\xi_m^+)<\infty,\qquad\forall n. $$ Now apply the submartingale convergence theorem to the stopped process. This yields a.s. convergence of $X_n$ on the event $\{T_K=\infty\}=\{\sup_mX_m\le K\}$. Finally, vary $K$.