This is an old Exercise from a Probability II Exam, there are several questions related to this one on this site, but since this is an old exam question (without solution provided) I'd like to understand the suggested approach. It goes like this:
Problem: Let $(E_n)_{n \geq 1}$ be a sequence of events with $E_n \in F_n$ for every $n \geq 1$. We Define $$ X_n := \sum_{k=1}^n 1_{\lbrace E_k\rbrace }, \ Y_n:= \sum_{k=1}^n \mathbb{P}(E_k \mid F_{k-1}) \text{ and } M_n:= X_n -Y_n $$ It is the goal to show that $$ \lbrace Y_\infty < \infty \rbrace = \lbrace X_\infty < \infty \rbrace \text{ almost surely} $$
The exam suggests to prove this by showing the following steps
1) Prove that $(X_n)_{n \geq 0}$ is a submartingale and $(M_n)_{n \geq 0}$ is a martingale (Check $\checkmark$)
2) Show that for $a>0$, $T_a := \inf \lbrace n \geq 0 : Y_{n+1} >a \rbrace$ is a stopping time (Check $\checkmark$)
3) Show that for every $a>0, n \geq 0, \ M_{n \wedge T_a}^{-} \leq a$ (Check $\checkmark$)
- 3.a) Deduce that $(M_{n \wedge T_a})_{n \geq 0}$ converges almost surely
- 3.b) Deduce that $\lbrace Y_ \infty < \infty \rbrace \subset \lbrace X_\infty < \infty\rbrace$ a.s.
Questions: As denoted above (by $\checkmark$) I managed to show the easy statements 1-3. But I have no clue on how to show the corollary statements 3a,3b.
I know that when $M_{n \wedge T_a}$ is bounded in $L^1$ then we have that the martingale $(M_{n \wedge T_a} )_{n \geq 0}$ converges a.s. with limit in $L^1$ which would help with the statement 3.b, but I don't see how $$M_{n \wedge T_a}^- = \max( - M_{n \wedge T_a},0) \leq a \in L^1 $$ helps me in anyway to make a statement about $M_{n \wedge T_a}$ being bounded.
I would appreciate hints to get me unstuck