I just learned martingale convergence theorem today which is the following:
If $X_n$ is a submartingale with $\sup EX_n^+<\infty$ then as $n\to\infty$, $X_n$ converges almost surely to a limit $X$ with $E|X|<\infty$.
I want to ask two questions and think of them intuitively (not technically):
(1) What does martingale convergence theorem tell us?
(2) Why martingale convergence should be true?
My thoughts are looking for simple and instructive examples. I loved the example where the terminology "martingale" comes from. We bet \$1 on fair coin flippings. Let $S_n$ be a symmetric simple random walk representing the money up to time $n$ with $S_0=1$ where $S_n=S_{n-1}+\xi_n$ and $P(\xi_i=1)=1/2=P(\xi_i=-1)$. Let $N=\inf\{n:S_n=0\}$ be the stopping time. Let $X_n=S_{N\land n}$ be the martingale representing that we will stop when we lose all money. Let $$H_m=\begin{cases}2H_{m-1}, & \xi_{m-1}=-1\\ 1, & \xi_{m-1}=1 \end{cases}$$ be the martingale gambling system.
Now, consider the martingale $Y_n:=(H\cdot X)_n$ which characterizes the amount of money up to time $n$ in the martingale gambling game. Assume that on average we will only have finite amount of money at any time $n$, then $Y_n$ converges to $0$ almost surely which tells us that we will lose all the money finally : ( As for question (2), the essential reason is that the condition of finite amount of money implies finiteness of upcrossings for any rational intervals $[a,b]$ which implies a convergence.
Please let me know if you have any ideas or comments. Thanks!