These feel like basic enough questions, but I don't know where to find the answer.


Let $X\_1,X\_2,X\_3,\dots$ be a supermartingale such that $|X\_{n+1} - X\_n| < K$ for all $n$ ($K$ fixed). Does the event $X\_n \rightarrow +\infty$ necessarily have probability zero? What if we also have the condition that given $X\_1,\dots,X\_n$ there are only $N$ possibilities for $X\_{n+1}$, each with probability $1/N$?


Suppose in addition that $\mathbb{E}(X\_{n+1}|X\_1,\dots,X\_n) \le X\_n - c\_n$, where $\sum c\_n \rightarrow +\infty$. Is $X\_n \rightarrow -\infty$ almost certain?



Supermartingales and convergence
For a martingale $M\_n$ with bounded increment, then, almost surely :


* either $M\_n$ converges to a finite limite.
* or $\limsup M\_n=\infty$ and $\liminf M\_n=-\infty$.


Sketch of the proof. One can assume that $M\_0=0$. Let $T$ be the first time at which the martingale goes below $-A$. Then $M\_{n \wedge N}$ is a martingale bounded from below, therefore it converges. Then $M\_n$ converges if $N$ is infinite etc.


For your problem consider the Doob decomposition.


