How to Prove Unboundedness Suppose I have a submartingale $X_k$, what results/theorems can be useful if I want to show that $X_k$ is unbounded in the limit. There are results (basically bounding $\mathbb{E}X_k$) for convergence of submartinagles. But, I wanted to show that a submartingale goes to infinity in the limit for my research. 
Take the submartingale $X_0=0, X_{k+1}=X_k+1$ for instance. 

If the submartingale is of the form $$X_k = \sum_{j=1}^k \xi_j$$ where $\xi_j \in L^1$ are independent identically distributed random variables, then, by the strong law of large numbers,
$$\frac{X_k}{k} \to \mathbb{E}\xi_1 \qquad \text{almost surely as $k \to \infty$.}$$
This means that $$\lim_{k \to \infty} X_k = \infty$$ whenenver $\mathbb{E}\xi_1 \neq  0$. (Note that $\mathbb{E}\xi_1 \geq 0$ holds in any case since $(X_k)_{k \in \mathbb{N}}$ is a submartingale.)
