Consider a submartingale $\{{X_k,\mathcal{F}_k}\}$. Doob's inequality dictates that for any $n \in \mathbb{Z}$ we have
\begin{equation} \mathbb{P}\bigg(\max\limits_{1\leq k \leq n} X_k \geq \alpha\bigg) \leq \frac{\mathbb{E}[\max\{X_n,0\}]}{\alpha} \end{equation} for all $\alpha >0$. For a problem I am working on I am trying to bound $\mathbb{P}\bigg(\max\limits_{1\leq k <\infty} X_k \geq \alpha\bigg) $. I was wondering if there is any way I can use Doob's inequality to bound this and under what extra assumptions. I hope I don't ask something trivial. Appreciate any help! Thanks!
The events $\{max_{1\leq k\leq n} X_k\geq \alpha\}$ increase, as $n$ increases to $\infty$, to the event $\{max_{1\leq k <\infty} X_k\geq \alpha\}$. Hence $P\{max_{1\leq k <\infty} X_k\geq \alpha\} \leq \lim \frac {E \max\{X_n,o\}} \alpha$. The inequality is useful if you know that the last limit is finite, in particular if $sup_n E|X_n| <\infty$.