Doob's maximal inequality

For a nonnegative submartingale $(X_t)$, Doob’s maximal inequality states that, for all $n\ge 1$ and $t>0$,

$$\mathbb{P}\left(\underset{1\le i\le n}{\max }{X}_i\ge t\right)\le \frac{\mathbb{E}{X}_n}{t}.$$

Like Ville’s inequality for supermartingales, this enables some concentration inequalities to be given for submartingales. But unlike Ville’s inequality, the time-horizon is finite (eg the inequality is not truly time-uniform), and the bound is based on $\mathbb{E}{X}_n$ not $\mathbb{E}{X}_0$.