Doob maximal inequality for a nonnegative submartingale

If $(X_m{)}_{m\le n}$ is a nonnegative submartingale, then for every $a>0$,Stop at the first crossing of $a$ and use the submartingale property on that event.

For independent centered random variables and $S_m={\sum }_{j\le m}{X}_j$,Apply the Doob maximal inequality for a nonnegative submartingale to $(S_m+c{)}^2$ and optimize at $c=\mathrm{Var}(S_n)/x$.