Doob's martingale inequality
= Doob's martingale inequality
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For a nonnegative submartingale $X$, the maximal tail bound is $\lambda\mathbb P(\max_{k\leq n}X_k\geq\lambda)\leq\mathbb E[X_n;\max_{k\leq n}X_k\geq\lambda]$. Applied to the absolute value of a martingale and integrated over $\lambda$, it gives the <Doob Lp maximal inequality>; at $p=2$ this is the <Doob L2 maximal inequality>.