= One-sided maximal inequality for a centered square-integrable martingale
{title2=$\mathbb P(\max_{k\leq n}M_k\geq\lambda)\leq v/(\lambda^2+v)$}
For a centered square-integrable <martingale> with $v=\mathbb E M_n^2$ and $\lambda>0$, apply the <Doob maximal inequality for a nonnegative submartingale> to $(M_k+c)^2$ for $c\geq0$. <Conditional Jensen inequality> gives the <submartingale> property, and crossing $\lambda$ forces its square above $(\lambda+c)^2$. The bound $(v+c^2)/(\lambda+c)^2$ is minimized by $c=v/\lambda$. This extends the <Cantelli inequality> from one <random variable> to a <martingale> maximum, without requiring <independent increments>.
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