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Suppose $\{X_n\}_{n \geq 1}$ is a square-integrable martingale with $E(X_1)=0$. Then for $c>0$:

$$P\left(\max_{i=1, \ldots, n} X_i \geq c\right) \leq \frac{\textrm{Var}(X_n)}{\textrm{Var}(X_n) + c^2}.$$

I imagine Doob's martingale inequality will come into play, but the details elude me.

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    Got something from the answer below?2012-08-11

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