Ville's inequality

In probability theory, Ville's inequality provides an upper bound on the probability that a supermartingale exceeds a certain value. The inequality is named after Jean Ville, who proved it in 1939. The inequality has applications in statistical testing.

Statement

Let $X_0,X_1,X_2,\dots$ be a non-negative supermartingale. Then, for any real number $a>0,$

$\mathrm{P}[\underset{n\ge 0}{\sup }{X}_n\ge a]\le \frac{\mathrm{E}[X_0]}{a}.$

The inequality is a generalization of Markov's inequality.

References

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