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 $$a > 0,$$



\operatorname{P} \left[ \sup_{n \ge 0} X_n \ge a \right] \le \frac{\operatorname{E}[X_0]}{a} \. $$

The inequality is a generalization of Markov's inequality.