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 $a>0,$

$\operatorname {P} \left[\sup _{n\geq 0}X_{n}\geq a\right]\leq {\frac {\operatorname {E} [X_{0}]}{a}}\ .$

The inequality is a generalization of Markov's inequality.