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 , … {\displaystyle X_{0},X_{1},X_{2},\dots } be a non-negative supermartingale. Then, for any real number a > 0 , {\displaystyle a>0,}
P [ sup n ≥ 0 X n ≥ a ] ≤ E [ X 0 ] a . {\displaystyle \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.
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