Ville's inequality
Probabilistic inequality
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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.[1][2][3][4] [5] The inequality has applications in statistical testing.
Statement
Let be a non-negative supermartingale. Then, for any real number
![{\displaystyle \operatorname {P} \left[\sup _{n\geq 0}X_{n}\geq a\right]\leq {\frac {\operatorname {E} [X_{0}]}{a}}\ .}](http://wikimedia.org/api/rest_v1/media/math/render/svg/bf057cd9898bdbe0baefa76e77d5b9fbc7c3adea)
The inequality is a generalization of Markov's inequality.
