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.[1][2][3][4] The inequality has applications in statistical testing.
Statement
[edit]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}}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf057cd9898bdbe0baefa76e77d5b9fbc7c3adea)
The inequality is a generalization of Markov's inequality.
References
[edit]- ^ Ville, Jean (1939). Etude Critique de la Notion de Collectif (PDF) (Thesis).
- ^
Durrett, Rick (2019). Probability Theory and Examples (Fifth ed.). Exercise 4.8.2: Cambridge University Press.
{{cite book}}: CS1 maint: location (link) - ^ Howard, Steven R. (2019). Sequential and Adaptive Inference Based on Martingale Concentration (Thesis).
- ^ Choi, K. P. (1988). "Some sharp inequalities for Martingale transforms". Transactions of the American Mathematical Society. 307 (1): 279–300. doi:10.1090/S0002-9947-1988-0936817-3. S2CID 121892687.
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