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Ville's 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] The inequality has applications in statistical testing.

Statement

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Let be a non-negative supermartingale. Then, for any real number

The inequality is a generalization of Markov's inequality.

References

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  1. ^ Ville, Jean (1939). Etude Critique de la Notion de Collectif (PDF) (Thesis).
  2. ^ Durrett, Rick (2019). Probability Theory and Examples (Fifth ed.). Exercise 4.8.2: Cambridge University Press.{{cite book}}: CS1 maint: location (link)
  3. ^ Howard, Steven R. (2019). Sequential and Adaptive Inference Based on Martingale Concentration (Thesis).
  4. ^ 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.