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] [5]} The inequality has applications in statistical testing.

Statement

Let ${\displaystyle X_{0},X_{1},X_{2},\dots }$ be a non-negative supermartingale. Then, for any real number ${\displaystyle a>0,}$

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

The inequality is a generalization of Markov's inequality.

References

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.
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.
5. Revuz, Daniel; Yor, Marc (2005). Continuous Martingales and Brownian Motion. Vol. 293 (3 ed.). Exercise 1.15. ISBN   978-3-642-08400-3.