Ville's inequality explained

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

Let

X _{0,} X _{1,} X _{2,} ... be a non-negative supermartingale. Then, for any real number a > 0 , \operatorname{P} \left[ \sup _{n} X _{n} \ge a \right] \le (\operatorname{E)/([X) _{0]}{a}} .

The inequality is a generalization of Markov's inequality.

Notes and References

1. Jean . Ville . Etude Critique de la Notion de Collectif . 1939 .
2. Book: Durrett , Rick . Probability Theory and Examples. Fifth. 2019. Cambridge University Press. Exercise 4.8.2.
3. Steven R. . Howard. Sequential and Adaptive Inference Based on Martingale Concentration. 2019.
4. K. P. . Choi. Some sharp inequalities for Martingale transforms. 1988. Transactions of the American Mathematical Society. 307. 1. 279–300. 10.1090/S0002-9947-1988-0936817-3. 121892687. free.