Ville's inequality
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Short description: Probabilistic 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
Let [math]\displaystyle{ X_0, X_1, X_2, \dots }[/math] be a non-negative supermartingale. Then, for any real number [math]\displaystyle{ a \gt 0, }[/math]
- [math]\displaystyle{ \operatorname{P} \left[ \sup_{n \ge 0} X_n \ge a \right] \le \frac{\operatorname{E}[X_0]}{a} \ . }[/math]
The inequality is a generalization of Markov's inequality.
References
- ↑ 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.
- ↑ 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.
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