# Ville's inequality

If is a test martingale (i.e, a non-negative martingale with initial value 1), then, for each positive constant ,

This was first demonstrated in Ville's 1939 book (page 100). This inequality is sometimes referred to as Doob's inequality.

### Bibliography

- J. Ville, Etude critique de la notion de collectif. Paris: Gauthier-Villars, 1939.