Ville's inequality

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

$$
  P(\sup_n S_n \ge \lambda) \le 1\slash{}\lambda.
$$

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.