Doob's Maximal Inequality/Discrete Time

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a non-negative $\sequence {\FF_n}_{n \mathop \ge 0}$-submartingale.

Let:


 * $\ds X_n^\ast = \max_{0 \mathop \le k \mathop \le n} X_k$

where $\max$ is the pointwise maximum.

Let $\lambda > 0$.

Then:


 * $\lambda \map \Pr {X_n^\ast \ge \lambda} \le \expect {X_n}$