Doob's Maximal Inequality

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Theorem

Discrete Time

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:

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


Also known as

Doob's Maximal Inequality is also known as:


Source of Name

This entry was named for Joseph Leo Doob.