Doob's Maximal Inequality
(Redirected from Doob's Martingale 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:
- Doob's Martingale Inequality
- Kolmogorov's Submartingale Inequality for Andrey Nikolaevich Kolmogorov
- Just the Submartingale Inequality
Source of Name
This entry was named for Joseph Leo Doob.