Theorem (Doob’s Maximal inequalities)

Let $(X_{n})_{n\in\mathbb{N}}$ be a $(\mathcal{F}_{n})_{n\in\mathbb{N}}$-supermartingale; for $k\in\mathbb{N}$ let $$\overline{X}_{k}=\sup_{0\le n\le k}X_{n}, \ \ \underline{X_{k}}=\inf_{0\le n\le k}X_{n}, \ \ X^{*}=\sup_{n\in\mathbb{N}}|X_{n}|$$let $\lambda>0$ then

1. $$\lambda P(\overline{X_{k}}\ge\lambda)\le E[X_{0}]+E[X_{k}^{-}]$$
2. $$\lambda P(\underline{X_{k}}\le-\lambda)\le E[X_{k}^{-}]$$
3. $$\lambda P(X^{*}\ge\lambda)\le3\|X\|_{1}$$where $$\|X_{n}\|_{p}=\sup_{n\in\mathbb{N}}E[|X_{n}|^{p}]^{\frac{1}{p}} \ \ (1\le p< \infty)$$
4. If $(X_{n})_{n\in\mathbb{N}}$ is $(\mathcal{F}_{n})_{n\in\mathbb{N}}$-martingale, then $$\lambda^{p}P(X^{*}\ge\lambda)\le(\|X\|_{p})^p$$