Doob's L2 Maximal Inequality in Discrete Time

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space, let $n$ be zero or a natural number, and let $0\le t_0<t_1<\dots<t_n$ be real numbers.

1. Let $M=(M_t)_{t\ge0}$ be a square-integrable submartingale with $M_t(\omega)\ge0$ for every $t\ge0$ and every $\omega\in\Omega$, and let $M^{*}(\omega)=\max_{0\le k\le n}M_{t_k}(\omega)$ be the running maximum, a square-integrable random variable by Doob's maximal inequality. Then

2. Let $M=(M_t)_{t\ge0}$ be a square-integrable martingale and define pointwise

Then $\overline{M}$ is a square-integrable random variable and