 Solution

The discrete-time Doob L2 maximal inequality states that a square-integrable martingale satisfiesEquivalently, the Lp norm of the maximum at $p=2$ is at most $2\Vert M_n{\Vert }_2$. The same assertion holds for a nonnegative square-integrable submartingale. No assumption $M_0=0$ is required; for a bound in terms of fluctuations from the initial value, apply it to $(M_k-M_0)$.