Theorem 9.6. Let $\{M_n = \sum_{i=1}^n d_i, \mathscr F_n, n\geqslant 1\}$ be a martingale. Then there exist finite positive constants $a_p$, $b_p$ depending only on $p$ such that $$ a_p \mathbb E\left[\sum_{i=1}^n d_i^2 \right]^{p/2} \leqslant \mathbb E\left[\max_{j\leqslant n}|M_j|^p \right]\leqslant b_p\mathbb E\left[\sum_{i=1}^n d_i^2 \right]^{p/2}\quad \text{ for all } p\geqslant 1. $$
I'm now studying some advanced probability topics. The books I read review some classic martingale theory.however the following theorem says that the p-th absolute moment of maximum of the martingale can be controlled. I have no idea how the come out. can you give me some thoughts or some materials? Thanks a lot.