I have a martingale difference sequence (MDS) $X_n$ wrt. some adapted filtration $\mathcal{F}_n$, i.e. $\mathbb{E}[|X_n|] < \infty$ and $\mathbb{E}[X_{n+1}\mid\mathcal{F}_n] = 0$ a.s. for all $n \geq 0$.
I'm interested in bounds on $\mathbb{E}[\max_{0 \leq n \leq N} |X_n|]$. If $X_n$ was a martingale then there are well-known results linking the expectation of the maximum to the expectation at time $N$, but I don't know of anything in the MDS case. It seems that since MDS structure is essentially a very weak form of dependence, it should be possible to get something almost as good as the case where the $X_n$ are independent and then there are well-known simple bounds in the literature on "order statistics" (well these are not quite so relevant here as the bounds I know of assume also identical distribution, which, of course, we do not have in general in the case of an MDS). If I'm not mistaken, in the iid maximum order statistic case, you expect a growth in the expectation of the maximum wrt. $N$ that is proportional to $\sqrt{N}$. I wonder if something with similar growth holds in the MDS case?
Anyone have any ideas or know of any relevant references?
Many thanks.