Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length: $$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window: $$R_n = \max_{n-10 \leq k \leq n} X_k - \min_{n-10 \leq k \leq n} X_k $$

I know Doob's inequality, but can we give more precise informations about $M_n$ or $R_n$ ? At least when $X_{n+1} - X_n$ has a normal distribution?