# Supremum of a martingale

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?

In your latter case ($X_{n+1}-X_n$ normal) you can use a large deviation principle for martingales (if $X_{n+1}-X_n$ is bounded it is called Azuma inequality but there are extensions with worse rates for martingale differences with finite exponential moments) together with a naive union bound to get a good estimate. This will work also for estimating $$\max_{k\in [cn,n]}X_k$$ with $c>0$.
• Thanks, I'll have a look. In the case $X_n$ is just a simple random walk with independent increments $+1$ or $-1$ with probability $1/2$, is there a well known distribution for $R_n$ ? It looks roughly like a log-normal...