Let $X_t$ be a stochastic process, for example a brownian motion (i.e. $X_{t+h} - X_t \sim \mathcal{N}(0,\sqrt{h}^2)$).
The difference between now's value $X_t$ and a past value $X_{t-100}$ is
$$M_t = X_{t} - X_{t-100} \sim \mathcal{N}(0,\sqrt{100}^2)$$
thus the absolute variation between this past value and now's value, i.e. $|X_{t} - X_{t-100}|$, has a folded normal distribution.
Question:
In a similar fashion, what's the distribution of the range over a window of width 100 : $$R_t = \max_{t - 100\leq s \leq t} X_s - \min_{t - 100\leq s \leq t} X_s$$
It looks like a log-normal distribution. Can we prove this?
Moreover, can we prove something when $d\,X_t$ is something else than a normal distribution?
Numerical evidence: histogram of $M_t$ (folded normal distribution)
Histogram of $R_t$ :
