Let $(X_t)_{t \in \mathbb{N}}$ be a real-valued martingale that is bounded, i.e., there are $a, b \in \mathbb{R}$ such that $a \leq X_t \leq b$ for all $t$.
Define the path length $L$ of $(X_t)_{t \in \mathbb{N}}$ as the distance the martingale covers, $$ L := \sum_{t=0}^\infty | X_t - X_{t+1} |. $$
My questions:
- Is $L$ almost surely finite? I.e., can there be a set of positive measure on which $L = \infty$?
- If yes to question 1, is $\mathbb{E}[L]$ finite?