I am reading this paper which defines a martingale small ball condition that I have never encountered in a standard probability book.
The definition is as follows (page 8, definition 2.1): Let $Z_t$, $t\ge 1$ be an $\mathbb{R}$ values process adapted to the filtration $\mathcal{F}_t$. $Z_t$ satisfies the $(k,v,p)$-block martingale small-ball condition if for any $j \ge 0$, $\frac{1}{k}\sum_{i=1}^k P(|Z_{j+i}| \ge v | \mathcal{F}_j) \ge p$ almost surely.
The intuition of this definition is not clear to me. What property of $Z_t$ is it trying to quantify? If the definition above is satisfied, does it imply that $Z_t$ is a martingale? Or if $Z_t$ is a martingale, does the above definition hold? It is not clear to me why "martingale" appears in the definition.
Thoughts/clarifications appreciated.