Let $(\Omega, \mathcal{F},P)$ be a probability space. Consider a martingale $M_n$ with filtration $\mathcal{F}_n$. Let $B \in \mathcal{F}$. On $B$, $a_n \leq |M_n - M_{n-1}| \leq b_n$ a.s. Can we find any concentration inequation for $$P(|M_n| \geq t | B)$$
if $B \in \mathcal{F}_1$, we can easily find such inequality. I am intersted in the case $B \notin \mathcal{F}_1$