Let $(X_n)_{n \geq 0}$ be a super-martingale, that is $\mathbb{E}[X_{n+1} \mid X_1, \dots, X_n] \leq X_n$. Let's further assume that $\Pr[|X_n - X_{n-1}| < c_n] \geq 1-\delta$. Does there exist any extension of the Azuma–Hoeffding inequality to this high probabilistic setting ?
Reminder, Azuma's inequality (i.e. with $\delta=0$),
$\Pr[X_n-X_0 \geq a] \leq \exp\big(\frac{-a^2}{\sum_{k=1}^n c_k^2}\big)~$