The setup is as follows. We are given a martingale $X_0,X_1,...,X_k$. The difference $X_i-X_{i-1}$ is always between $[-1,1]$. Variance $D^2(X_i-X_{i-1}| X_{i-1})$ is something, but we can show that after taking expectation on $X_{i-1}$ then we get that $D^2(X_i-X_{i-1}) \leq X_0/k$. Therefore, I would like to get a tail bound $P( X_k - X_0 > t) \leq \exp(-t^2/ X_0 )$; constants in the exponent don't matter. There exists a similar bound $P( X_k - X_0 > t) \leq \exp(-\frac{t^2}{\sum_{i=1}^k c_i^2} )$, but here $c_i^2$ is the bound on supremum of $D^2(X_i-X_{i-1}| X_{i-1})$ for the worst-case $X_{i-1}$, so this one does not imply what I want.

How can I get a hold on a bound like that?