The Azuma inequality states that if we have a martingale $X_1,\ldots,X_N$ that satisfies a bounded difference condition: $$|X_k - X_{k-1}| \leq c_k$$ Then: $$\Pr\left[X_N - X_0 \geq \sqrt{2\sum_kc_k^2 \ln(1/\delta)}\right] \leq \delta$$

My question is:

Does the same inequality hold if the constants $c_k$ are not fixed up front, but are themselves a function of the realizations of the previous terms in the martingale $X_1,\ldots,X_{k-1}$?

For concreteness, consider the following coin flipping game. An adversary has $n$ coins, which he flips in sequence (represented by random variables $B_i$ taking values uniformly in $\{-1,1\}$). After observing the outcomes of $B_1,\ldots,B_{i-1}$ (but crucially, not $B_i$), he chooses a non-negative weight $c_i$. At the end of the $n$ coin flips, we compute the quantity: $$X_n = \sum_{i=1}^n c_i\cdot B_i$$

Can we now say that: $$\Pr\left[X_N \geq \sqrt{2\sum_kc_k^2 \ln(1/\delta)}\right] \leq \delta$$

noting that here, the values $c_k$ are now random variables themselves?