TITLE: Adaptive version of the Azuma–Hoeffding inequality
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: There is no such inequality even if we further restrict $c_k$ to be in $\{0,1\}$ and weaken the inequality to include a constant factor. (I think it is natural to add the condition that the $c_k$ values are uniformly bounded.) Suppose $c_k \in \{0,1\}$ and without loss of generality no $1$ follows a $0$. The choice of $c_k$ is equivalent to a stopping rule $\tau \le N$. We bet a constant amount on each toss of a coin and decide to stop after $\tau$ flips: $c_k = 0$ when $k \gt \tau$, so $c_k$ only depends on $X_1,...,X_{k-1}$.
Choose a constant $s \gt 0$. Can we bound 
$$P\left(X_N \ge s \sqrt{\sum_{k=1}^N c_k^2}\right) = P(X_\tau \ge s \sqrt{\tau})?$$
The law of the iterated logarithm implies that for an infinite sequence $\{B_i\}$, the maximum number of standard deviations of $\sum B_i$ above the mean is almost surely unbounded, hence as $N\to \infty$ the maximum exceeds $s$ with probability approaching $1$. We can let $\tau$ be the minimum of $N$ and the first time $t$ so that $\sum_{i=1}^{t} B_i > s\sqrt{t}$.
If you let the bound depend on $N$, something like 
$$P(\max \frac{X_t}{\sqrt{t}} \ge s) \le f(s,N),$$
then there are nontrivial bounds possible.