In Azuma's Inequality, is the statement true when $|X\_k - X\_{k-1}| < c\_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the above inequality (for each $k$) holds with high probability?



Azuma's Inequality when the conditions hold with high probability?
There is a large literature on variations of Azuma's inequality. One lemma that is similar to what you ask is Lemma 3.1 of [this old paper of Wormald and myself](http://cs.anu.edu.au/~bdm/papers/degseq1.pdf). It considers the case where $|X\_k-X\_{k-1}|$ is within one bound with very high probability and within some wider bound always. There are lots of such results.


