The Azuma Hoeffding inequality says that if $ X_1, X_2, ldots $ is a martingale and the differences are limited by constants, $ | X_i – X_ {i-1} | le 1 $ we say, then we should not expect the difference $ | X_N – X_0 | $ to grow *too fast*, Formally we have

$ P (| X_N -X_0 |> epsilon N) le exp Big ( frac {- epsilon ^ 2 N} {2} Big) $ for each $ epsilon> 0 $,

Note that inequality has nothing to do with it *to distribute* the variables $ X_i $ are. It is only used as the differences are limited. For all $ X_i $ With small deviations we expect stronger concentration results.

Suppose the $ A_1, A_2, ldots $ are drawn independently of $ mathcal N (0, sigma ^ 2) $ Distributions are cut off outside $[-1,1]$ and each one $ X_i = A_1 + ldots + A_i $, The above inequality does not distinguish between the cases in which $ sigma ^ 2 $ is big and small. If it is smaller, we should expect a higher concentration around the mean. In the degenerate case when $ sigma ^ 2 = 0 $ and everything $ A_i equiv 0 $ The left side is exactly zero.

Are there any modifications by Azuma Hoeffding that take into account the deviations of the conditional variables? $ X_ {i + 1} | X_i, ldots, X_1 $ ? So far I have found this paper only in information theory. Theorem 2 is a version of AH that includes the variance. However, this paper is fairly new and it is likely that probabilists have considered the problem in the past.

Can someone show me the right direction?