The Azuma-Hoeffding Inequality says that if $X\_1,X\_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X\_i - X\_{i-1}\| \le 1$ say, then we should not expect the difference $\|X\_N - X\_0\|$ to grow *too fast*. Formally we have


$$P\left(|X\_N -X\_0| > \epsilon N\right) \le \exp\Big ( \frac{- \epsilon^2 N}{2 }\Big) \mbox{ for any }\epsilon > 0.$$


Note the inequality has nothing to do with how *spread out* the variables $X\_i$ are. It only uses how the differences are bounded. In case all $X\_i$ have small variance we'd expect stronger concentration results. 


For example suppose the $A\_1 , A\_2, \ldots$ are drawn independently from $\mathcal N(0,\sigma^2)$ distributions cut off outside $[-1,1]$ and each $X\_i = A\_1 + \ldots + A\_i$. The above inequality does not distinguish between the cases when $\sigma^2$ is large and small. When it is smaller we should expect a stronger concentration around the mean. In the degenerate case when $\sigma^2=0$ and all $A\_i \equiv 0$ the left-hand-side will be exactly zero.


Are there any modifications of Azuma-Hoeffding that take into account the variances of the conditined variiables $ X\_{i+1} | X\_i, \ldots, X\_1$ ? So far I have only found [this](https://arxiv.org/pdf/1111.1977.pdf) paper in information theory. The theorem 2 is a version of AH that involves the variance. However that paper is quite recent, and it seems likely the problem has been considered by probabilists in the past. 


Can anyone point me in the right direction?



Can we do better than Azuma-Hoeffding when the variance is small?
Exponential inequalities for sums of independent random variables (r.v.'s) can be extended to martingales in a standard and completely general manner; see Theorem 8.5 or Theorem 8.1 for real-valued martingales, and Theorem 3.1 or Theorem 3.2 for martingales with values in 2-smooth Banach spaces in [this paper](https://projecteuclid.org/euclid.aop/1176988477). 


In particular, Theorem 8.7 in the same paper implies the following martingale version of the [Bennett (8b)](https://www.jstor.org/stable/2282438?seq=1#page_scan_tab_contents)--[Hoeffding (2.9)](https://www.jstor.org/stable/2282952?origin=JSTOR-pdf&seq=1#page_scan_tab_contents) inequality given for sums of independent r.v.'s: 



> 
> If $(X\_j)\_{j=0}^n$ is a real-valued martingale with respect to a filter $(F\_j)\_{j=0}^n$ of $\sigma$-algebras such that for $d\_j:=X\_j-X\_{j-1}$ we have 
>  $|d\_j|\le a$ and $\sum\_1^n E(d\_j^2|F\_{j-1})\le b^2$ for some real $a,b>0$ and all $j$, then 
>  \begin{equation}
>  P(X\_j-X\_0\ge r)\le\exp\Big\{\frac{b^2}{a^2}\,\psi\Big(\frac{ra}{b^2}\Big)\Big\}
> \end{equation}
>  for $r\ge0$, where $\psi(u):=u-(1+u)\ln(1+u)$. 
> 
> 
> 


According to [Theorem 3](https://epubs.siam.org/doi/10.1137/1134032?mobileUi=0), this bound is the best possible exponential bound on $P(X\_j-X\_0\ge r)$ in terms of $a,b^2,r$. 


