According to [1]



> 
> Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let
>  $S(\mathcal{X})$ denote the class of all sequences
>  $f=(f\_j)=(f\_0,f\_1,...)$ of Bochner-integrable random vectors in
>  $\mathcal{X}$, with $f\_0=0$ defined on a probability space
>  $(\Omega,F,P)$.
> 
> 
> For $f\in S(\mathcal{X})$, put $f^\*=\sup\{||f\_j||,j=0,1,...\}$, $d\_0=d\_0(f\_0)$, $d\_j=d\_j(f)=f\_j-f\_{j-1}$. $\mathcal{M}(\mathcal{X})$ denotes the class of all sequences $(f\_j)\in S(\mathcal{X})$ that are martingales.
> 
> 
> 


Theorem 3 says:



> 
> Suppose that $\sum\_{n=1}^\infty \mathrm{ess} \sup ||d\_n||^2\leq1$, and either $\mathcal{X}\in D(A\_1,A\_2)$ or $\mathcal{X}=L^p,p\leq 2$. Then
>  \begin{align}
> P(f^\*\geq r) \leq 2 \exp\left(-\frac{r^2}{2B}\right), r \leq 0,\quad (1)
> \end{align}
>  where $B=A\_1^2+A\_2$ for $\mathcal{X}\in D(A\_1,A\_2)$ and $B=p-1$ for $\mathcal{X}=L^p$.
> 
> 
> 


I am interested in the following statement:



> 
> In the case $\mathcal{X}=\mathbb{R}$, Theorem 3 is another result due to Hoeffding (1963), often ascribed to Azuma (1967).
> 
> 
> 


This is not clear to me at all how [Azuma's inequality](http://en.wikipedia.org/wiki/Azuma%27s_inequality) follows from this. I could set $f\_j=\sum\_{i=1}^jX\_i$, which is clearly a martingal and $d\_j=X\_j$, however even if $X\_j\in[a\_j,b\_j]$, the assumption $\sum\_{n=1}^\infty \mathrm{ess} \sup ||d\_n||^2\leq1$ does not hold.


Furthermore, the left hand side of Eq. (1) would be
$$
P\left(\sup\left\{||\sum\_{i=1}^nX\_i||,n=0,1,...\right\} \geq r\right)
$$


[1] [An Approach to Inequalities for the Distributions of Infinite-Dimensional Martingales, Iosif Pinelis](http://books.google.de/books?id=_qGsljjxgsUC&pg=PA128&lpg=PA128&dq=pinelis+an+approach+to+inequalities+for+the&source=bl&ots=RCKr4SGZn5&sig=fCs_zRA_1ncFTginpX2_1DeO8nY&hl=en&sa=X&ei=ryRjVPe0LNjtaOu9gfAM&redir_esc=y#v=onepage&q&f=false) 



How does Azuma's Inequality result from Pinelis Inequality?
For a fixed integer $n$, we consider $d\_j=X\_j$ if $j\leqslant n$ and $d\_j=0$ otherwise. If $|X\_j|\leqslant r\_j$ almost surely, then we have 
$\sum\_{j=1}^\infty\mathrm{ess\,sup}\lVert X\_j\rVert^2\leqslant \sum\_{j=1}^nc\_j^2=:c$, hence we use the result with $d'\_j:=d\_j/c$ and $r':=r/c$.


