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 ess \sup ||d_n||^2\leq1$$\sum_{n=1}^\infty \mathrm{ess} \sup ||d_n||^2\leq1$, and either $\mathcal{X}\in D(A_1,A_2)$ or $\mathcal{X}=L^p,p\leq 2$. Then \begin{align} P(f^*\geq r) \leq 2 \exp\left(-\frac{r^2}{2B}\right), r \leq 0,\quad (1) \end{align} where $B=A_1^2+A_2$ for $\mathcal{X}\in D(A_1,A_2)$ and $B=p-1$ for $\mathcal{X}=L^p$.
I am interested in the following statement:
In the case $\mathcal{X}=\mathbb{R}$, Theorem 3 is another result due to Hoeffding (1963), often ascribed to Azuma (1967).
This is not clear to me at all how Azuma's inequality follows from this. I could set $f_j=\sum_{i=1}^jX_i$, which is clearly a martingal and $d_j=X_j$, however even if $X_j\in[a_j,b_j]$, the assumption $\sum_{n=1}^\infty ess \sup ||d_n||^2\leq1$$\sum_{n=1}^\infty \mathrm{ess} \sup ||d_n||^2\leq1$ does not hold.
Furthermore, the left hand side of Eq. (1) would be $$ P(\sup\{||\sum_{i=1}^nX_i||,n=0,1,...\} \geq r) $$$$ P\left(\sup\left\{||\sum_{i=1}^nX_i||,n=0,1,...\right\} \geq r\right) $$
[1] An Approach to Inequalities for the Distributions of Infinite-Dimensional Martingales, Iosif Pinelis
