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 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


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 $r':=r/c$.


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